Revised5Report on the Algorithmic Language Scheme

Revised

Report on the Algorithmic Language

Scheme

RICHARD KELSEY, WILLIAM CLINGER, AND JONATHAN REES (Editors)

H. ABELSON R. K. DYBVIG C. T. HAYNES G. J. ROZAS

N. I. ADAMS IV D. P. FRIEDMAN E. KOHLBECKER G. L. STEELE JR.

D. H. BARTLEY R. HALSTEAD D. OXLEY G. J. SUSSMAN

G. BROOKS C. HANSON K. M. PITMAN M. WAND

Dedicated to the Memory of Robert Hieb

20 February 1998

SUMMARY

The report gives a deﬁning description of the program-

ming language Scheme. Scheme is a statically scoped and

properly tail-recursive dialect of the Lisp programming

language invented by Guy Lewis Steele Jr. and Gerald

Jay Sussman. It was designed to have an exceptionally

clear and simple semantics and few diﬀerent ways to form

expressions. A wide variety of programming paradigms, in-

cluding imperative, functional, and message passing styles,

ﬁnd convenient expression in Scheme.

The introduction oﬀers a brief history of the language and

of the report.

The ﬁrst three chapters present the fundamental ideas of

the language and describe the notational conventions used

for describing the language and for writing programs in the

language.

Chapters 4 and 5 describe the syntax and semantics of

expressions, programs, and deﬁnitions.

Chapter 6 describes Scheme’s built-in procedures, which

include all of the language’s data manipulation and in-

put/output primitives.

Chapter 7 provides a formal syntax for Scheme written in

extended BNF, along with a formal denotational semantics.

An example of the use of the language follows the formal

syntax and semantics.

The report concludes with a list of references and an al-

phabetic index.

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Overview of Scheme . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Notation and terminology . . . . . . . . . . . . . . . . 3

2 Lexical conventions . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Identiﬁers . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Whitespace and comments . . . . . . . . . . . . . . . . 5

2.3 Other notations . . . . . . . . . . . . . . . . . . . . . . 5

3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1 Variables, syntactic keywords, and regions . . . . . . . 6

3.2 Disjointness of types . . . . . . . . . . . . . . . . . . . 6

3.3 External representations . . . . . . . . . . . . . . . . . 6

3.4 Storage model . . . . . . . . . . . . . . . . . . . . . . . 7

3.5 Proper tail recursion . . . . . . . . . . . . . . . . . . . 7

4 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1 Primitive expression types . . . . . . . . . . . . . . . . 8

4.2 Derived expression types . . . . . . . . . . . . . . . . . 10

4.3 Macros . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Program structure . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3 Syntax deﬁnitions . . . . . . . . . . . . . . . . . . . . 17

6 Standard procedures . . . . . . . . . . . . . . . . . . . . . . 17

6.1 Equivalence predicates . . . . . . . . . . . . . . . . . . 17

6.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.3 Other data types . . . . . . . . . . . . . . . . . . . . . 25

6.4 Control features . . . . . . . . . . . . . . . . . . . . . . 31

6.5 Eval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.6 Input and output . . . . . . . . . . . . . . . . . . . . . 35

7 Formal syntax and semantics . . . . . . . . . . . . . . . . . . 38

7.1 Formal syntax . . . . . . . . . . . . . . . . . . . . . . . 38

7.2 Formal semantics . . . . . . . . . . . . . . . . . . . . . 40

7.3 Derived expression types . . . . . . . . . . . . . . . . . 43

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Additional material . . . . . . . . . . . . . . . . . . . . . . . . 45

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Alphabetic index of deﬁnitions of concepts,

keywords, and procedures . . . . . . . . . . . . . . . . 48

2 Revised

Scheme

INTRODUCTION

Programming languages should be designed not by piling

feature on top of feature, but by removing the weaknesses

and restrictions that make additional features appear nec-

essary. Scheme demonstrates that a very small number of

rules for forming expressions, with no restrictions on how

they are composed, suﬃce to form a practical and eﬃcient

programming language that is ﬂexible enough to support

most of the major programming paradigms in use today.

Scheme was one of the ﬁrst programming languages to in-

corporate ﬁrst class procedures as in the lambda calculus,

thereby proving the usefulness of static scope rules and

block structure in a dynamically typed language. Scheme

was the ﬁrst major dialect of Lisp to distinguish procedures

from lambda expressions and symbols, to use a single lex-

ical environment for all variables, and to evaluate the op-

erator position of a procedure call in the same way as an

operand position. By relying entirely on procedure calls

to express iteration, Scheme emphasized the fact that tail-

recursive procedure calls are essentially goto’s that pass

arguments. Scheme was the ﬁrst widely used program-

ming language to embrace ﬁrst class escape procedures,

from which all previously known sequential control struc-

tures can be synthesized. A subsequent version of Scheme

introduced the concept of exact and inexact numbers, an

extension of Common Lisp’s generic arithmetic. More re-

cently, Scheme became the ﬁrst programming language to

support hygienic macros, which permit the syntax of a

block-structured language to be extended in a consistent

and reliable manner.

Background

The ﬁrst description of Scheme was written in 1975 [28]. A

revised report [25] appeared in 1978, which described the

evolution of the language as its MIT implementation was

upgraded to support an innovative compiler [26]. Three

distinct projects began in 1981 and 1982 to use variants

of Scheme for courses at MIT, Yale, and Indiana Univer-

sity [21, 17, 10]. An introductory computer science text-

book using Scheme was published in 1984 [1].

As Scheme became more widespread, local dialects be-

gan to diverge until students and researchers occasion-

ally found it diﬃcult to understand code written at other

sites. Fifteen representatives of the major implementations

of Scheme therefore met in October 1984 to work toward

a better and more widely accepted standard for Scheme.

Their report [4] was published at MIT and Indiana Univer-

sity in the summer of 1985. Further revision took place in

the spring of 1986 [23], and in the spring of 1988 [6]. The

present report reﬂects further revisions agreed upon in a

meeting at Xerox PARC in June 1992.

We intend this report to belong to the entire Scheme com-

munity, and so we grant permission to copy it in whole or in

part without fee. In particular, we encourage implementors

of Scheme to use this report as a starting point for manuals

and other documentation, modifying it as necessary.

Acknowledgements

We would like to thank the following people for their

help: Alan Bawden, Michael Blair, George Carrette, Andy

Cromarty, Pavel Curtis, Jeﬀ Dalton, Olivier Danvy, Ken

Dickey, Bruce Duba, Marc Feeley, Andy Freeman, Richard

Gabriel, Yekta G¨ursel, Ken Haase, Robert Hieb, Paul

Hudak, Morry Katz, Chris Lindblad, Mark Meyer, Jim

Miller, Jim Philbin, John Ramsdell, Mike Shaﬀ, Jonathan

Shapiro, Julie Sussman, Perry Wagle, Daniel Weise, Henry

Wu, and Ozan Yigit. We thank Carol Fessenden, Daniel

Friedman, and Christopher Haynes for permission to use

text from the Scheme 311 version 4 reference manual.

We thank Texas Instruments, Inc. for permission to use

text from the TI Scheme Language Reference Manual[30].

We gladly acknowledge the inﬂuence of manuals for MIT

Scheme[17], T[22], Scheme 84[11],Common Lisp[27], and

Algol 60[18].

We also thank Betty Dexter for the extreme eﬀort she put

into setting this report in T

X, and Donald Knuth for de-

signing the program that caused her troubles.

The Artiﬁcial Intelligence Laboratory of the Massachusetts

Institute of Technology, the Computer Science Department

of Indiana University, the Computer and Information Sci-

ences Department of the University of Oregon, and the

NEC Research Institute supported the preparation of this

report. Support for the MIT work was provided in part by

the Advanced Research Projects Agency of the Department

of Defense under Oﬃce of Naval Research contract N00014-

80-C-0505. Support for the Indiana University work was

provided by NSF grants NCS 83-04567 and NCS 83-03325.

1. Overview of Scheme 3

DESCRIPTION OF THE LANGUAGE

1. Overview of Scheme

1.1. Semantics

This section gives an overview of Scheme’s semantics. A

detailed informal semantics is the subject of chapters 3

through 6. For reference purposes, section 7.2 provides a

formal semantics of Scheme.

Following Algol, Scheme is a statically scoped program-

ming language. Each use of a variable is associated with a

lexically apparent binding of that variable.

Scheme has latent as opposed to manifest types. Types

are associated with values (also called objects) rather than

with variables. (Some authors refer to languages with

latent types as weakly typed or dynamically typed lan-

guages.) Other languages with latent types are APL,

Snobol, and other dialects of Lisp. Languages with mani-

fest types (sometimes referred to as strongly typed or stat-

ically typed languages) include Algol 60, Pascal, and C.

All objects created in the course of a Scheme computation,

including procedures and continuations, have unlimited ex-

tent. No Scheme object is ever destroyed. The reason that

implementations of Scheme do not (usually!) run out of

storage is that they are permitted to reclaim the storage

occupied by an object if they can prove that the object

cannot possibly matter to any future computation. Other

languages in which most objects have unlimited extent in-

clude APL and other Lisp dialects.

Implementations of Scheme are required to be properly

tail-recursive. This allows the execution of an iterative

computation in constant space, even if the iterative compu-

tation is described by a syntactically recursive procedure.

Thus with a properly tail-recursive implementation, iter-

ation can be expressed using the ordinary procedure-call

mechanics, so that special iteration constructs are useful

only as syntactic sugar. See section 3.5.

Scheme procedures are objects in their own right. Pro-

cedures can be created dynamically, stored in data struc-

tures, returned as results of procedures, and so on. Other

languages with these properties include Common Lisp and

ML.

One distinguishing feature of Scheme is that continuations,

which in most other languages only operate behind the

scenes, also have “ﬁrst-class” status. Continuations are

useful for implementing a wide variety of advanced control

constructs, including non-local exits, backtracking, and

coroutines. See section 6.4.

Arguments to Scheme procedures are always passed by

value, which means that the actual argument expressions

are evaluated before the procedure gains control, whether

the procedure needs the result of the evaluation or not.

ML, C, and APL are three other languages that always

pass arguments by value. This is distinct from the lazy-

evaluation semantics of Haskell, or the call-by-name se-

mantics of Algol 60, where an argument expression is not

evaluated unless its value is needed by the procedure.

Scheme’s model of arithmetic is designed to remain as in-

dependent as possible of the particular ways in which num-

bers are represented within a computer. In Scheme, every

integer is a rational number, every rational is a real, and

every real is a complex number. Thus the distinction be-

tween integer and real arithmetic, so important to many

programming languages, does not appear in Scheme. In its

place is a distinction between exact arithmetic, which cor-

responds to the mathematical ideal, and inexact arithmetic

on approximations. As in Common Lisp, exact arithmetic

is not limited to integers.

1.2. Syntax

Scheme, like most dialects of Lisp, employs a fully paren-

thesized preﬁx notation for programs and (other) data; the

grammar of Scheme generates a sublanguage of the lan-

guage used for data. An important consequence of this sim-

ple, uniform representation is the susceptibility of Scheme

programs and data to uniform treatment by other Scheme

programs. For example, the eval procedure evaluates a

Scheme program expressed as data.

The read procedure performs syntactic as well as lexical

decomposition of the data it reads. The read procedure

parses its input as data (section 7.1.2), not as program.

The formal syntax of Scheme is described in section 7.1.

1.3. Notation and terminology

1.3.1. Primitive, library, and optional features

It is required that every implementation of Scheme support

all features that are not marked as being optional. Imple-

mentations are free to omit optional features of Scheme

or to add extensions, provided the extensions are not in

conﬂict with the language reported here. In particular,

implementations must support portable code by providing

a syntactic mode that preempts no lexical conventions of

this report.

To aid in understanding and implementing Scheme, some

features are marked as library. These can be easily imple-

mented in terms of the other, primitive, features. They are

redundant in the strict sense of the word, but they capture

common patterns of usage, and are therefore provided as

convenient abbreviations.

4 Revised

Scheme

1.3.2. Error situations and unspeciﬁed behavior

When speaking of an error situation, this report uses the

phrase “an error is signalled” to indicate that implemen-

tations must detect and report the error. If such wording

does not appear in the discussion of an error, then imple-

mentations are not required to detect or report the error,

though they are encouraged to do so. An error situation

that implementations are not required to detect is usually

referred to simply as “an error.”

For example, it is an error for a procedure to be passed an

argument that the procedure is not explicitly speciﬁed to

handle, even though such domain errors are seldom men-

tioned in this report. Implementations may extend a pro-

cedure’s domain of deﬁnition to include such arguments.

This report uses the phrase “may report a violation of an

implementation restriction” to indicate circumstances un-

der which an implementation is permitted to report that

it is unable to continue execution of a correct program be-

cause of some restriction imposed by the implementation.

Implementation restrictions are of course discouraged, but

implementations are encouraged to report violations of im-

plementation restrictions.

For example, an implementation may report a violation of

an implementation restriction if it does not have enough

storage to run a program.

If the value of an expression is said to be “unspeciﬁed,”

then the expression must evaluate to some object without

signalling an error, but the value depends on the imple-

mentation; this report explicitly does not say what value

should be returned.

1.3.3. Entry format

Chapters 4 and 6 are organized into entries. Each entry de-

scribes one language feature or a group of related features,

where a feature is either a syntactic construct or a built-in

procedure. An entry begins with one or more header lines

of the form

template category

for required, primitive features, or

template qualiﬁer category

where qualiﬁer is either “library” or “optional” as deﬁned

in section 1.3.1.

If category is “syntax”, the entry describes an expression

type, and the template gives the syntax of the expression

type. Components of expressions are designated by syn-

tactic variables, which are written using angle brackets,

for example, hexpressioni, hvariablei. Syntactic variables

should be understood to denote segments of program text;

for example, hexpressioni stands for any string of charac-

ters which is a syntactically valid expression. The notation

hthing

i . . .

indicates zero or more occurrences of a hthingi, and

hthing

i hthing

i . . .

indicates one or more occurrences of a hthingi.

If category is “procedure”, then the entry describes a pro-

cedure, and the header line gives a template for a call to the

procedure. Argument names in the template are italicized .

Thus the header line

(vector-ref vector k ) procedure

indicates that the built-in procedure vector-ref takes two

arguments, a vector vector and an exact non-negative in-

teger k (see below). The header lines

(make-vector k) procedure

(make-vector k ﬁll ) procedure

indicate that the make-vector procedure must be deﬁned

to take either one or two arguments.

It is an error for an operation to be presented with an ar-

gument that it is not speciﬁed to handle. For succinctness,

we follow the convention that if an argument name is also

the name of a type listed in section 3.2, then that argu-

ment must be of the named type. For example, the header

line for vector-ref given above dictates that the ﬁrst ar-

gument to vector-ref must be a vector. The following

naming conventions also imply type restrictions:

obj any object

list, list

, . . . list

, . . . list (see section 6.3.2)

z, z

, . . . z

, . . . complex number

x, x

, . . . x

, . . . real number

y, y

, . . . y

, . . . real number

q, q

, . . . q

, . . . rational number

n, n

, . . . n

, . . . integer

k, k

, . . . k

, . . . exact non-negative integer

1.3.4. Evaluation examples

The symbol “=⇒” used in program examples should be

read “evaluates to.” For example,

(* 5 8) =⇒ 40

means that the expression (* 5 8) evaluates to the ob-

ject 40. Or, more precisely: the expression given by the

sequence of characters “(* 5 8)” evaluates, in the initial

environment, to an object that may be represented exter-

nally by the sequence of characters “40”. See section 3.3

for a discussion of external representations of objects.

1.3.5. Naming conventions

By convention, the names of procedures that always return

a boolean value usually end in “?”. Such procedures are

called predicates.

2. Lexical conventions 5

By convention, the names of procedures that store values

into previously allocated locations (see section 3.4) usually

end in “!”. Such procedures are called mutation proce-

dures. By convention, the value returned by a mutation

procedure is unspeciﬁed.

By convention, “->” appears within the names of proce-

dures that take an object of one type and return an anal-

ogous object of another type. For example, list->vector

takes a list and returns a vector whose elements are the

same as those of the list.

2. Lexical conventions

This section gives an informal account of some of the lexical

conventions used in writing Scheme programs. For a formal

syntax of Scheme, see section 7.1.

Upper and lower case forms of a letter are never distin-

guished except within character and string constants. For

example, Foo is the same identiﬁer as FOO, and #x1AB is

the same number as #X1ab.

2.1. Identiﬁers

Most identiﬁers allowed by other programming languages

are also acceptable to Scheme. The precise rules for form-

ing identiﬁers vary among implementations of Scheme, but

in all implementations a sequence of letters, digits, and “ex-

tended alphabetic characters” that begins with a character

that cannot begin a number is an identiﬁer. In addition,

+, -, and ... are identiﬁers. Here are some examples of

identiﬁers:

lambda q

list->vector soup

+ V17a

<=? a34kTMNs

the-word-recursion-has-many-meanings

Extended alphabetic characters may be used within iden-

tiﬁers as if they were letters. The following are extended

alphabetic characters:

! $ % & * + - . / : < = > ? @ ^ _ ~

See section 7.1.1 for a formal syntax of identiﬁers.

Identiﬁers have two uses within Scheme programs:

• Any identiﬁer may be used as a variable or as a syn-

tactic keyword (see sections 3.1 and 4.3).

• When an identiﬁer appears as a literal or within a

literal (see section 4.1.2), it is being used to denote a

symbol (see section 6.3.3).

2.2. Whitespace and comments

Whitespace characters are spaces and newlines. (Imple-

mentations typically provide additional whitespace char-

acters such as tab or page break.) Whitespace is used for

improved readability and as necessary to separate tokens

from each other, a token being an indivisible lexical unit

such as an identiﬁer or number, but is otherwise insigniﬁ-

cant. Whitespace may occur between any two tokens, but

not within a token. Whitespace may also occur inside a

string, where it is signiﬁcant.

A semicolon (;) indicates the start of a comment. The

comment continues to the end of the line on which the

semicolon appears. Comments are invisible to Scheme, but

the end of the line is visible as whitespace. This prevents a

comment from appearing in the middle of an identiﬁer or

number.

;;; The FACT procedure computes the factorial

;;; of a non-negative integer.

(define fact

(lambda (n)

(if (= n 0)

1 ;Base case: return 1

(* n (fact (- n 1))))))

2.3. Other notations

For a description of the notations used for numbers, see

section 6.2.

. + - These are used in numbers, and may also occur

anywhere in an identiﬁer except as the ﬁrst charac-

ter. A delimited plus or minus sign by itself is also an

identiﬁer. A delimited period (not occurring within a

number or identiﬁer) is used in the notation for pairs

(section 6.3.2), and to indicate a rest-parameter in a

formal parameter list (section 4.1.4). A delimited se-

quence of three successive periods is also an identiﬁer.

( ) Parentheses are used for grouping and to notate lists

(section 6.3.2).

’ The single quote character is used to indicate literal data

(section 4.1.2).

` The backquote character is used to indicate almost-

constant data (section 4.2.6).

, ,@ The character comma and the sequence comma at-

sign are used in conjunction with backquote (sec-

tion 4.2.6).

" The double quote character is used to delimit strings

(section 6.3.5).

6 Revised

Scheme

\ Backslash is used in the syntax for character constants

(section 6.3.4) and as an escape character within string

constants (section 6.3.5).

[ ] { } | Left and right square brackets and curly braces

and vertical bar are reserved for possible future exten-

sions to the language.

# Sharp sign is used for a variety of purposes depending

on the character that immediately follows it:

#t #f These are the boolean constants (section 6.3.1).

#\ This introduces a character constant (section 6.3.4).

#( This introduces a vector constant (section 6.3.6). Vec-

tor constants are terminated by ) .

#e #i #b #o #d #x These are used in the notation for

numbers (section 6.2.4).

3. Basic concepts

3.1. Variables, syntactic keywords, and re-

gions

An identiﬁer may name a type of syntax, or it may name

a location where a value can be stored. An identiﬁer that

names a type of syntax is called a syntactic keyword and is

said to be bound to that syntax. An identiﬁer that names

a location is called a variable and is said to be bound to

that location. The set of all visible bindings in eﬀect at

some point in a program is known as the environment in

eﬀect at that point. The value stored in the location to

which a variable is bound is called the variable’s value.

By abuse of terminology, the variable is sometimes said

to name the value or to be bound to the value. This is

not quite accurate, but confusion rarely results from this

practice.

Certain expression types are used to create new kinds of

syntax and bind syntactic keywords to those new syntaxes,

while other expression types create new locations and bind

variables to those locations. These expression types are

called binding constructs. Those that bind syntactic key-

words are listed in section 4.3. The most fundamental of

the variable binding constructs is the lambda expression,

because all other variable binding constructs can be ex-

plained in terms of lambda expressions. The other variable

binding constructs are let, let*, letrec, and do expres-

sions (see sections 4.1.4, 4.2.2, and 4.2.4).

Like Algol and Pascal, and unlike most other dialects of

Lisp except for Common Lisp, Scheme is a statically scoped

language with block structure. To each place where an

identiﬁer is bound in a program there corresponds a region

of the program text within which the binding is visible.

The region is determined by the particular binding con-

struct that establishes the binding; if the binding is estab-

lished by a lambda expression, for example, then its region

is the entire lambda expression. Every mention of an iden-

tiﬁer refers to the binding of the identiﬁer that established

the innermost of the regions containing the use. If there is

no binding of the identiﬁer whose region contains the use,

then the use refers to the binding for the variable in the

top level environment, if any (chapters 4 and 6); if there is

no binding for the identiﬁer, it is said to be unbound.

3.2. Disjointness of types

No object satisﬁes more than one of the following predi-

cates:

boolean? pair?

symbol? number?

char? string?

vector? port?

procedure?

These predicates deﬁne the types boolean, pair, symbol,

number, char (or character), string, vector, port, and pro-

cedure. The empty list is a special object of its own type;

it satisﬁes none of the above predicates.

Although there is a separate boolean type, any Scheme

value can be used as a boolean value for the purpose of a

conditional test. As explained in section 6.3.1, all values

count as true in such a test except for #f. This report uses

the word “true” to refer to any Scheme value except #f,

and the word “false” to refer to #f.

3.3. External representations

An important concept in Scheme (and Lisp) is that of the

external representation of an object as a sequence of char-

acters. For example, an external representation of the inte-

ger 28 is the sequence of characters “28”, and an external

representation of a list consisting of the integers 8 and 13

is the sequence of characters “(8 13)”.

The external representation of an object is not neces-

sarily unique. The integer 28 also has representations

“#e28.000” and “#x1c”, and the list in the previous para-

graph also has the representations “( 08 13 )” and “(8

. (13 . ()))” (see section 6.3.2).

Many objects have standard external representations, but

some, such as procedures, do not have standard represen-

tations (although particular implementations may deﬁne

representations for them).

An external representation may be written in a program to

obtain the corresponding object (see quote, section 4.1.2).

3. Basic concepts 7

External representations can also be used for input and

output. The procedure read (section 6.6.2) parses external

representations, and the procedure write (section 6.6.3)

generates them. Together, they provide an elegant and

powerful input/output facility.

Note that the sequence of characters “(+ 2 6)” is not an

external representation of the integer 8, even though it is an

expression evaluating to the integer 8; rather, it is an exter-

nal representation of a three-element list, the elements of

which are the symbol + and the integers 2 and 6. Scheme’s

syntax has the property that any sequence of characters

that is an expression is also the external representation of

some object. This can lead to confusion, since it may not

be obvious out of context whether a given sequence of char-

acters is intended to denote data or program, but it is also

a source of power, since it facilitates writing programs such

as interpreters and compilers that treat programs as data

(or vice versa).

The syntax of external representations of various kinds of

objects accompanies the description of the primitives for

manipulating the objects in the appropriate sections of

chapter 6.

3.4. Storage model

Variables and objects such as pairs, vectors, and strings

implicitly denote locations or sequences of locations. A

string, for example, denotes as many locations as there

are characters in the string. (These locations need not

correspond to a full machine word.) A new value may be

stored into one of these locations using the string-set!

procedure, but the string continues to denote the same

locations as before.

An object fetched from a location, by a variable reference or

by a procedure such as car, vector-ref, or string-ref,

is equivalent in the sense of eqv? (section 6.1) to the object

last stored in the location before the fetch.

Every location is marked to show whether it is in use. No

variable or object ever refers to a location that is not in use.

Whenever this report speaks of storage being allocated for

a variable or object, what is meant is that an appropriate

number of locations are chosen from the set of locations

that are not in use, and the chosen locations are marked

to indicate that they are now in use before the variable or

object is made to denote them.

In many systems it is desirable for constants (i.e. the val-

ues of literal expressions) to reside in read-only-memory.

To express this, it is convenient to imagine that every

object that denotes locations is associated with a ﬂag

telling whether that object is mutable or immutable. In

such systems literal constants and the strings returned by

symbol->string are immutable objects, while all objects

created by the other procedures listed in this report are

mutable. It is an error to attempt to store a new value

into a location that is denoted by an immutable object.

3.5. Proper tail recursion

Implementations of Scheme are required to be properly tail-

recursive. Procedure calls that occur in certain syntactic

contexts deﬁned below are ‘tail calls’. A Scheme imple-

mentation is properly tail-recursive if it supports an un-

bounded number of active tail calls. A call is active if

the called procedure may still return. Note that this in-

cludes calls that may be returned from either by the cur-

rent continuation or by continuations captured earlier by

call-with-current-continuation that are later invoked.

In the absence of captured continuations, calls could return

at most once and the active calls would be those that had

not yet returned. A formal deﬁnition of proper tail recur-

sion can be found in [8].

Rationale:

Intuitively, no space is needed for an active tail call because the

continuation that is used in the tail call has the same semantics

as the continuation passed to the procedure containing the call.

Although an improper implementation might use a new con-

tinuation in the call, a return to this new continuation would

be followed immediately by a return to the continuation passed

to the procedure. A properly tail-recursive implementation re-

turns to that continuation directly.

Proper tail recursion was one of the central ideas in Steele and

Sussman’s original version of Scheme. Their ﬁrst Scheme in-

terpreter implemented both functions and actors. Control ﬂow

was expressed using actors, which diﬀered from functions in

that they passed their results on to another actor instead of

returning to a caller. In the terminology of this section, each

actor ﬁnished with a tail call to another actor.

Steele and Sussman later observed that in their interpreter the

code for dealing with actors was identical to that for functions

and thus there was no need to include both in the language.

A tail call is a procedure call that occurs in a tail con-

text. Tail contexts are deﬁned inductively. Note that a tail

context is always determined with respect to a particular

lambda expression.

• The last expression within the body of a lambda ex-

pression, shown as htail expressioni below, occurs in a

tail context.

(lambda hformalsi

hdeﬁnitioni* hexpressioni* htail expressioni)

• If one of the following expressions is in a tail context,

then the subexpressions shown as htail expressioni are

in a tail context. These were derived from rules in

8 Revised

Scheme

the grammar given in chapter 7 by replacing some oc-

currences of hexpressioni with htail expressioni. Only

those rules that contain tail contexts are shown here.

(if hexpressioni htail expressioni htail expressioni)

(if hexpressioni htail expressioni)

(cond hcond clausei

)

(cond hcond clausei* (else htail sequencei))

(case hexpressioni

hcase clausei

)

(case hexpressioni

hcase clausei*

(else htail sequencei))

(and hexpressioni* htail expressioni)

(or hexpressioni* htail expressioni)

(let (hbinding speci*) htail bodyi)

(let hvariablei (hbinding speci*) htail bodyi)

(let* (hbinding speci*) htail bodyi)

(letrec (hbinding speci*) htail bodyi)

(let-syntax (hsyntax speci*) htail bodyi)

(letrec-syntax (hsyntax speci*) htail bodyi)

(begin htail sequencei)

(do (hiteration speci*)

(htesti htail sequencei)

hexpressioni*)

where

hcond clausei −→ (htesti htail sequencei)

hcase clausei −→ ((hdatumi*) htail sequencei)

htail bodyi −→ hdeﬁnitioni* htail sequencei

htail sequencei −→ hexpressioni* htail expressioni

• If a cond expression is in a tail context, and has a

clause of the form (hexpression

i => hexpression

then the (implied) call to the procedure that results

from the evaluation of hexpression

i is in a tail context.

hexpression

i itself is not in a tail context.

Certain built-in procedures are also required to perform

tail calls. The ﬁrst argument passed to apply and to

call-with-current-continuation, and the second argu-

ment passed to call-with-values, must be called via a

tail call. Similarly, eval must evaluate its argument as if

it were in tail position within the eval procedure.

In the following example the only tail call is the call to f.

None of the calls to g or h are tail calls. The reference to

x is in a tail context, but it is not a call and thus is not a

tail call.

(lambda ()

(if (g)

(let ((x (h)))

(and (g) (f))))

Note: Implementations are allowed, but not required, to recog-

nize that some non-tail calls, such as the call to h above, can be

evaluated as though they were tail calls. In the example above,

the let expression could be compiled as a tail call to h. (The

possibility of h returning an unexpected number of values can

be ignored, because in that case the eﬀect of the let is explicitly

unspeciﬁed and implementation-dependent.)

4. Expressions

Expression types are categorized as primitive or derived.

Primitive expression types include variables and procedure

calls. Derived expression types are not semantically prim-

itive, but can instead be deﬁned as macros. With the ex-

ception of quasiquote, whose macro deﬁnition is complex,

the derived expressions are classiﬁed as library features.

Suitable deﬁnitions are given in section 7.3.

4.1. Primitive expression types

4.1.1. Variable references

hvariablei syntax

An expression consisting of a variable (section 3.1) is a

variable reference. The value of the variable reference is

the value stored in the location to which the variable is

bound. It is an error to reference an unbound variable.

(define x 28)

x =⇒ 28

4.1.2. Literal expressions

(quote hdatumi) syntax

’hdatumi syntax

hconstanti syntax

(quote hdatumi) evaluates to hdatumi. hDatumi may be

any external representation of a Scheme object (see sec-

tion 3.3). This notation is used to include literal constants

in Scheme code.

(quote a) =⇒ a

(quote #(a b c)) =⇒ #(a b c)

(quote (+ 1 2)) =⇒ (+ 1 2)

4. Expressions 9

(quote hdatumi) may be abbreviated as ’hdatumi. The

two notations are equivalent in all respects.

’a =⇒ a

’#(a b c) =⇒ #(a b c)

’() =⇒ ()

’(+ 1 2) =⇒ (+ 1 2)

’(quote a) =⇒ (quote a)

’’a =⇒ (quote a)

Numerical constants, string constants, character constants,

and boolean constants evaluate “to themselves”; they need

not be quoted.

’"abc" =⇒ "abc"

"abc" =⇒ "abc"

’145932 =⇒ 145932

145932 =⇒ 145932

’#t =⇒ #t

#t =⇒ #t

As noted in section 3.4, it is an error to alter a constant

(i.e. the value of a literal expression) using a mutation pro-

cedure like set-car! or string-set!.

4.1.3. Procedure calls

(hoperatori hoperand

i . . . ) syntax

A procedure call is written by simply enclosing in paren-

theses expressions for the procedure to be called and the

arguments to be passed to it. The operator and operand

expressions are evaluated (in an unspeciﬁed order) and the

resulting procedure is passed the resulting arguments.

(+ 3 4) =⇒ 7

((if #f + *) 3 4) =⇒ 12

A number of procedures are available as the values of vari-

ables in the initial environment; for example, the addition

and multiplication procedures in the above examples are

the values of the variables + and *. New procedures are cre-

ated by evaluating lambda expressions (see section 4.1.4).

Procedure calls may return any number of values (see

values in section 6.4). With the exception of values the

procedures available in the initial environment return one

value or, for procedures such as apply, pass on the values

returned by a call to one of their arguments.

Procedure calls are also called combinations.

Note: In contrast to other dialects of Lisp, the order of

evaluation is unspeciﬁed, and the operator expression and the

operand expressions are always evaluated with the same evalu-

ation rules.

Note: Although the order of evaluation is otherwise unspeci-

ﬁed, the eﬀect of any concurrent evaluation of the operator and

operand expressions is constrained to be consistent with some

sequential order of evaluation. The order of evaluation may be

chosen diﬀerently for each procedure call.

Note: In many dialects of Lisp, the empty combination, (),

is a legitimate expression. In Scheme, combinations must have

at least one subexpression, so () is not a syntactically valid

expression.

4.1.4. Procedures

(lambda hformalsi hbodyi) syntax

Syntax: hFormalsi should be a formal arguments list as

described below, and hbodyi should be a sequence of one

or more expressions.

Semantics: A lambda expression evaluates to a procedure.

The environment in eﬀect when the lambda expression was

evaluated is remembered as part of the procedure. When

the procedure is later called with some actual arguments,

the environment in which the lambda expression was evalu-

ated will be extended by binding the variables in the formal

argument list to fresh locations, the corresponding actual

argument values will be stored in those locations, and the

expressions in the body of the lambda expression will be

evaluated sequentially in the extended environment. The

result(s) of the last expression in the body will be returned

as the result(s) of the procedure call.

(lambda (x) (+ x x)) =⇒ a procedure

((lambda (x) (+ x x)) 4) =⇒ 8

(define reverse-subtract

(lambda (x y) (- y x)))

(reverse-subtract 7 10) =⇒ 3

(define add4

(let ((x 4))

(lambda (y) (+ x y))))

(add4 6) =⇒ 10

hFormalsi should have one of the following forms:

• (hvariable

i . . . ): The procedure takes a ﬁxed num-

ber of arguments; when the procedure is called, the

arguments will be stored in the bindings of the corre-

sponding variables.

• hvariablei: The procedure takes any number of ar-

guments; when the procedure is called, the sequence

of actual arguments is converted into a newly allo-

cated list, and the list is stored in the binding of the

hvariablei.

• (hvariable

i . . . hvariable

i . hvariable

n+1

i): If a

space-delimited period precedes the last variable, then

the procedure takes n or more arguments, where n

is the number of formal arguments before the period

(there must be at least one). The value stored in the

binding of the last variable will be a newly allocated

list of the actual arguments left over after all the other

actual arguments have been matched up against the

other formal arguments.

10 Revised

Scheme

It is an error for a hvariablei to appear more than once in

hformalsi.

((lambda x x) 3 4 5 6) =⇒ (3 4 5 6)

((lambda (x y . z) z)

3 4 5 6) =⇒ (5 6)

Each procedure created as the result of evaluating a lambda

expression is (conceptually) tagged with a storage location,

in order to make eqv? and eq? work on procedures (see

section 6.1).

4.1.5. Conditionals

(if htesti hconsequenti halternatei) syntax

(if htesti hconsequenti) syntax

Syntax: hTesti, hconsequenti, and halternatei may be arbi-

trary expressions.

Semantics: An if expression is evaluated as follows: ﬁrst,

htesti is evaluated. If it yields a true value (see sec-

tion 6.3.1), then hconsequenti is evaluated and its value(s)

is(are) returned. Otherwise halternatei is evaluated and its

value(s) is(are) returned. If htesti yields a false value and

no halternatei is speciﬁed, then the result of the expression

is unspeciﬁed.

(if (> 3 2) ’yes ’no) =⇒ yes

(if (> 2 3) ’yes ’no) =⇒ no

(if (> 3 2)

(- 3 2)

(+ 3 2)) =⇒ 1

4.1.6. Assignments

(set! hvariablei hexpressioni) syntax

hExpressioni is evaluated, and the resulting value is stored

in the location to which hvariablei is bound. hVariablei

must be bound either in some region enclosing the set!

expression or at top level. The result of the set! expression

is unspeciﬁed.

(define x 2)

(+ x 1) =⇒ 3

(set! x 4) =⇒ unspeciﬁed

(+ x 1) =⇒ 5

4.2. Derived expression types

The constructs in this section are hygienic, as discussed

in section 4.3. For reference purposes, section 7.3 gives

macro deﬁnitions that will convert most of the constructs

described in this section into the primitive constructs de-

scribed in the previous section.

4.2.1. Conditionals

(cond hclause

i hclause

i . . . ) library syntax

Syntax: Each hclausei should be of the form

(htesti hexpression

i . . . )

where htesti is any expression. Alternatively, a hclausei

may be of the form

(htesti => hexpressioni)

The last hclausei may be an “else clause,” which has the

form

(else hexpression

i hexpression

i . . . ).

Semantics: A cond expression is evaluated by evaluating

the htesti expressions of successive hclauseis in order until

one of them evaluates to a true value (see section 6.3.1).

When a htesti evaluates to a true value, then the remain-

ing hexpressionis in its hclausei are evaluated in order,

and the result(s) of the last hexpressioni in the hclausei

is(are) returned as the result(s) of the entire cond expres-

sion. If the selected hclausei contains only the htesti and no

hexpressionis, then the value of the htesti is returned as the

result. If the selected hclausei uses the => alternate form,

then the hexpressioni is evaluated. Its value must be a pro-

cedure that accepts one argument; this procedure is then

called on the value of the htesti and the value(s) returned

by this procedure is(are) returned by the cond expression.

If all htestis evaluate to false values, and there is no else

clause, then the result of the conditional expression is un-

speciﬁed; if there is an else clause, then its hexpressionis are

evaluated, and the value(s) of the last one is(are) returned.

(cond ((> 3 2) ’greater)

((< 3 2) ’less)) =⇒ greater

(cond ((> 3 3) ’greater)

((< 3 3) ’less)

(else ’equal)) =⇒ equal

(cond ((assv ’b ’((a 1) (b 2))) => cadr)

(else #f)) =⇒ 2

(case hkeyi hclause

i hclause

i . . . ) library syntax

Syntax: hKeyi may be any expression. Each hclausei

should have the form

((hdatum

i . . . ) hexpression

i hexpression

i . . . ),

where each hdatumi is an external representation of some

object. All the hdatumis must be distinct. The last

hclausei may be an “else clause,” which has the form

(else hexpression

i hexpression

i . . . ).

Semantics: A case expression is evaluated as follows.

hKeyi is evaluated and its result is compared against each

hdatumi. If the result of evaluating hkeyi is equivalent

(in the sense of eqv?; see section 6.1) to a hdatumi, then

the expressions in the corresponding hclausei are evaluated

from left to right and the result(s) of the last expression in

4. Expressions 11

the hclausei is(are) returned as the result(s) of the case ex-

pression. If the result of evaluating hkeyi is diﬀerent from

every hdatumi, then if there is an else clause its expres-

sions are evaluated and the result(s) of the last is(are) the

result(s) of the case expression; otherwise the result of the

case expression is unspeciﬁed.

(case (* 2 3)

((2 3 5 7) ’prime)

((1 4 6 8 9) ’composite)) =⇒ composite

(case (car ’(c d))

((a) ’a)

((b) ’b)) =⇒ unspeciﬁed

(case (car ’(c d))

((a e i o u) ’vowel)

((w y) ’semivowel)

(else ’consonant)) =⇒ consonant

(and htest

i . . . ) library syntax

The htesti expressions are evaluated from left to right, and

the value of the ﬁrst expression that evaluates to a false

value (see section 6.3.1) is returned. Any remaining ex-

pressions are not evaluated. If all the expressions evaluate

to true values, the value of the last expression is returned.

If there are no expressions then #t is returned.

(and (= 2 2) (> 2 1)) =⇒ #t

(and (= 2 2) (< 2 1)) =⇒ #f

(and 1 2 ’c ’(f g)) =⇒ (f g)

(and) =⇒ #t

(or htest

i . . . ) library syntax

The htesti expressions are evaluated from left to right, and

the value of the ﬁrst expression that evaluates to a true

value (see section 6.3.1) is returned. Any remaining ex-

pressions are not evaluated. If all expressions evaluate to

false values, the value of the last expression is returned. If

there are no expressions then #f is returned.

(or (= 2 2) (> 2 1)) =⇒ #t

(or (= 2 2) (< 2 1)) =⇒ #t

(or #f #f #f) =⇒ #f

(or (memq ’b ’(a b c))

(/ 3 0)) =⇒ (b c)

4.2.2. Binding constructs

The three binding constructs let, let*, and letrec give

Scheme a block structure, like Algol 60. The syntax of the

three constructs is identical, but they diﬀer in the regions

they establish for their variable bindings. In a let ex-

pression, the initial values are computed before any of the

variables become bound; in a let* expression, the bind-

ings and evaluations are performed sequentially; while in a

letrec expression, all the bindings are in eﬀect while their

initial values are being computed, thus allowing mutually

recursive deﬁnitions.

(let hbindingsi hbodyi) library syntax

Syntax: hBindingsi should have the form

((hvariable

i hinit

i) . . . ),

where each hiniti is an expression, and hbodyi should be a

sequence of one or more expressions. It is an error for a

hvariablei to appear more than once in the list of variables

being bound.

Semantics: The hinitis are evaluated in the current envi-

ronment (in some unspeciﬁed order), the hvariableis are

bound to fresh locations holding the results, the hbodyi is

evaluated in the extended environment, and the value(s) of

the last expression of hbodyi is(are) returned. Each bind-

ing of a hvariablei has hbodyi as its region.

(let ((x 2) (y 3))

(* x y)) =⇒ 6

(let ((x 2) (y 3))

(let ((x 7)

(z (+ x y)))

(* z x))) =⇒ 35

See also named let, section 4.2.4.

(let* hbindingsi hbodyi) library syntax

Syntax: hBindingsi should have the form

((hvariable

i hinit

i) . . . ),

and hbodyi should be a sequence of one or more expres-

sions.

Semantics: Let* is similar to let, but the bindings are

performed sequentially from left to right, and the region of

a binding indicated by (hvariablei hiniti) is that part of

the let* expression to the right of the binding. Thus the

second binding is done in an environment in which the ﬁrst

binding is visible, and so on.

(let ((x 2) (y 3))

(let* ((x 7)

(z (+ x y)))

(* z x))) =⇒ 70

(letrec hbindingsi hbodyi) library syntax

Syntax: hBindingsi should have the form

((hvariable

i hinit

i) . . . ),

and hbodyi should be a sequence of one or more expres-

sions. It is an error for a hvariablei to appear more than

once in the list of variables being bound.

Semantics: The hvariableis are bound to fresh locations

holding undeﬁned values, the hinitis are evaluated in the

12 Revised

Scheme

resulting environment (in some unspeciﬁed order), each

hvariablei is assigned to the result of the corresponding

hiniti, the hbodyi is evaluated in the resulting environment,

and the value(s) of the last expression in hbodyi is(are) re-

turned. Each binding of a hvariablei has the entire letrec

expression as its region, making it possible to deﬁne mutu-

ally recursive procedures.

(letrec ((even?

(lambda (n)

(if (zero? n)

(odd? (- n 1)))))

(odd?

(lambda (n)

(if (zero? n)

(even? (- n 1))))))

(even? 88))

=⇒ #t

One restriction on letrec is very important: it must be

possible to evaluate each hiniti without assigning or refer-

ring to the value of any hvariablei. If this restriction is

violated, then it is an error. The restriction is necessary

because Scheme passes arguments by value rather than by

name. In the most common uses of letrec, all the hinitis

are lambda expressions and the restriction is satisﬁed au-

tomatically.

4.2.3. Sequencing

(begin hexpression

i hexpression

i . . . ) library syntax

The hexpressionis are evaluated sequentially from left to

right, and the value(s) of the last hexpressioni is(are) re-

turned. This expression type is used to sequence side ef-

fects such as input and output.

(define x 0)

(begin (set! x 5)

(+ x 1)) =⇒ 6

(begin (display "4 plus 1 equals ")

(display (+ 4 1))) =⇒ unspeciﬁed

and prints 4 plus 1 equals 5

4.2.4. Iteration

(do ((hvariable

i hinit

i hstep

i) library syntax

. . . )

(htesti hexpressioni . . . )

hcommandi . . . )

Do is an iteration construct. It speciﬁes a set of variables

to be bound, how they are to be initialized at the start,

and how they are to be updated on each iteration. When a

termination condition is met, the loop exits after evaluating

the hexpressionis.

Do expressions are evaluated as follows: The hiniti ex-

pressions are evaluated (in some unspeciﬁed order), the

hvariableis are bound to fresh locations, the results of

the hiniti expressions are stored in the bindings of the

hvariableis, and then the iteration phase begins.

Each iteration begins by evaluating htesti; if the result is

false (see section 6.3.1), then the hcommandi expressions

are evaluated in order for eﬀect, the hstepi expressions

are evaluated in some unspeciﬁed order, the hvariableis

are bound to fresh locations, the results of the hstepis are

stored in the bindings of the hvariableis, and the next iter-

ation begins.

If htesti evaluates to a true value, then the hexpressionis

are evaluated from left to right and the value(s) of the

last hexpressioni is(are) returned. If no hexpressionis are

present, then the value of the do expression is unspeciﬁed.

The region of the binding of a hvariablei consists of the

entire do expression except for the hinitis. It is an error

for a hvariablei to appear more than once in the list of do

variables.

A hstepi may be omitted, in which case the eﬀect is the

same as if (hvariablei hiniti hvariablei) had been written

instead of (hvariablei hiniti).

(do ((vec (make-vector 5))

(i 0 (+ i 1)))

((= i 5) vec)

(vector-set! vec i i)) =⇒ #(0 1 2 3 4)

(let ((x ’(1 3 5 7 9)))

(do ((x x (cdr x))

(sum 0 (+ sum (car x))))

((null? x) sum))) =⇒ 25

(let hvariablei hbindingsi hbodyi) library syntax

“Named let” is a variant on the syntax of let which pro-

vides a more general looping construct than do and may

also be used to express recursions. It has the same syn-

tax and semantics as ordinary let except that hvariablei

is bound within hbodyi to a procedure whose formal argu-

ments are the bound variables and whose body is hbodyi.

Thus the execution of hbodyi may be repeated by invoking

the procedure named by hvariablei.

(let loop ((numbers ’(3 -2 1 6 -5))

(nonneg ’())

(neg ’()))

(cond ((null? numbers) (list nonneg neg))

((>= (car numbers) 0)

(loop (cdr numbers)

(cons (car numbers) nonneg)

neg))

((< (car numbers) 0)

4. Expressions 13

(loop (cdr numbers)

nonneg

(cons (car numbers) neg)))))

=⇒ ((6 1 3) (-5 -2))

4.2.5. Delayed evaluation

(delay hexpressioni) library syntax

The delay construct is used together with the proce-

dure force to implement lazy evaluation or call by need.

(delay hexpressioni) returns an object called a promise

which at some point in the future may be asked (by the

force procedure) to evaluate hexpressioni, and deliver the

resulting value. The eﬀect of hexpressioni returning multi-

ple values is unspeciﬁed.

See the description of force (section 6.4) for a more com-

plete description of delay.

4.2.6. Quasiquotation

(quasiquote hqq templatei) syntax

`hqq templatei syntax

“Backquote” or “quasiquote” expressions are useful for

constructing a list or vector structure when most but not

all of the desired structure is known in advance. If no

commas appear within the hqq templatei, the result of

evaluating `hqq templatei is equivalent to the result of

evaluating ’hqq templatei. If a comma appears within

the hqq templatei, however, the expression following the

comma is evaluated (“unquoted”) and its result is inserted

into the structure instead of the comma and the expres-

sion. If a comma appears followed immediately by an at-

sign (@), then the following expression must evaluate to

a list; the opening and closing parentheses of the list are

then “stripped away” and the elements of the list are in-

serted in place of the comma at-sign expression sequence.

A comma at-sign should only appear within a list or vector

hqq templatei.

`(list ,(+ 1 2) 4) =⇒ (list 3 4)

(let ((name ’a)) `(list ,name ’,name))

=⇒ (list a (quote a))

`(a ,(+ 1 2) ,@(map abs ’(4 -5 6)) b)

=⇒ (a 3 4 5 6 b)

`(( foo ,(- 10 3)) ,@(cdr ’(c)) . ,(car ’(cons)))

=⇒ ((foo 7) . cons)

`#(10 5 ,(sqrt 4) ,@(map sqrt ’(16 9)) 8)

=⇒ #(10 5 2 4 3 8)

Quasiquote forms may be nested. Substitutions are made

only for unquoted components appearing at the same nest-

ing level as the outermost backquote. The nesting level in-

creases by one inside each successive quasiquotation, and

decreases by one inside each unquotation.

`(a `(b ,(+ 1 2) ,(foo ,(+ 1 3) d) e) f)

=⇒ (a `(b ,(+ 1 2) ,(foo 4 d) e) f)

(let ((name1 ’x)

(name2 ’y))

`(a `(b ,,name1 ,’,name2 d) e))

=⇒ (a `(b ,x ,’y d) e)

The two notations `hqq templatei and (quasiquote

hqq templatei) are identical in all respects. ,hexpressioni

is identical to (unquote hexpressioni), and ,@hexpressioni

is identical to (unquote-splicing hexpressioni). The ex-

ternal syntax generated by write for two-element lists

whose car is one of these symbols may vary between im-

plementations.

(quasiquote (list (unquote (+ 1 2)) 4))

=⇒ (list 3 4)

’(quasiquote (list (unquote (+ 1 2)) 4))

=⇒ `(list ,(+ 1 2) 4)

i.e., (quasiquote (list (unquote (+ 1 2)) 4))

Unpredictable behavior can result if any of the symbols

quasiquote, unquote, or unquote-splicing appear in po-

sitions within a hqq templatei otherwise than as described

above.

4.3. Macros

Scheme programs can deﬁne and use new derived expres-

sion types, called macros. Program-deﬁned expression

types have the syntax

(hkeywordi hdatumi ...)

where hkeywordi is an identiﬁer that uniquely determines

the expression type. This identiﬁer is called the syntactic

keyword, or simply keyword, of the macro. The number of

the hdatumis, and their syntax, depends on the expression

type.

Each instance of a macro is called a use of the macro. The

set of rules that speciﬁes how a use of a macro is transcribed

into a more primitive expression is called the transformer

of the macro.

The macro deﬁnition facility consists of two parts:

• A set of expressions used to establish that certain iden-

tiﬁers are macro keywords, associate them with macro

transformers, and control the scope within which a

macro is deﬁned, and

• a pattern language for specifying macro transformers.

The syntactic keyword of a macro may shadow variable

bindings, and local variable bindings may shadow keyword

bindings. All macros deﬁned using the pattern language

are “hygienic” and “referentially transparent” and thus

preserve Scheme’s lexical scoping [14, 15, 2, 7, 9]:

14 Revised

Scheme

• If a macro transformer inserts a binding for an identi-

ﬁer (variable or keyword), the identiﬁer will in eﬀect be

renamed throughout its scope to avoid conﬂicts with

other identiﬁers. Note that a define at top level may

or may not introduce a binding; see section 5.2.

• If a macro transformer inserts a free reference to an

identiﬁer, the reference refers to the binding that was

visible where the transformer was speciﬁed, regardless

of any local bindings that may surround the use of the

macro.

4.3.1. Binding constructs for syntactic keywords

Let-syntax and letrec-syntax are analogous to let and

letrec, but they bind syntactic keywords to macro trans-

formers instead of binding variables to locations that con-

tain values. Syntactic keywords may also be bound at top

level; see section 5.3.

(let-syntax hbindingsi hbodyi) syntax

Syntax: hBindingsi should have the form

((hkeywordi htransformer speci) . . . )

Each hkeywordi is an identiﬁer, each htransformer speci

is an instance of syntax-rules, and hbodyi should be a

sequence of one or more expressions. It is an error for a

hkeywordi to appear more than once in the list of keywords

being bound.

Semantics: The hbodyi is expanded in the syntactic envi-

ronment obtained by extending the syntactic environment

of the let-syntax expression with macros whose keywords

are the hkeywordis, bound to the speciﬁed transformers.

Each binding of a hkeywordi has hbodyi as its region.

(let-syntax ((when (syntax-rules ()

((when test stmt1 stmt2 ...)

(if test

(begin stmt1

stmt2 ...))))))

(let ((if #t))

(when if (set! if ’now))

if)) =⇒ now

(let ((x ’outer))

(let-syntax ((m (syntax-rules () ((m) x))))

(let ((x ’inner))

(m)))) =⇒ outer

(letrec-syntax hbindingsi hbodyi) syntax

Syntax: Same as for let-syntax.

Semantics: The hbodyi is expanded in the syntactic envi-

ronment obtained by extending the syntactic environment

of the letrec-syntax expression with macros whose key-

words are the hkeywordis, bound to the speciﬁed trans-

formers. Each binding of a hkeywordi has the hbindingsi

as well as the hbodyi within its region, so the transformers

can transcribe expressions into uses of the macros intro-

duced by the letrec-syntax expression.

(letrec-syntax

((my-or (syntax-rules ()

((my-or) #f)

((my-or e) e)

((my-or e1 e2 ...)

(let ((temp e1))

(if temp

temp

(my-or e2 ...)))))))

(let ((x #f)

(y 7)

(temp 8)

(let odd?)

(if even?))

(my-or x

(let temp)

(if y)

y))) =⇒ 7

4.3.2. Pattern language

A htransformer speci has the following form:

(syntax-rules hliteralsi hsyntax rulei . . . )

Syntax: hLiteralsi is a list of identiﬁers and each

hsyntax rulei should be of the form

(hpatterni htemplatei)

The hpatterni in a hsyntax rulei is a list hpatterni that

begins with the keyword for the macro.

A hpatterni is either an identiﬁer, a constant, or one of the

following

(hpatterni ...)

(hpatterni hpatterni ... . hpatterni)

(hpatterni ... hpatterni hellipsisi)

#(hpatterni ...)

#(hpatterni ... hpatterni hellipsisi)

and a template is either an identiﬁer, a constant, or one of

the following

(helementi ...)

(helementi helementi ... . htemplatei)

#(helementi ...)

where an helementi is a htemplatei optionally followed by

an hellipsisi and an hellipsisi is the identiﬁer “...” (which

cannot be used as an identiﬁer in either a template or a

pattern).

Semantics: An instance of syntax-rules produces a new

macro transformer by specifying a sequence of hygienic

4. Expressions 15

rewrite rules. A use of a macro whose keyword is associated

with a transformer speciﬁed by syntax-rules is matched

against the patterns contained in the hsyntax ruleis, be-

ginning with the leftmost hsyntax rulei. When a match is

found, the macro use is transcribed hygienically according

to the template.

An identiﬁer that appears in the pattern of a hsyntax rulei

is a pattern variable, unless it is the keyword that begins

the pattern, is listed in hliteralsi, or is the identiﬁer “...”.

Pattern variables match arbitrary input elements and are

used to refer to elements of the input in the template. It

is an error for the same pattern variable to appear more

than once in a hpatterni.

The keyword at the beginning of the pattern in a

hsyntax rulei is not involved in the matching and is not

considered a pattern variable or literal identiﬁer.

Rationale: The scope of the keyword is determined by the

expression or syntax deﬁnition that binds it to the associated

macro transformer. If the keyword were a pattern variable or

literal identiﬁer, then the template that follows the pattern

would be within its scope regardless of whether the keyword

were bound by let-syntax or by letrec-syntax.

Identiﬁers that appear in hliteralsi are interpreted as literal

identiﬁers to be matched against corresponding subforms

of the input. A subform in the input matches a literal

identiﬁer if and only if it is an identiﬁer and either both its

occurrence in the macro expression and its occurrence in

the macro deﬁnition have the same lexical binding, or the

two identiﬁers are equal and both have no lexical binding.

A subpattern followed by ... can match zero or more el-

ements of the input. It is an error for ... to appear in

hliteralsi. Within a pattern the identiﬁer ... must follow

the last element of a nonempty sequence of subpatterns.

More formally, an input form F matches a pattern P if and

only if:

• P is a non-literal identiﬁer; or

• P is a literal identiﬁer and F is an identiﬁer with the

same binding; or

• P is a list (P

. . . P

) and F is a list of n forms that

match P

through P

, respectively; or

• P is an improper list (P

. . . P

. P

n+1

) and

F is a list or improper list of n or more forms that

match P

through P

, respectively, and whose nth

“cdr” matches P

n+1

; or

• P is of the form (P

. . . P

n+1

hellipsisi) where

hellipsisi is the identiﬁer ... and F is a proper list

of at least n forms, the ﬁrst n of which match P

through P

, respectively, and each remaining element

of F matches P

n+1

; or

• P is a vector of the form #(P

. . . P

) and F is a

vector of n forms that match P

through P

; or

• P is of the form #(P

. . . P

n+1

hellipsisi) where

hellipsisi is the identiﬁer ... and F is a vector of n or

more forms the ﬁrst n of which match P

through P

respectively, and each remaining element of F matches

n+1

; or

• P is a datum and F is equal to P in the sense of the

equal? procedure.

It is an error to use a macro keyword, within the scope of

its binding, in an expression that does not match any of

the patterns.

When a macro use is transcribed according to the template

of the matching hsyntax rulei, pattern variables that occur

in the template are replaced by the subforms they match

in the input. Pattern variables that occur in subpatterns

followed by one or more instances of the identiﬁer ... are

allowed only in subtemplates that are followed by as many

instances of .... They are replaced in the output by all

of the subforms they match in the input, distributed as

indicated. It is an error if the output cannot be built up

as speciﬁed.

Identiﬁers that appear in the template but are not pattern

variables or the identiﬁer ... are inserted into the output

as literal identiﬁers. If a literal identiﬁer is inserted as a

free identiﬁer then it refers to the binding of that identiﬁer

within whose scope the instance of syntax-rules appears.

If a literal identiﬁer is inserted as a bound identiﬁer then

it is in eﬀect renamed to prevent inadvertent captures of

free identiﬁers.

As an example, if let and cond are deﬁned as in section 7.3

then they are hygienic (as required) and the following is not

an error.

(let ((=> #f))

(cond (#t => ’ok))) =⇒ ok

The macro transformer for cond recognizes => as a local

variable, and hence an expression, and not as the top-level

identiﬁer =>, which the macro transformer treats as a syn-

tactic keyword. Thus the example expands into

(let ((=> #f))

(if #t (begin => ’ok)))

instead of

(let ((=> #f))

(let ((temp #t))

(if temp (’ok temp))))

which would result in an invalid procedure call.

16 Revised

Scheme

5. Program structure

5.1. Programs

A Scheme program consists of a sequence of expressions,

deﬁnitions, and syntax deﬁnitions. Expressions are de-

scribed in chapter 4; deﬁnitions and syntax deﬁnitions are

the subject of the rest of the present chapter.

Programs are typically stored in ﬁles or entered inter-

actively to a running Scheme system, although other

paradigms are possible; questions of user interface lie out-

side the scope of this report. (Indeed, Scheme would still be

useful as a notation for expressing computational methods

even in the absence of a mechanical implementation.)

Deﬁnitions and syntax deﬁnitions occurring at the top level

of a program can be interpreted declaratively. They cause

bindings to be created in the top level environment or mod-

ify the value of existing top-level bindings. Expressions

occurring at the top level of a program are interpreted im-

peratively; they are executed in order when the program

is invoked or loaded, and typically perform some kind of

initialization.

At the top level of a program (begin hform

i . . . ) is

equivalent to the sequence of expressions, deﬁnitions, and

syntax deﬁnitions that form the body of the begin.

5.2. Deﬁnitions

Deﬁnitions are valid in some, but not all, contexts where

expressions are allowed. They are valid only at the top

level of a hprogrami and at the beginning of a hbodyi.

A deﬁnition should have one of the following forms:

• (define hvariablei hexpressioni)

• (define (hvariablei hformalsi) hbodyi)

hFormalsi should be either a sequence of zero or more

variables, or a sequence of one or more variables fol-

lowed by a space-delimited period and another vari-

able (as in a lambda expression). This form is equiv-

alent to

(define hvariablei

(lambda (hformalsi) hbodyi)).

• (define (hvariablei . hformali) hbodyi)

hFormali should be a single variable. This form is

equivalent to

(define hvariablei

(lambda hformali hbodyi)).

5.2.1. Top level deﬁnitions

At the top level of a program, a deﬁnition

(define hvariablei hexpressioni)

has essentially the same eﬀect as the assignment expres-

sion

(set! hvariablei hexpressioni)

if hvariablei is bound. If hvariablei is not bound, however,

then the deﬁnition will bind hvariablei to a new location

before performing the assignment, whereas it would be an

error to perform a set! on an unbound variable.

(define add3

(lambda (x) (+ x 3)))

(add3 3) =⇒ 6

(define first car)

(first ’(1 2)) =⇒ 1

Some implementations of Scheme use an initial environ-

ment in which all possible variables are bound to locations,

most of which contain undeﬁned values. Top level deﬁni-

tions in such an implementation are truly equivalent to

assignments.

5.2.2. Internal deﬁnitions

Deﬁnitions may occur at the beginning of a hbodyi (that

is, the body of a lambda, let, let*, letrec, let-syntax,

or letrec-syntax expression or that of a deﬁnition of an

appropriate form). Such deﬁnitions are known as internal

deﬁnitions as opposed to the top level deﬁnitions described

above. The variable deﬁned by an internal deﬁnition is

local to the hbodyi. That is, hvariablei is bound rather

than assigned, and the region of the binding is the entire

hbodyi. For example,

(let ((x 5))

(define foo (lambda (y) (bar x y)))

(define bar (lambda (a b) (+ (* a b) a)))

(foo (+ x 3))) =⇒ 45

A hbodyi containing internal deﬁnitions can always be con-

verted into a completely equivalent letrec expression. For

example, the let expression in the above example is equiv-

alent to

(let ((x 5))

(letrec ((foo (lambda (y) (bar x y)))

(bar (lambda (a b) (+ (* a b) a))))

(foo (+ x 3))))

Just as for the equivalent letrec expression, it must be

possible to evaluate each hexpressioni of every internal def-

inition in a hbodyi without assigning or referring to the

value of any hvariablei being deﬁned.

Wherever an internal deﬁnition may occur (begin

hdeﬁnition

i . . . ) is equivalent to the sequence of deﬁni-

tions that form the body of the begin.

6. Standard procedures 17

5.3. Syntax deﬁnitions

Syntax deﬁnitions are valid only at the top level of a

hprogrami. They have the following form:

(define-syntax hkeywordi htransformer speci)

hKeywordi is an identiﬁer, and the htransformer speci

should be an instance of syntax-rules. The top-level syn-

tactic environment is extended by binding the hkeywordi

to the speciﬁed transformer.

There is no define-syntax analogue of internal deﬁni-

tions.

Although macros may expand into deﬁnitions and syntax

deﬁnitions in any context that permits them, it is an error

for a deﬁnition or syntax deﬁnition to shadow a syntactic

keyword whose meaning is needed to determine whether

some form in the group of forms that contains the shad-

owing deﬁnition is in fact a deﬁnition, or, for internal def-

initions, is needed to determine the boundary between the

group and the expressions that follow the group. For ex-

ample, the following are errors:

(define define 3)

(begin (define begin list))

(let-syntax

((foo (syntax-rules ()

((foo (proc args ...) body ...)

(define proc

(lambda (args ...)

body ...))))))

(let ((x 3))

(foo (plus x y) (+ x y))

(define foo x)

(plus foo x)))

6. Standard procedures

This chapter describes Scheme’s built-in procedures. The

initial (or “top level”) Scheme environment starts out with

a number of variables bound to locations containing useful

values, most of which are primitive procedures that ma-

nipulate data. For example, the variable abs is bound to

(a location initially containing) a procedure of one argu-

ment that computes the absolute value of a number, and

the variable + is bound to a procedure that computes sums.

Built-in procedures that can easily be written in terms of

other built-in procedures are identiﬁed as “library proce-

dures”.

A program may use a top-level deﬁnition to bind any vari-

able. It may subsequently alter any such binding by an

assignment (see 4.1.6). These operations do not modify

the behavior of Scheme’s built-in procedures. Altering any

top-level binding that has not been introduced by a deﬁni-

tion has an unspeciﬁed eﬀect on the behavior of the built-in

procedures.

6.1. Equivalence predicates

A predicate is a procedure that always returns a boolean

value (#t or #f). An equivalence predicate is the compu-

tational analogue of a mathematical equivalence relation

(it is symmetric, reﬂexive, and transitive). Of the equiva-

lence predicates described in this section, eq? is the ﬁnest

or most discriminating, and equal? is the coarsest. Eqv?

is slightly less discriminating than eq?.

(eqv? obj

obj

) procedure

The eqv? procedure deﬁnes a useful equivalence relation

on objects. Brieﬂy, it returns #t if obj

and obj

should

normally be regarded as the same object. This relation is

left slightly open to interpretation, but the following par-

tial speciﬁcation of eqv? holds for all implementations of

Scheme.

The eqv? procedure returns #t if:

• obj

and obj

are both #t or both #f.

• obj

and obj

are both symbols and

(string=? (symbol->string obj1)

(symbol->string obj2))

=⇒ #t

Note: This assumes that neither obj

nor obj

is an “un-

interned symbol” as alluded to in section 6.3.3. This re-

port does not presume to specify the behavior of eqv? on

implementation-dependent extensions.

• obj

and obj

are both numbers, are numerically equal

(see =, section 6.2), and are either both exact or both

inexact.

• obj

and obj

are both characters and are the same

character according to the char=? procedure (sec-

tion 6.3.4).

• both obj

and obj

are the empty list.

• obj

and obj

are pairs, vectors, or strings that denote

the same locations in the store (section 3.4).

• obj

and obj

are procedures whose location tags are

equal (section 4.1.4).

The eqv? procedure returns #f if:

• obj

and obj

are of diﬀerent types (section 3.2).

18 Revised

Scheme

• one of obj

and obj

is #t but the other is #f.

• obj

and obj

are symbols but

(string=? (symbol->string obj

)

(symbol->string obj

))

=⇒ #f

• one of obj

and obj

is an exact number but the other

is an inexact number.

• obj

and obj

are numbers for which the = procedure

returns #f.

• obj

and obj

are characters for which the char=? pro-

cedure returns #f.

• one of obj

and obj

is the empty list but the other is

not.

• obj

and obj

are pairs, vectors, or strings that denote

distinct locations.

• obj

and obj

are procedures that would behave diﬀer-

ently (return diﬀerent value(s) or have diﬀerent side

eﬀects) for some arguments.

(eqv? ’a ’a) =⇒ #t

(eqv? ’a ’b) =⇒ #f

(eqv? 2 2) =⇒ #t

(eqv? ’() ’()) =⇒ #t

(eqv? 100000000 100000000) =⇒ #t

(eqv? (cons 1 2) (cons 1 2))=⇒ #f

(eqv? (lambda () 1)

(lambda () 2)) =⇒ #f

(eqv? #f ’nil) =⇒ #f

(let ((p (lambda (x) x)))

(eqv? p p)) =⇒ #t

The following examples illustrate cases in which the above

rules do not fully specify the behavior of eqv?. All that

can be said about such cases is that the value returned by

eqv? must be a boolean.

(eqv? "" "") =⇒ unspeciﬁed

(eqv? ’#() ’#()) =⇒ unspeciﬁed

(eqv? (lambda (x) x)

(lambda (x) x)) =⇒ unspeciﬁed

(eqv? (lambda (x) x)

(lambda (y) y)) =⇒ unspeciﬁed

The next set of examples shows the use of eqv? with pro-

cedures that have local state. Gen-counter must return a

distinct procedure every time, since each procedure has its

own internal counter. Gen-loser, however, returns equiv-

alent procedures each time, since the local state does not

aﬀect the value or side eﬀects of the procedures.

(define gen-counter

(lambda ()

(let ((n 0))

(lambda () (set! n (+ n 1)) n))))

(let ((g (gen-counter)))

(eqv? g g)) =⇒ #t

(eqv? (gen-counter) (gen-counter))

=⇒ #f

(define gen-loser

(lambda ()

(let ((n 0))

(lambda () (set! n (+ n 1)) 27))))

(let ((g (gen-loser)))

(eqv? g g)) =⇒ #t

(eqv? (gen-loser) (gen-loser))

=⇒ unspeciﬁed

(letrec ((f (lambda () (if (eqv? f g) ’both ’f)))

(g (lambda () (if (eqv? f g) ’both ’g))))

(eqv? f g))

=⇒ unspeciﬁed

(letrec ((f (lambda () (if (eqv? f g) ’f ’both)))

(g (lambda () (if (eqv? f g) ’g ’both))))

(eqv? f g))

=⇒ #f

Since it is an error to modify constant objects (those re-

turned by literal expressions), implementations are per-

mitted, though not required, to share structure between

constants where appropriate. Thus the value of eqv? on

constants is sometimes implementation-dependent.

(eqv? ’(a) ’(a)) =⇒ unspeciﬁed

(eqv? "a" "a") =⇒ unspeciﬁed

(eqv? ’(b) (cdr ’(a b))) =⇒ unspeciﬁed

(let ((x ’(a)))

(eqv? x x)) =⇒ #t

Rationale: The above deﬁnition of eqv? allows implementa-

tions latitude in their treatment of procedures and literals: im-

plementations are free either to detect or to fail to detect that

two procedures or two literals are equivalent to each other, and

can decide whether or not to merge representations of equivalent

objects by using the same pointer or bit pattern to represent

both.

(eq? obj

obj

) procedure

Eq? is similar to eqv? except that in some cases it is capable

of discerning distinctions ﬁner than those detectable by

eqv?.

Eq? and eqv? are guaranteed to have the same behavior on

symbols, booleans, the empty list, pairs, procedures, and

non-empty strings and vectors. Eq?’s behavior on numbers

and characters is implementation-dependent, but it will al-

ways return either true or false, and will return true only

when eqv? would also return true. Eq? may also behave

diﬀerently from eqv? on empty vectors and empty strings.

6. Standard procedures 19

(eq? ’a ’a) =⇒ #t

(eq? ’(a) ’(a)) =⇒ unspeciﬁed

(eq? (list ’a) (list ’a)) =⇒ #f

(eq? "a" "a") =⇒ unspeciﬁed

(eq? "" "") =⇒ unspeciﬁed

(eq? ’() ’()) =⇒ #t

(eq? 2 2) =⇒ unspeciﬁed

(eq? #\A #\A) =⇒ unspeciﬁed

(eq? car car) =⇒ #t

(let ((n (+ 2 3)))

(eq? n n)) =⇒ unspeciﬁed

(let ((x ’(a)))

(eq? x x)) =⇒ #t

(let ((x ’#()))

(eq? x x)) =⇒ #t

(let ((p (lambda (x) x)))

(eq? p p)) =⇒ #t

Rationale: It will usually be possible to implement eq? much

more eﬃciently than eqv?, for example, as a simple pointer com-

parison instead of as some more complicated operation. One

reason is that it may not be possible to compute eqv? of two

numbers in constant time, whereas eq? implemented as pointer

comparison will always ﬁnish in constant time. Eq? may be used

like eqv? in applications using procedures to implement objects

with state since it obeys the same constraints as eqv?.

(equal? obj

obj

) library procedure

Equal? recursively compares the contents of pairs, vectors,

and strings, applying eqv? on other objects such as num-

bers and symbols. A rule of thumb is that objects are

generally equal? if they print the same. Equal? may fail

to terminate if its arguments are circular data structures.

(equal? ’a ’a) =⇒ #t

(equal? ’(a) ’(a)) =⇒ #t

(equal? ’(a (b) c)

’(a (b) c)) =⇒ #t

(equal? "abc" "abc") =⇒ #t

(equal? 2 2) =⇒ #t

(equal? (make-vector 5 ’a)

(make-vector 5 ’a)) =⇒ #t

(equal? (lambda (x) x)

(lambda (y) y)) =⇒ unspeciﬁed

6.2. Numbers

Numerical computation has traditionally been neglected

by the Lisp community. Until Common Lisp there was

no carefully thought out strategy for organizing numerical

computation, and with the exception of the MacLisp sys-

tem [20] little eﬀort was made to execute numerical code

eﬃciently. This report recognizes the excellent work of the

Common Lisp committee and accepts many of their rec-

ommendations. In some ways this report simpliﬁes and

generalizes their proposals in a manner consistent with the

purposes of Scheme.

It is important to distinguish between the mathemati-

cal numbers, the Scheme numbers that attempt to model

them, the machine representations used to implement the

Scheme numbers, and notations used to write numbers.

This report uses the types number, complex, real, rational,

and integer to refer to both mathematical numbers and

Scheme numbers. Machine representations such as ﬁxed

point and ﬂoating point are referred to by names such as

ﬁxnum and ﬂonum.

6.2.1. Numerical types

Mathematically, numbers may be arranged into a tower of

subtypes in which each level is a subset of the level above

it:

number

complex

real

rational

integer

For example, 3 is an integer. Therefore 3 is also a rational,

a real, and a complex. The same is true of the Scheme

numbers that model 3. For Scheme numbers, these types

are deﬁned by the predicates number?, complex?, real?,

rational?, and integer?.

There is no simple relationship between a number’s type

and its representation inside a computer. Although most

implementations of Scheme will oﬀer at least two diﬀerent

representations of 3, these diﬀerent representations denote

the same integer.

Scheme’s numerical operations treat numbers as abstract

data, as independent of their representation as possible.

Although an implementation of Scheme may use ﬁxnum,

ﬂonum, and perhaps other representations for numbers,

this should not be apparent to a casual programmer writing

simple programs.

It is necessary, however, to distinguish between numbers

that are represented exactly and those that may not be.

For example, indexes into data structures must be known

exactly, as must some polynomial coeﬃcients in a symbolic

algebra system. On the other hand, the results of measure-

ments are inherently inexact, and irrational numbers may

be approximated by rational and therefore inexact approx-

imations. In order to catch uses of inexact numbers where

exact numbers are required, Scheme explicitly distinguishes

exact from inexact numbers. This distinction is orthogonal

to the dimension of type.

6.2.2. Exactness

Scheme numbers are either exact or inexact. A number is

exact if it was written as an exact constant or was derived

from exact numbers using only exact operations. A number

20 Revised

Scheme

is inexact if it was written as an inexact constant, if it

was derived using inexact ingredients, or if it was derived

using inexact operations. Thus inexactness is a contagious

property of a number.

If two implementations produce exact results for a com-

putation that did not involve inexact intermediate results,

the two ultimate results will be mathematically equivalent.

This is generally not true of computations involving inex-

act numbers since approximate methods such as ﬂoating

point arithmetic may be used, but it is the duty of each

implementation to make the result as close as practical to

the mathematically ideal result.

Rational operations such as + should always produce ex-

act results when given exact arguments. If the operation

is unable to produce an exact result, then it may either

report the violation of an implementation restriction or it

may silently coerce its result to an inexact value. See sec-

tion 6.2.3.

With the exception of inexact->exact, the operations de-

scribed in this section must generally return inexact results

when given any inexact arguments. An operation may,

however, return an exact result if it can prove that the

value of the result is unaﬀected by the inexactness of its

arguments. For example, multiplication of any number by

an exact zero may produce an exact zero result, even if the

other argument is inexact.

6.2.3. Implementation restrictions

Implementations of Scheme are not required to implement

the whole tower of subtypes given in section 6.2.1, but

they must implement a coherent subset consistent with

both the purposes of the implementation and the spirit

of the Scheme language. For example, an implementation

in which all numbers are real may still be quite useful.

Implementations may also support only a limited range of

numbers of any type, subject to the requirements of this

section. The supported range for exact numbers of any

type may be diﬀerent from the supported range for inex-

act numbers of that type. For example, an implementation

that uses ﬂonums to represent all its inexact real numbers

may support a practically unbounded range of exact inte-

gers and rationals while limiting the range of inexact reals

(and therefore the range of inexact integers and rationals)

to the dynamic range of the ﬂonum format. Furthermore

the gaps between the representable inexact integers and ra-

tionals are likely to be very large in such an implementation

as the limits of this range are approached.

An implementation of Scheme must support exact integers

throughout the range of numbers that may be used for

indexes of lists, vectors, and strings or that may result

from computing the length of a list, vector, or string. The

length, vector-length, and string-length procedures

must return an exact integer, and it is an error to use

anything but an exact integer as an index. Furthermore

any integer constant within the index range, if expressed

by an exact integer syntax, will indeed be read as an exact

integer, regardless of any implementation restrictions that

may apply outside this range. Finally, the procedures listed

below will always return an exact integer result provided all

their arguments are exact integers and the mathematically

expected result is representable as an exact integer within

the implementation:

+ - *

quotient remainder modulo

max min abs

numerator denominator gcd

lcm floor ceiling

truncate round rationalize

expt

Implementations are encouraged, but not required, to sup-

port exact integers and exact rationals of practically unlim-

ited size and precision, and to implement the above proce-

dures and the / procedure in such a way that they always

return exact results when given exact arguments. If one of

these procedures is unable to deliver an exact result when

given exact arguments, then it may either report a vio-

lation of an implementation restriction or it may silently

coerce its result to an inexact number. Such a coercion

may cause an error later.

An implementation may use ﬂoating point and other ap-

proximate representation strategies for inexact numbers.

This report recommends, but does not require, that the

IEEE 32-bit and 64-bit ﬂoating point standards be followed

by implementations that use ﬂonum representations, and

that implementations using other representations should

match or exceed the precision achievable using these ﬂoat-

ing point standards [12].

In particular, implementations that use ﬂonum represen-

tations must follow these rules: A ﬂonum result must be

represented with at least as much precision as is used to

express any of the inexact arguments to that operation. It

is desirable (but not required) for potentially inexact oper-

ations such as sqrt, when applied to exact arguments, to

produce exact answers whenever possible (for example the

square root of an exact 4 ought to be an exact 2). If, how-

ever, an exact number is operated upon so as to produce an

inexact result (as by sqrt), and if the result is represented

as a ﬂonum, then the most precise ﬂonum format available

must be used; but if the result is represented in some other

way then the representation must have at least as much

precision as the most precise ﬂonum format available.

Although Scheme allows a variety of written notations for

numbers, any particular implementation may support only

some of them. For example, an implementation in which

all numbers are real need not support the rectangular and

6. Standard procedures 21

polar notations for complex numbers. If an implementa-

tion encounters an exact numerical constant that it cannot

represent as an exact number, then it may either report a

violation of an implementation restriction or it may silently

represent the constant by an inexact number.

6.2.4. Syntax of numerical constants

The syntax of the written representations for numbers is

described formally in section 7.1.1. Note that case is not

signiﬁcant in numerical constants.

A number may be written in binary, octal, decimal, or hex-

adecimal by the use of a radix preﬁx. The radix preﬁxes

are #b (binary), #o (octal), #d (decimal), and #x (hexadec-

imal). With no radix preﬁx, a number is assumed to be

expressed in decimal.

A numerical constant may be speciﬁed to be either exact or

inexact by a preﬁx. The preﬁxes are #e for exact, and #i

for inexact. An exactness preﬁx may appear before or after

any radix preﬁx that is used. If the written representation

of a number has no exactness preﬁx, the constant may be

either inexact or exact. It is inexact if it contains a decimal

point, an exponent, or a “#” character in the place of a

digit, otherwise it is exact.

In systems with inexact numbers of varying precisions it

may be useful to specify the precision of a constant. For

this purpose, numerical constants may be written with an

exponent marker that indicates the desired precision of the

inexact representation. The letters s, f, d, and l specify

the use of short, single, double, and long precision, respec-

tively. (When fewer than four internal inexact represen-

tations exist, the four size speciﬁcations are mapped onto

those available. For example, an implementation with two

internal representations may map short and single together

and long and double together.) In addition, the exponent

marker e speciﬁes the default precision for the implemen-

tation. The default precision has at least as much precision

as double, but implementations may wish to allow this de-

fault to be set by the user.

3.14159265358979F0

Round to single — 3.141593

0.6L0

Extend to long — .600000000000000

6.2.5. Numerical operations

The reader is referred to section 1.3.3 for a summary of

the naming conventions used to specify restrictions on the

types of arguments to numerical routines. The examples

used in this section assume that any numerical constant

written using an exact notation is indeed represented as

an exact number. Some examples also assume that certain

numerical constants written using an inexact notation can

be represented without loss of accuracy; the inexact con-

stants were chosen so that this is likely to be true in imple-

mentations that use ﬂonums to represent inexact numbers.

(number? obj ) procedure

(complex? obj ) procedure

(real? obj ) procedure

(rational? obj ) procedure

(integer? obj ) procedure

These numerical type predicates can be applied to any kind

of argument, including non-numbers. They return #t if the

object is of the named type, and otherwise they return #f.

In general, if a type predicate is true of a number then

all higher type predicates are also true of that number.

Consequently, if a type predicate is false of a number, then

all lower type predicates are also false of that number.

If z is an inexact complex number, then (real? z) is true

if and only if (zero? (imag-part z)) is true. If x is an

inexact real number, then (integer? x) is true if and only

if (= x (round x)).

(complex? 3+4i) =⇒ #t

(complex? 3) =⇒ #t

(real? 3) =⇒ #t

(real? -2.5+0.0i) =⇒ #t

(real? #e1e10) =⇒ #t

(rational? 6/10) =⇒ #t

(rational? 6/3) =⇒ #t

(integer? 3+0i) =⇒ #t

(integer? 3.0) =⇒ #t

(integer? 8/4) =⇒ #t

Note: The behavior of these type predicates on inexact num-

bers is unreliable, since any inaccuracy may aﬀect the result.

Note: In many implementations the rational? procedure will

be the same as real?, and the complex? procedure will be the

same as number?, but unusual implementations may be able

to represent some irrational numbers exactly or may extend the

number system to support some kind of non-complex numbers.

(exact? z) procedure

(inexact? z) procedure

These numerical predicates provide tests for the exactness

of a quantity. For any Scheme number, precisely one of

these predicates is true.

(= z

. . . ) procedure

(< x

. . . ) procedure

(> x

. . . ) procedure

(<= x

. . . ) procedure

(>= x

. . . ) procedure

These procedures return #t if their arguments are (respec-

tively): equal, monotonically increasing, monotonically de-

creasing, monotonically nondecreasing, or monotonically

nonincreasing.

22 Revised

Scheme

These predicates are required to be transitive.

Note: The traditional implementations of these predicates in

Lisp-like languages are not transitive.

Note: While it is not an error to compare inexact numbers

using these predicates, the results may be unreliable because a

small inaccuracy may aﬀect the result; this is especially true of

= and zero?. When in doubt, consult a numerical analyst.

(zero? z) library procedure

(positive? x) library procedure

(negative? x) library procedure

(odd? n) library procedure

(even? n) library procedure

These numerical predicates test a number for a particular

property, returning #t or #f. See note above.

(max x

. . . ) library procedure

(min x

. . . ) library procedure

These procedures return the maximum or minimum of their

arguments.

(max 3 4) =⇒ 4 ; exact

(max 3.9 4) =⇒ 4.0 ; inexact

Note: If any argument is inexact, then the result will also be

inexact (unless the procedure can prove that the inaccuracy is

not large enough to aﬀect the result, which is possible only in

unusual implementations). If min or max is used to compare

numbers of mixed exactness, and the numerical value of the

result cannot be represented as an inexact number without loss

of accuracy, then the procedure may report a violation of an

implementation restriction.

(+ z

. . . ) procedure

(* z

. . . ) procedure

These procedures return the sum or product of their argu-

ments.

(+ 3 4) =⇒ 7

(+ 3) =⇒ 3

(+) =⇒ 0

(* 4) =⇒ 4

(*) =⇒ 1

(- z

) procedure

(- z) procedure

(- z

. . . ) optional procedure

(/ z

) procedure

(/ z) procedure

(/ z

. . . ) optional procedure

With two or more arguments, these procedures return the

diﬀerence or quotient of their arguments, associating to the

left. With one argument, however, they return the additive

or multiplicative inverse of their argument.

(- 3 4) =⇒ -1

(- 3 4 5) =⇒ -6

(- 3) =⇒ -3

(/ 3 4 5) =⇒ 3/20

(/ 3) =⇒ 1/3

(abs x ) library procedure

Abs returns the absolute value of its argument.

(abs -7) =⇒ 7

(quotient n

) procedure

(remainder n

) procedure

(modulo n

) procedure

These procedures implement number-theoretic (integer) di-

vision. n

should be non-zero. All three procedures return

integers. If n

is an integer:

(quotient n

) =⇒ n

(remainder n

) =⇒ 0

(modulo n

) =⇒ 0

If n

is not an integer:

(quotient n

) =⇒ n

(remainder n

) =⇒ n

(modulo n

) =⇒ n

where n

is n

rounded towards zero, 0 < |n

| < |n

0 < |n

| < |n

|, n

and n

diﬀer from n

by a multiple of

, n

has the same sign as n

, and n

has the same sign

as n

From this we can conclude that for integers n

and n

with

not equal to 0,

(= n

(+ (* n

(quotient n

))

(remainder n

)))

=⇒ #t

provided all numbers involved in that computation are ex-

act.

(modulo 13 4) =⇒ 1

(remainder 13 4) =⇒ 1

(modulo -13 4) =⇒ 3

(remainder -13 4) =⇒ -1

(modulo 13 -4) =⇒ -3

(remainder 13 -4) =⇒ 1

(modulo -13 -4) =⇒ -1

(remainder -13 -4) =⇒ -1

(remainder -13 -4.0) =⇒ -1.0 ; inexact

6. Standard procedures 23

(gcd n

. . . ) library procedure

(lcm n

. . . ) library procedure

These procedures return the greatest common divisor or

least common multiple of their arguments. The result is

always non-negative.

(gcd 32 -36) =⇒ 4

(gcd) =⇒ 0

(lcm 32 -36) =⇒ 288

(lcm 32.0 -36) =⇒ 288.0 ; inexact

(lcm) =⇒ 1

(numerator q) procedure

(denominator q) procedure

These procedures return the numerator or denominator of

their argument; the result is computed as if the argument

was represented as a fraction in lowest terms. The denom-

inator is always positive. The denominator of 0 is deﬁned

to be 1.

(numerator (/ 6 4)) =⇒ 3

(denominator (/ 6 4)) =⇒ 2

(denominator

(exact->inexact (/ 6 4))) =⇒ 2.0

(floor x ) procedure

(ceiling x ) procedure

(truncate x ) procedure

(round x ) procedure

These procedures return integers. Floor returns the

largest integer not larger than x. Ceiling returns the

smallest integer not smaller than x. Truncate returns the

integer closest to x whose absolute value is not larger than

the absolute value of x. Round returns the closest inte-

ger to x, rounding to even when x is halfway between two

integers.

Rationale: Round rounds to even for consistency with the de-

fault rounding mode speciﬁed by the IEEE ﬂoating point stan-

dard.

Note: If the argument to one of these procedures is inexact,

then the result will also be inexact. If an exact value is needed,

the result should be passed to the inexact->exact procedure.

(floor -4.3) =⇒ -5.0

(ceiling -4.3) =⇒ -4.0

(truncate -4.3) =⇒ -4.0

(round -4.3) =⇒ -4.0

(floor 3.5) =⇒ 3.0

(ceiling 3.5) =⇒ 4.0

(truncate 3.5) =⇒ 3.0

(round 3.5) =⇒ 4.0 ; inexact

(round 7/2) =⇒ 4 ; exact

(round 7) =⇒ 7

(rationalize x y) library procedure

Rationalize returns the simplest rational number diﬀer-

ing from x by no more than y. A rational number r

simpler than another rational number r

if r

= p

and

= p

(in lowest terms) and |p

| ≤ |p

| and |q

| ≤ |q

Thus 3/5 is simpler than 4/7. Although not all rationals

are comparable in this ordering (consider 2/7 and 3/5) any

interval contains a rational number that is simpler than ev-

ery other rational number in that interval (the simpler 2/5

lies between 2/7 and 3/5). Note that 0 = 0/1 is the sim-

plest rational of all.

(rationalize

(inexact->exact .3) 1/10) =⇒ 1/3 ; exact

(rationalize .3 1/10) =⇒ #i1/3 ; inexact

(exp z) procedure

(log z) procedure

(sin z) procedure

(cos z) procedure

(tan z) procedure

(asin z) procedure

(acos z) procedure

(atan z) procedure

(atan y x) procedure

These procedures are part of every implementation that

supports general real numbers; they compute the usual

transcendental functions. Log computes the natural log-

arithm of z (not the base ten logarithm). Asin, acos,

and atan compute arcsine (sin

−1

), arccosine (cos

−1

), and

arctangent (tan

−1

), respectively. The two-argument vari-

ant of atan computes (angle (make-rectangular x y))

(see below), even in implementations that don’t support

general complex numbers.

In general, the mathematical functions log, arcsine, arc-

cosine, and arctangent are multiply deﬁned. The value of

log z is deﬁned to be the one whose imaginary part lies in

the range from −π (exclusive) to π (inclusive). log 0 is un-

deﬁned. With log deﬁned this way, the values of sin

−1

cos

−1

z, and tan

−1

z are according to the following for-

mulæ:

sin

−1

z = −i log(iz +

1 − z

)

cos

−1

z = π/2 − sin

−1

tan

−1

z = (log(1 + iz) − log(1 − iz))/(2i)

The above speciﬁcation follows [27], which in turn

cites [19]; refer to these sources for more detailed discussion

of branch cuts, boundary conditions, and implementation

of these functions. When it is possible these procedures

produce a real result from a real argument.

24 Revised

Scheme

(sqrt z) procedure

Returns the principal square root of z. The result will have

either positive real part, or zero real part and non-negative

imaginary part.

(expt z

) procedure

Returns z

raised to the power z

. For z

6= 0

= e

log z

is 1 if z = 0 and 0 otherwise.

(make-rectangular x

) procedure

(make-polar x

) procedure

(real-part z) procedure

(imag-part z) procedure

(magnitude z) procedure

(angle z) procedure

These procedures are part of every implementation that

supports general complex numbers. Suppose x

, x

and x

are real numbers and z is a complex number such

that

z = x

+ x

i = x

· e

Then

(make-rectangular x

) =⇒ z

(make-polar x

) =⇒ z

(real-part z) =⇒ x

(imag-part z) =⇒ x

(magnitude z) =⇒ |x

(angle z) =⇒ x

angle

where −π < x

ang le

≤ π with x

ang le

= x

+ 2πn for some

integer n.

Rationale: Magnitude is the same as abs for a real argu-

ment, but abs must be present in all implementations, whereas

magnitude need only be present in implementations that sup-

port general complex numbers.

(exact->inexact z) procedure

(inexact->exact z) procedure

Exact->inexact returns an inexact representation of z.

The value returned is the inexact number that is numeri-

cally closest to the argument. If an exact argument has no

reasonably close inexact equivalent, then a violation of an

implementation restriction may be reported.

Inexact->exact returns an exact representation of z. The

value returned is the exact number that is numerically clos-

est to the argument. If an inexact argument has no rea-

sonably close exact equivalent, then a violation of an im-

plementation restriction may be reported.

These procedures implement the natural one-to-one corre-

spondence between exact and inexact integers throughout

an implementation-dependent range. See section 6.2.3.

6.2.6. Numerical input and output

(number->string z ) procedure

(number->string z radix ) procedure

Radix must be an exact integer, either 2, 8, 10, or 16. If

omitted, radix defaults to 10. The procedure number->

string takes a number and a radix and returns as a string

an external representation of the given number in the given

radix such that

(let ((number number)

(radix radix))

(eqv? number

(string->number (number->string number

radix)

radix)))

is true. It is an error if no possible result makes this ex-

pression true.

If z is inexact, the radix is 10, and the above expression

can be satisﬁed by a result that contains a decimal point,

then the result contains a decimal point and is expressed

using the minimum number of digits (exclusive of exponent

and trailing zeroes) needed to make the above expression

true [3, 5]; otherwise the format of the result is unspeciﬁed.

The result returned by number->string never contains an

explicit radix preﬁx.

Note: The error case can occur only when z is not a complex

number or is a complex number with a non-rational real or

imaginary part.

Rationale: If z is an inexact number represented using ﬂonums,

and the radix is 10, then the above expression is normally satis-

ﬁed by a result containing a decimal point. The unspeciﬁed case

allows for inﬁnities, NaNs, and non-ﬂonum representations.

(string->number string) procedure

(string->number string radix ) procedure

Returns a number of the maximally precise representation

expressed by the given string. Radix must be an exact

integer, either 2, 8, 10, or 16. If supplied, radix is a default

radix that may be overridden by an explicit radix preﬁx in

string (e.g. "#o177"). If radix is not supplied, then the

default radix is 10. If string is not a syntactically valid

notation for a number, then string->number returns #f.

(string->number "100") =⇒ 100

(string->number "100" 16) =⇒ 256

(string->number "1e2") =⇒ 100.0

(string->number "15##") =⇒ 1500.0

Note: The domain of string->number may be restricted by

implementations in the following ways. String->number is per-

mitted to return #f whenever string contains an explicit radix

preﬁx. If all numbers supported by an implementation are real,

6. Standard procedures 25

then string->number is permitted to return #f whenever string

uses the polar or rectangular notations for complex numbers. If

all numbers are integers, then string->number may return #f

whenever the fractional notation is used. If all numbers are

exact, then string->number may return #f whenever an ex-

ponent marker or explicit exactness preﬁx is used, or if a #

appears in place of a digit. If all inexact numbers are integers,

then string->number may return #f whenever a decimal point

is used.

6.3. Other data types

This section describes operations on some of Scheme’s non-

numeric data types: booleans, pairs, lists, symbols, char-

acters, strings and vectors.

6.3.1. Booleans

The standard boolean objects for true and false are written

as #t and #f. What really matters, though, are the objects

that the Scheme conditional expressions (if, cond, and,

or, do) treat as true or false. The phrase “a true value”

(or sometimes just “true”) means any object treated as

true by the conditional expressions, and the phrase “a false

value” (or “false”) means any object treated as false by the

conditional expressions.

Of all the standard Scheme values, only #f counts as false

in conditional expressions. Except for #f, all standard

Scheme values, including #t, pairs, the empty list, sym-

bols, numbers, strings, vectors, and procedures, count as

true.

Note: Programmers accustomed to other dialects of Lisp

should be aware that Scheme distinguishes both #f and the

empty list from the symbol nil.

Boolean constants evaluate to themselves, so they do not

need to be quoted in programs.

#t =⇒ #t

#f =⇒ #f

’#f =⇒ #f

(not obj ) library procedure

Not returns #t if obj is false, and returns #f otherwise.

(not #t) =⇒ #f

(not 3) =⇒ #f

(not (list 3)) =⇒ #f

(not #f) =⇒ #t

(not ’()) =⇒ #f

(not (list)) =⇒ #f

(not ’nil) =⇒ #f

(boolean? obj ) library procedure

Boolean? returns #t if obj is either #t or #f and returns

#f otherwise.

(boolean? #f) =⇒ #t

(boolean? 0) =⇒ #f

(boolean? ’()) =⇒ #f

6.3.2. Pairs and lists

A pair (sometimes called a dotted pair) is a record structure

with two ﬁelds called the car and cdr ﬁelds (for historical

reasons). Pairs are created by the procedure cons. The

car and cdr ﬁelds are accessed by the procedures car and

cdr. The car and cdr ﬁelds are assigned by the procedures

set-car! and set-cdr!.

Pairs are used primarily to represent lists. A list can be

deﬁned recursively as either the empty list or a pair whose

cdr is a list. More precisely, the set of lists is deﬁned as

the smallest set X such that

• The empty list is in X .

• If list is in X , then any pair whose cdr ﬁeld contains

list is also in X .

The objects in the car ﬁelds of successive pairs of a list are

the elements of the list. For example, a two-element list

is a pair whose car is the ﬁrst element and whose cdr is a

pair whose car is the second element and whose cdr is the

empty list. The length of a list is the number of elements,

which is the same as the number of pairs.

The empty list is a special object of its own type (it is not

a pair); it has no elements and its length is zero.

Note: The above deﬁnitions imply that all lists have ﬁnite

length and are terminated by the empty list.

The most general notation (external representation) for

Scheme pairs is the “dotted” notation (c

. c

) where c

is the value of the car ﬁeld and c

is the value of the cdr

ﬁeld. For example (4 . 5) is a pair whose car is 4 and

whose cdr is 5. Note that (4 . 5) is the external repre-

sentation of a pair, not an expression that evaluates to a

pair.

A more streamlined notation can be used for lists: the

elements of the list are simply enclosed in parentheses and

separated by spaces. The empty list is written () . For

example,

(a b c d e)

and

(a . (b . (c . (d . (e . ())))))

26 Revised

Scheme

are equivalent notations for a list of symbols.

A chain of pairs not ending in the empty list is called an

improper list. Note that an improper list is not a list.

The list and dotted notations can be combined to represent

improper lists:

(a b c . d)

is equivalent to

(a . (b . (c . d)))

Whether a given pair is a list depends upon what is stored

in the cdr ﬁeld. When the set-cdr! procedure is used, an

object can be a list one moment and not the next:

(define x (list ’a ’b ’c))

(define y x)

y =⇒ (a b c)

(list? y) =⇒ #t

(set-cdr! x 4) =⇒ unspeciﬁed

x =⇒ (a . 4)

(eqv? x y) =⇒ #t

y =⇒ (a . 4)

(list? y) =⇒ #f

(set-cdr! x x) =⇒ unspeciﬁed

(list? x) =⇒ #f

Within literal expressions and representations of ob-

jects read by the read procedure, the forms ’hdatumi,

`hdatumi, ,hdatumi, and ,@hdatumi denote two-ele-

ment lists whose ﬁrst elements are the symbols quote,

quasiquote, unquote, and unquote-splicing, respec-

tively. The second element in each case is hdatumi. This

convention is supported so that arbitrary Scheme pro-

grams may be represented as lists. That is, according

to Scheme’s grammar, every hexpressioni is also a hdatumi

(see section 7.1.2). Among other things, this permits the

use of the read procedure to parse Scheme programs. See

section 3.3.

(pair? obj ) procedure

Pair? returns #t if obj is a pair, and otherwise returns #f.

(pair? ’(a . b)) =⇒ #t

(pair? ’(a b c)) =⇒ #t

(pair? ’()) =⇒ #f

(pair? ’#(a b)) =⇒ #f

(cons obj

obj

) procedure

Returns a newly allocated pair whose car is obj

and whose

cdr is obj

. The pair is guaranteed to be diﬀerent (in the

sense of eqv?) from every existing object.

(cons ’a ’()) =⇒ (a)

(cons ’(a) ’(b c d)) =⇒ ((a) b c d)

(cons "a" ’(b c)) =⇒ ("a" b c)

(cons ’a 3) =⇒ (a . 3)

(cons ’(a b) ’c) =⇒ ((a b) . c)

(car pair ) procedure

Returns the contents of the car ﬁeld of pair. Note that it

is an error to take the car of the empty list.

(car ’(a b c)) =⇒ a

(car ’((a) b c d)) =⇒ (a)

(car ’(1 . 2)) =⇒ 1

(car ’()) =⇒ error

(cdr pair ) procedure

Returns the contents of the cdr ﬁeld of pair. Note that it

is an error to take the cdr of the empty list.

(cdr ’((a) b c d)) =⇒ (b c d)

(cdr ’(1 . 2)) =⇒ 2

(cdr ’()) =⇒ error

(set-car! pair obj ) procedure

Stores obj in the car ﬁeld of pair. The value returned by

set-car! is unspeciﬁed.

(define (f) (list ’not-a-constant-list))

(define (g) ’(constant-list))

(set-car! (f) 3) =⇒ unspeciﬁed

(set-car! (g) 3) =⇒ error

(set-cdr! pair obj ) procedure

Stores obj in the cdr ﬁeld of pair . The value returned by

set-cdr! is unspeciﬁed.

(caar pair ) library procedure

(cadr pair ) library procedure

(cdddar pair ) library procedure

(cddddr pair ) library procedure

These procedures are compositions of car and cdr, where

for example caddr could be deﬁned by

(define caddr (lambda (x) (car (cdr (cdr x))))).

Arbitrary compositions, up to four deep, are provided.

There are twenty-eight of these procedures in all.

(null? obj ) library procedure

Returns #t if obj is the empty list, otherwise returns #f.

(list? obj ) library procedure

Returns #t if obj is a list, otherwise returns #f. By deﬁni-

tion, all lists have ﬁnite length and are terminated by the

empty list.

6. Standard procedures 27

(list? ’(a b c)) =⇒ #t

(list? ’()) =⇒ #t

(list? ’(a . b)) =⇒ #f

(let ((x (list ’a)))

(set-cdr! x x)

(list? x)) =⇒ #f

(list obj . . . ) library procedure

Returns a newly allocated list of its arguments.

(list ’a (+ 3 4) ’c) =⇒ (a 7 c)

(list) =⇒ ()

(length list) library procedure

Returns the length of list.

(length ’(a b c)) =⇒ 3

(length ’(a (b) (c d e))) =⇒ 3

(length ’()) =⇒ 0

(append list . . . ) library procedure

Returns a list consisting of the elements of the ﬁrst list

followed by the elements of the other lists.

(append ’(x) ’(y)) =⇒ (x y)

(append ’(a) ’(b c d)) =⇒ (a b c d)

(append ’(a (b)) ’((c))) =⇒ (a (b) (c))

The resulting list is always newly allocated, except that

it shares structure with the last list argument. The last

argument may actually be any object; an improper list

results if the last argument is not a proper list.

(append ’(a b) ’(c . d)) =⇒ (a b c . d)

(append ’() ’a) =⇒ a

(reverse list) library procedure

Returns a newly allocated list consisting of the elements of

list in reverse order.

(reverse ’(a b c)) =⇒ (c b a)

(reverse ’(a (b c) d (e (f))))

=⇒ ((e (f)) d (b c) a)

(list-tail list k) library procedure

Returns the sublist of list obtained by omitting the ﬁrst k

elements. It is an error if list has fewer than k elements.

List-tail could be deﬁned by

(define list-tail

(lambda (x k)

(if (zero? k)

(list-tail (cdr x) (- k 1)))))

(list-ref list k) library procedure

Returns the kth element of list. (This is the same as the

car of (list-tail list k).) It is an error if list has fewer

than k elements.

(list-ref ’(a b c d) 2) =⇒ c

(list-ref ’(a b c d)

(inexact->exact (round 1.8)))

=⇒ c

(memq obj list) library procedure

(memv obj list) library procedure

(member obj list) library procedure

These procedures return the ﬁrst sublist of list whose car

is obj , where the sublists of list are the non-empty lists

returned by (list-tail list k) for k less than the length

of list. If obj does not occur in list, then #f (not the empty

list) is returned. Memq uses eq? to compare obj with the

elements of list , while memv uses eqv? and member uses

equal?.

(memq ’a ’(a b c)) =⇒ (a b c)

(memq ’b ’(a b c)) =⇒ (b c)

(memq ’a ’(b c d)) =⇒ #f

(memq (list ’a) ’(b (a) c)) =⇒ #f

(member (list ’a)

’(b (a) c)) =⇒ ((a) c)

(memq 101 ’(100 101 102)) =⇒ unspeciﬁed

(memv 101 ’(100 101 102)) =⇒ (101 102)

(assq obj alist) library procedure

(assv obj alist) library procedure

(assoc obj alist) library procedure

Alist (for “association list”) must be a list of pairs. These

procedures ﬁnd the ﬁrst pair in alist whose car ﬁeld is obj ,

and returns that pair. If no pair in alist has obj as its car,

then #f (not the empty list) is returned. Assq uses eq? to

compare obj with the car ﬁelds of the pairs in alist, while

assv uses eqv? and assoc uses equal?.

(define e ’((a 1) (b 2) (c 3)))

(assq ’a e) =⇒ (a 1)

(assq ’b e) =⇒ (b 2)

(assq ’d e) =⇒ #f

(assq (list ’a) ’(((a)) ((b)) ((c))))

=⇒ #f

(assoc (list ’a) ’(((a)) ((b)) ((c))))

=⇒ ((a))

(assq 5 ’((2 3) (5 7) (11 13)))

=⇒ unspeciﬁed

(assv 5 ’((2 3) (5 7) (11 13)))

=⇒ (5 7)

Rationale: Although they are ordinarily used as predicates,

memq, memv, member, assq, assv, and assoc do not have question

marks in their names because they return useful values rather

than just #t or #f.

28 Revised

Scheme

6.3.3. Symbols

Symbols are objects whose usefulness rests on the fact that

two symbols are identical (in the sense of eqv?) if and only

if their names are spelled the same way. This is exactly the

property needed to represent identiﬁers in programs, and

so most implementations of Scheme use them internally for

that purpose. Symbols are useful for many other applica-

tions; for instance, they may be used the way enumerated

values are used in Pascal.

The rules for writing a symbol are exactly the same as the

rules for writing an identiﬁer; see sections 2.1 and 7.1.1.

It is guaranteed that any symbol that has been returned as

part of a literal expression, or read using the read proce-

dure, and subsequently written out using the write proce-

dure, will read back in as the identical symbol (in the sense

of eqv?). The string->symbol procedure, however, can

create symbols for which this write/read invariance may

not hold because their names contain special characters or

letters in the non-standard case.

Note: Some implementations of Scheme have a feature known

as “slashiﬁcation” in order to guarantee write/read invariance

for all symbols, but historically the most important use of this

feature has been to compensate for the lack of a string data

type.

Some implementations also have “uninterned symbols”, which

defeat write/read invariance even in implementations with

slashiﬁcation, and also generate exceptions to the rule that two

symbols are the same if and only if their names are spelled the

same.

(symbol? obj ) procedure

Returns #t if obj is a symbol, otherwise returns #f.

(symbol? ’foo) =⇒ #t

(symbol? (car ’(a b))) =⇒ #t

(symbol? "bar") =⇒ #f

(symbol? ’nil) =⇒ #t

(symbol? ’()) =⇒ #f

(symbol? #f) =⇒ #f

(symbol->string symbol ) procedure

Returns the name of symbol as a string. If the symbol was

part of an object returned as the value of a literal expres-

sion (section 4.1.2) or by a call to the read procedure, and

its name contains alphabetic characters, then the string

returned will contain characters in the implementation’s

preferred standard case—some implementations will prefer

upper case, others lower case. If the symbol was returned

by string->symbol, the case of characters in the string

returned will be the same as the case in the string that

was passed to string->symbol. It is an error to apply

mutation procedures like string-set! to strings returned

by this procedure.

The following examples assume that the implementation’s

standard case is lower case:

(symbol->string ’flying-fish)

=⇒ "flying-fish"

(symbol->string ’Martin) =⇒ "martin"

(symbol->string

(string->symbol "Malvina"))

=⇒ "Malvina"

(string->symbol string) procedure

Returns the symbol whose name is string. This procedure

can create symbols with names containing special charac-

ters or letters in the non-standard case, but it is usually

a bad idea to create such symbols because in some imple-

mentations of Scheme they cannot be read as themselves.

See symbol->string.

The following examples assume that the implementation’s

standard case is lower case:

(eq? ’mISSISSIppi ’mississippi)

=⇒ #t

(string->symbol "mISSISSIppi")

=⇒ the symbol with name "mISSISSIppi"

(eq? ’bitBlt (string->symbol "bitBlt"))

=⇒ #f

(eq? ’JollyWog

(string->symbol

(symbol->string ’JollyWog)))

=⇒ #t

(string=? "K. Harper, M.D."

(symbol->string

(string->symbol "K. Harper, M.D.")))

=⇒ #t

6.3.4. Characters

Characters are objects that represent printed characters

such as letters and digits. Characters are written using the

notation #\hcharacteri or #\hcharacter namei. For exam-

ple:

#\a ; lower case letter

#\A ; upper case letter

#\( ; left parenthesis

#\ ; the space character

#\space ; the preferred way to write a space

#\newline ; the newline character

Case is signiﬁcant in #\hcharacteri, but not in #\hcharacter

namei. If hcharacteri in #\hcharacteri is alphabetic, then

the character following hcharacteri must be a delimiter

character such as a space or parenthesis. This rule resolves

the ambiguous case where, for example, the sequence of

6. Standard procedures 29

characters “#\space” could be taken to be either a repre-

sentation of the space character or a representation of the

character “#\s” followed by a representation of the symbol

“pace.”

Characters written in the #\ notation are self-evaluating.

That is, they do not have to be quoted in programs.

Some of the procedures that operate on characters ignore

the diﬀerence between upper case and lower case. The pro-

cedures that ignore case have “-ci” (for “case insensitive”)

embedded in their names.

(char? obj ) procedure

Returns #t if obj is a character, otherwise returns #f.

(char=? char

char

) procedure

(char<? char

char

) procedure

(char>? char

char

) procedure

(char<=? char

char

) procedure

(char>=? char

char

) procedure

These procedures impose a total ordering on the set of

characters. It is guaranteed that under this ordering:

• The upper case characters are in order. For example,

(char<? #\A #\B) returns #t.

• The lower case characters are in order. For example,

(char<? #\a #\b) returns #t.

• The digits are in order. For example, (char<? #\0

#\9) returns #t.

• Either all the digits precede all the upper case letters,

or vice versa.

• Either all the digits precede all the lower case letters,

or vice versa.

Some implementations may generalize these procedures to

take more than two arguments, as with the corresponding

numerical predicates.

(char-ci=? char

char

) library procedure

(char-ci<? char

char

) library procedure

(char-ci>? char

char

) library procedure

(char-ci<=? char

char

) library procedure

(char-ci>=? char

char

) library procedure

These procedures are similar to char=? et cetera, but they

treat upper case and lower case letters as the same. For

example, (char-ci=? #\A #\a) returns #t. Some imple-

mentations may generalize these procedures to take more

than two arguments, as with the corresponding numerical

predicates.

(char-alphabetic? char) library procedure

(char-numeric? char) library procedure

(char-whitespace? char) library procedure

(char-upper-case? letter) library procedure

(char-lower-case? letter) library procedure

These procedures return #t if their arguments are alpha-

betic, numeric, whitespace, upper case, or lower case char-

acters, respectively, otherwise they return #f. The follow-

ing remarks, which are speciﬁc to the ASCII character set,

are intended only as a guide: The alphabetic characters are

the 52 upper and lower case letters. The numeric charac-

ters are the ten decimal digits. The whitespace characters

are space, tab, line feed, form feed, and carriage return.

(char->integer char) procedure

(integer->char n) procedure

Given a character, char->integer returns an exact inte-

ger representation of the character. Given an exact inte-

ger that is the image of a character under char->integer,

integer->char returns that character. These procedures

implement order-preserving isomorphisms between the set

of characters under the char<=? ordering and some subset

of the integers under the <= ordering. That is, if

(char<=? a b) =⇒ #t and (<= x y) =⇒ #t

and x and y are in the domain of integer->char, then

(<= (char->integer a)

(char->integer b)) =⇒ #t

(char<=? (integer->char x)

(integer->char y)) =⇒ #t

(char-upcase char) library procedure

(char-downcase char) library procedure

These procedures return a character char

such that

(char-ci=? char char

). In addition, if char is alpha-

betic, then the result of char-upcase is upper case and

the result of char-downcase is lower case.

6.3.5. Strings

Strings are sequences of characters. Strings are written

as sequences of characters enclosed within doublequotes

("). A doublequote can be written inside a string only by

escaping it with a backslash (\), as in

"The word \"recursion\" has many meanings."

A backslash can be written inside a string only by escaping

it with another backslash. Scheme does not specify the

eﬀect of a backslash within a string that is not followed by

a doublequote or backslash.

30 Revised

Scheme

A string constant may continue from one line to the next,

but the exact contents of such a string are unspeciﬁed.

The length of a string is the number of characters that it

contains. This number is an exact, non-negative integer

that is ﬁxed when the string is created. The valid indexes

of a string are the exact non-negative integers less than

the length of the string. The ﬁrst character of a string has

index 0, the second has index 1, and so on.

In phrases such as “the characters of string beginning with

index start and ending with index end ,” it is understood

that the index start is inclusive and the index end is ex-

clusive. Thus if start and end are the same index, a null

substring is referred to, and if start is zero and end is the

length of string, then the entire string is referred to.

Some of the procedures that operate on strings ignore the

diﬀerence between upper and lower case. The versions that

ignore case have “-ci” (for “case insensitive”) embedded

in their names.

(string? obj ) procedure

Returns #t if obj is a string, otherwise returns #f.

(make-string k) procedure

(make-string k char) procedure

Make-string returns a newly allocated string of length k.

If char is given, then all elements of the string are ini-

tialized to char , otherwise the contents of the string are

unspeciﬁed.

(string char . . . ) library procedure

Returns a newly allocated string composed of the argu-

ments.

(string-length string) procedure

Returns the number of characters in the given string.

(string-ref string k) procedure

k must be a valid index of string. String-ref returns

character k of string using zero-origin indexing.

(string-set! string k char) procedure

k must be a valid index of string. String-set! stores char

in element k of string and returns an unspeciﬁed value.

(define (f) (make-string 3 #\*))

(define (g) "***")

(string-set! (f) 0 #\?) =⇒ unspeciﬁed

(string-set! (g) 0 #\?) =⇒ error

(string-set! (symbol->string ’immutable)

#\?) =⇒ error

(string=? string

string

) library procedure

(string-ci=? string

string

) library procedure

Returns #t if the two strings are the same length and con-

tain the same characters in the same positions, otherwise

returns #f. String-ci=? treats upper and lower case let-

ters as though they were the same character, but string=?

treats upper and lower case as distinct characters.

(string<? string

string

) library procedure

(string>? string

string

) library procedure

(string<=? string

string

) library procedure

(string>=? string

string

) library procedure

(string-ci<? string

string

) library procedure

(string-ci>? string

string

) library procedure

(string-ci<=? string

string

) library procedure

(string-ci>=? string

string

) library procedure

These procedures are the lexicographic extensions to

strings of the corresponding orderings on characters. For

example, string<? is the lexicographic ordering on strings

induced by the ordering char<? on characters. If two

strings diﬀer in length but are the same up to the length

of the shorter string, the shorter string is considered to be

lexicographically less than the longer string.

Implementations may generalize these and the string=?

and string-ci=? procedures to take more than two argu-

ments, as with the corresponding numerical predicates.

(substring string start end) library procedure

String must be a string, and start and end must be exact

integers satisfying

0 ≤ start ≤ end ≤ (string-length string).

Substring returns a newly allocated string formed from

the characters of string beginning with index start (inclu-

sive) and ending with index end (exclusive).

(string-append string . . . ) library procedure

Returns a newly allocated string whose characters form the

concatenation of the given strings.

(string->list string) library procedure

(list->string list) library procedure

String->list returns a newly allocated list of the charac-

ters that make up the given string. List->string returns

a newly allocated string formed from the characters in the

list list, which must be a list of characters. String->list

and list->string are inverses so far as equal? is con-

cerned.

(string-copy string) library procedure

Returns a newly allocated copy of the given string.

6. Standard procedures 31

(string-fill! string char) library procedure

Stores char in every element of the given string and returns

an unspeciﬁed value.

6.3.6. Vectors

Vectors are heterogenous structures whose elements are in-

dexed by integers. A vector typically occupies less space

than a list of the same length, and the average time re-

quired to access a randomly chosen element is typically

less for the vector than for the list.

The length of a vector is the number of elements that it

contains. This number is a non-negative integer that is

ﬁxed when the vector is created. The valid indexes of a

vector are the exact non-negative integers less than the

length of the vector. The ﬁrst element in a vector is indexed

by zero, and the last element is indexed by one less than

the length of the vector.

Vectors are written using the notation #(obj . . . ). For

example, a vector of length 3 containing the number zero

in element 0, the list (2 2 2 2) in element 1, and the

string "Anna" in element 2 can be written as following:

#(0 (2 2 2 2) "Anna")

Note that this is the external representation of a vector, not

an expression evaluating to a vector. Like list constants,

vector constants must be quoted:

’#(0 (2 2 2 2) "Anna")

=⇒ #(0 (2 2 2 2) "Anna")

(vector? obj ) procedure

Returns #t if obj is a vector, otherwise returns #f.

(make-vector k) procedure

(make-vector k ﬁll ) procedure

Returns a newly allocated vector of k elements. If a second

argument is given, then each element is initialized to ﬁll .

Otherwise the initial contents of each element is unspeci-

ﬁed.

(vector obj . . . ) library procedure

Returns a newly allocated vector whose elements contain

the given arguments. Analogous to list.

(vector ’a ’b ’c) =⇒ #(a b c)

(vector-length vector ) procedure

Returns the number of elements in vector as an exact in-

teger.

(vector-ref vector k ) procedure

k must be a valid index of vector . Vector-ref returns the

contents of element k of vector .

(vector-ref ’#(1 1 2 3 5 8 13 21)

=⇒ 8

(vector-ref ’#(1 1 2 3 5 8 13 21)

(let ((i (round (* 2 (acos -1)))))

(if (inexact? i)

(inexact->exact i)

i)))

=⇒ 13

(vector-set! vector k obj ) procedure

k must be a valid index of vector. Vector-set! stores obj

in element k of vector . The value returned by vector-set!

is unspeciﬁed.

(let ((vec (vector 0 ’(2 2 2 2) "Anna")))

(vector-set! vec 1 ’("Sue" "Sue"))

vec)

=⇒ #(0 ("Sue" "Sue") "Anna")

(vector-set! ’#(0 1 2) 1 "doe")

=⇒ error ; constant vector

(vector->list vector ) library procedure

(list->vector list) library procedure

Vector->list returns a newly allocated list of the objects

contained in the elements of vector. List->vector returns

a newly created vector initialized to the elements of the list

list.

(vector->list ’#(dah dah didah))

=⇒ (dah dah didah)

(list->vector ’(dididit dah))

=⇒ #(dididit dah)

(vector-fill! vector ﬁll) library procedure

Stores ﬁll in every element of vector. The value returned

by vector-fill! is unspeciﬁed.

6.4. Control features

This chapter describes various primitive procedures which

control the ﬂow of program execution in special ways. The

procedure? predicate is also described here.

(procedure? obj ) procedure

Returns #t if obj is a procedure, otherwise returns #f.

(procedure? car) =⇒ #t

(procedure? ’car) =⇒ #f

(procedure? (lambda (x) (* x x)))

=⇒ #t

(procedure? ’(lambda (x) (* x x)))

=⇒ #f

(call-with-current-continuation procedure?)

=⇒ #t

32 Revised

Scheme

(apply proc arg

. . . args) procedure

Proc must be a procedure and args must be a list. Calls

proc with the elements of the list (append (list arg

. . . ) args) as the actual arguments.

(apply + (list 3 4)) =⇒ 7

(define compose

(lambda (f g)

(lambda args

(f (apply g args)))))

((compose sqrt *) 12 75) =⇒ 30

(map proc list

list

. . . ) library procedure

The list s must be lists, and proc must be a procedure taking

as many arguments as there are lists and returning a single

value. If more than one list is given, then they must all

be the same length. Map applies proc element-wise to the

elements of the lists and returns a list of the results, in

order. The dynamic order in which proc is applied to the

elements of the lists is unspeciﬁed.

(map cadr ’((a b) (d e) (g h)))

=⇒ (b e h)

(map (lambda (n) (expt n n))

’(1 2 3 4 5))

=⇒ (1 4 27 256 3125)

(map + ’(1 2 3) ’(4 5 6)) =⇒ (5 7 9)

(let ((count 0))

(map (lambda (ignored)

(set! count (+ count 1))

count)

’(a b))) =⇒ (1 2) or (2 1)

(for-each proc list

list

. . . ) library procedure

The arguments to for-each are like the arguments to map,

but for-each calls proc for its side eﬀects rather than for

its values. Unlike map, for-each is guaranteed to call proc

on the elements of the lists in order from the ﬁrst ele-

ment(s) to the last, and the value returned by for-each is

unspeciﬁed.

(let ((v (make-vector 5)))

(for-each (lambda (i)

(vector-set! v i (* i i)))

’(0 1 2 3 4))

v) =⇒ #(0 1 4 9 16)

(force promise) library procedure

Forces the value of promise (see delay, section 4.2.5). If no

value has been computed for the promise, then a value is

computed and returned. The value of the promise is cached

(or “memoized”) so that if it is forced a second time, the

previously computed value is returned.

(force (delay (+ 1 2))) =⇒ 3

(let ((p (delay (+ 1 2))))

(list (force p) (force p)))

=⇒ (3 3)

(define a-stream

(letrec ((next

(lambda (n)

(cons n (delay (next (+ n 1)))))))

(next 0)))

(define head car)

(define tail

(lambda (stream) (force (cdr stream))))

(head (tail (tail a-stream)))

=⇒ 2

Force and delay are mainly intended for programs written

in functional style. The following examples should not be

considered to illustrate good programming style, but they

illustrate the property that only one value is computed for

a promise, no matter how many times it is forced.

(define count 0)

(define p

(delay (begin (set! count (+ count 1))

(if (> count x)

count

(force p)))))

(define x 5)

p =⇒ a promise

(force p) =⇒ 6

p =⇒ a promise, still

(begin (set! x 10)

(force p)) =⇒ 6

Here is a possible implementation of delay and force.

Promises are implemented here as procedures of no argu-

ments, and force simply calls its argument:

(define force

(lambda (object)

(object)))

We deﬁne the expression

(delay hexpressioni)

to have the same meaning as the procedure call

(make-promise (lambda () hexpressioni))

as follows

(define-syntax delay

(syntax-rules ()

((delay expression)

(make-promise (lambda () expression))))),

6. Standard procedures 33

where make-promise is deﬁned as follows:

(define make-promise

(lambda (proc)

(let ((result-ready? #f)

(result #f))

(lambda ()

(if result-ready?

result

(let ((x (proc)))

(if result-ready?

result

(begin (set! result-ready? #t)

(set! result x)

result))))))))

Rationale: A promise may refer to its own value, as in the

last example above. Forcing such a promise may cause the

promise to be forced a second time before the value of the ﬁrst

force has been computed. This complicates the deﬁnition of

make-promise.

Various extensions to this semantics of delay and force

are supported in some implementations:

• Calling force on an object that is not a promise may

simply return the object.

• It may be the case that there is no means by which

a promise can be operationally distinguished from its

forced value. That is, expressions like the following

may evaluate to either #t or to #f, depending on the

implementation:

(eqv? (delay 1) 1) =⇒ unspeciﬁed

(pair? (delay (cons 1 2))) =⇒ unspeciﬁed

• Some implementations may implement “implicit forc-

ing,” where the value of a promise is forced by primi-

tive procedures like cdr and +:

(+ (delay (* 3 7)) 13) =⇒ 34

(call-with-current-continuation proc) procedure

Proc must be a procedure of one argument. The procedure

call-with-current-continuation packages up the cur-

rent continuation (see the rationale below) as an “escape

procedure” and passes it as an argument to proc. The es-

cape procedure is a Scheme procedure that, if it is later

called, will abandon whatever continuation is in eﬀect at

that later time and will instead use the continuation that

was in eﬀect when the escape procedure was created. Call-

ing the escape procedure may cause the invocation of before

and after thunks installed using dynamic-wind.

The escape procedure accepts the same number of ar-

guments as the continuation to the original call to

call-with-current-continuation. Except for continua-

tions created by the call-with-values procedure, all con-

tinuations take exactly one value. The eﬀect of passing no

value or more than one value to continuations that were

not created by call-with-values is unspeciﬁed.

The escape procedure that is passed to proc has unlimited

extent just like any other procedure in Scheme. It may be

stored in variables or data structures and may be called as

many times as desired.

The following examples show only the most common ways

in which call-with-current-continuation is used. If

all real uses were as simple as these examples, there

would be no need for a procedure with the power of

call-with-current-continuation.

(call-with-current-continuation

(lambda (exit)

(for-each (lambda (x)

(if (negative? x)

(exit x)))

’(54 0 37 -3 245 19))

#t)) =⇒ -3

(define list-length

(lambda (obj)

(call-with-current-continuation

(lambda (return)

(letrec ((r

(lambda (obj)

(cond ((null? obj) 0)

((pair? obj)

(+ (r (cdr obj)) 1))

(else (return #f))))))

(r obj))))))

(list-length ’(1 2 3 4)) =⇒ 4

(list-length ’(a b . c)) =⇒ #f

Rationale:

A common use of call-with-current-continuation is for

structured, non-local exits from loops or procedure bodies, but

in fact call-with-current-continuation is extremely useful

for implementing a wide variety of advanced control structures.

Whenever a Scheme expression is evaluated there is a contin-

uation wanting the result of the expression. The continuation

represents an entire (default) future for the computation. If the

expression is evaluated at top level, for example, then the con-

tinuation might take the result, print it on the screen, prompt

for the next input, evaluate it, and so on forever. Most of the

time the continuation includes actions speciﬁed by user code,

as in a continuation that will take the result, multiply it by the

value stored in a local variable, add seven, and give the answer

to the top level continuation to be printed. Normally these

ubiquitous continuations are hidden behind the scenes and pro-

grammers do not think much about them. On rare occasions,

however, a programmer may need to deal with continuations ex-

plicitly. Call-with-current-continuation allows Scheme pro-

34 Revised

Scheme

grammers to do that by creating a procedure that acts just like

the current continuation.

Most programming languages incorporate one or more special-

purpose escape constructs with names like exit, return, or

even goto. In 1965, however, Peter Landin [16] invented a

general purpose escape operator called the J-operator. John

Reynolds [24] described a simpler but equally powerful con-

struct in 1972. The catch special form described by Sussman

and Steele in the 1975 report on Scheme is exactly the same as

Reynolds’s construct, though its name came from a less general

construct in MacLisp. Several Scheme implementors noticed

that the full power of the catch construct could be provided by

a procedure instead of by a special syntactic construct, and the

name call-with-current-continuation was coined in 1982.

This name is descriptive, but opinions diﬀer on the merits of

such a long name, and some people use the name call/cc in-

stead.

(values obj . . .) procedure

Delivers all of its arguments to its continuation. Except

for continuations created by the call-with-values pro-

cedure, all continuations take exactly one value. Values

might be deﬁned as follows:

(define (values . things)

(call-with-current-continuation

(lambda (cont) (apply cont things))))

(call-with-values producer consumer) procedure

Calls its producer argument with no values and a contin-

uation that, when passed some values, calls the consumer

procedure with those values as arguments. The continua-

tion for the call to consumer is the continuation of the call

to call-with-values.

(call-with-values (lambda () (values 4 5))

(lambda (a b) b))

=⇒ 5

(call-with-values * -) =⇒ -1

(dynamic-wind before thunk after) procedure

Calls thunk without arguments, returning the result(s) of

this call. Before and after are called, also without ar-

guments, as required by the following rules (note that

in the absence of calls to continuations captured using

call-with-current-continuation the three arguments

are called once each, in order). Before is called whenever

execution enters the dynamic extent of the call to thunk

and after is called whenever it exits that dynamic extent.

The dynamic extent of a procedure call is the period be-

tween when the call is initiated and when it returns. In

Scheme, because of call-with-current-continuation,

the dynamic extent of a call may not be a single, connected

time period. It is deﬁned as follows:

• The dynamic extent is entered when execution of the

body of the called procedure begins.

• The dynamic extent is also entered when exe-

cution is not within the dynamic extent and a

continuation is invoked that was captured (using

call-with-current-continuation) during the dy-

namic extent.

• It is exited when the called procedure returns.

• It is also exited when execution is within the dynamic

extent and a continuation is invoked that was captured

while not within the dynamic extent.

If a second call to dynamic-wind occurs within the dynamic

extent of the call to thunk and then a continuation is in-

voked in such a way that the after s from these two invoca-

tions of dynamic-wind are both to be called, then the after

associated with the second (inner) call to dynamic-wind is

called ﬁrst.

If a second call to dynamic-wind occurs within the dy-

namic extent of the call to thunk and then a continua-

tion is invoked in such a way that the befores from these

two invocations of dynamic-wind are both to be called,

then the before associated with the ﬁrst (outer) call to

dynamic-wind is called ﬁrst.

If invoking a continuation requires calling the before from

one call to dynamic-wind and the after from another, then

the after is called ﬁrst.

The eﬀect of using a captured continuation to enter or exit

the dynamic extent of a call to before or after is undeﬁned.

(let ((path ’())

(c #f))

(let ((add (lambda (s)

(set! path (cons s path)))))

(dynamic-wind

(lambda () (add ’connect))

(lambda ()

(add (call-with-current-continuation

(lambda (c0)

(set! c c0)

’talk1))))

(lambda () (add ’disconnect)))

(if (< (length path) 4)

(c ’talk2)

(reverse path))))

=⇒ (connect talk1 disconnect

connect talk2 disconnect)

6. Standard procedures 35

6.5. Eval

(eval expression environment-speciﬁer ) procedure

Evaluates expression in the speciﬁed environment and re-

turns its value. Expression must be a valid Scheme expres-

sion represented as data, and environment-speciﬁer must

be a value returned by one of the three procedures de-

scribed below. Implementations may extend eval to allow

non-expression programs (deﬁnitions) as the ﬁrst argument

and to allow other values as environments, with the re-

striction that eval is not allowed to create new bindings

in the environments associated with null-environment or

scheme-report-environment.

(eval ’(* 7 3) (scheme-report-environment 5))

=⇒ 21

(let ((f (eval ’(lambda (f x) (f x x))

(null-environment 5))))

(f + 10))

=⇒ 20

(scheme-report-environment version) procedure

(null-environment version) procedure

Version must be the exact integer 5, corresponding to this

revision of the Scheme report (the Revised

Report on

Scheme). Scheme-report-environment returns a speciﬁer

for an environment that is empty except for all bindings de-

ﬁned in this report that are either required or both optional

and supported by the implementation. Null-environment

returns a speciﬁer for an environment that is empty except

for the (syntactic) bindings for all syntactic keywords de-

ﬁned in this report that are either required or both optional

and supported by the implementation.

Other values of version can be used to specify environments

matching past revisions of this report, but their support is

not required. An implementation will signal an error if

version is neither 5 nor another value supported by the

implementation.

The eﬀect of assigning (through the use of eval) a vari-

able bound in a scheme-report-environment (for exam-

ple car) is unspeciﬁed. Thus the environments speciﬁed

by scheme-report-environment may be immutable.

(interaction-environment) optional procedure

This procedure returns a speciﬁer for the environment that

contains implementation-deﬁned bindings, typically a su-

perset of those listed in the report. The intent is that this

procedure will return the environment in which the imple-

mentation would evaluate expressions dynamically typed

by the user.

6.6. Input and output

6.6.1. Ports

Ports represent input and output devices. To Scheme, an

input port is a Scheme object that can deliver characters

upon command, while an output port is a Scheme object

that can accept characters.

(call-with-input-file string proc) library procedure

(call-with-output-file string proc) library procedure

String should be a string naming a ﬁle, and proc

should be a procedure that accepts one argument. For

call-with-input-file, the ﬁle should already exist; for

call-with-output-file, the eﬀect is unspeciﬁed if the

ﬁle already exists. These procedures call proc with one ar-

gument: the port obtained by opening the named ﬁle for

input or output. If the ﬁle cannot be opened, an error is

signalled. If proc returns, then the port is closed automati-

cally and the value(s) yielded by the proc is(are) returned.

If proc does not return, then the port will not be closed

automatically unless it is possible to prove that the port

will never again be used for a read or write operation.

Rationale: Because Scheme’s escape procedures have un-

limited extent, it is possible to escape from the current con-

tinuation but later to escape back in. If implementations

were permitted to close the port on any escape from the

current continuation, then it would be impossible to write

portable code using both call-with-current-continuation

and call-with-input-file or call-with-output-file.

(input-port? obj ) procedure

(output-port? obj ) procedure

Returns #t if obj is an input port or output port respec-

tively, otherwise returns #f.

(current-input-port) procedure

(current-output-port) procedure

Returns the current default input or output port.

(with-input-from-file string thunk)

optional procedure

(with-output-to-file string thunk)

optional procedure

String should be a string naming a ﬁle, and proc should be

a procedure of no arguments. For with-input-from-file,

the ﬁle should already exist; for with-output-to-file,

the eﬀect is unspeciﬁed if the ﬁle already exists. The

ﬁle is opened for input or output, an input or output

port connected to it is made the default value returned

by current-input-port or current-output-port (and is

36 Revised

Scheme

used by (read), (write obj ), and so forth), and the thunk

is called with no arguments. When the thunk returns,

the port is closed and the previous default is restored.

With-input-from-file and with-output-to-file re-

turn(s) the value(s) yielded by thunk . If an escape pro-

cedure is used to escape from the continuation of these

procedures, their behavior is implementation dependent.

(open-input-file ﬁlename) procedure

Takes a string naming an existing ﬁle and returns an input

port capable of delivering characters from the ﬁle. If the

ﬁle cannot be opened, an error is signalled.

(open-output-file ﬁlename) procedure

Takes a string naming an output ﬁle to be created and

returns an output port capable of writing characters to a

new ﬁle by that name. If the ﬁle cannot be opened, an

error is signalled. If a ﬁle with the given name already

exists, the eﬀect is unspeciﬁed.

(close-input-port port) procedure

(close-output-port port) procedure

Closes the ﬁle associated with port, rendering the port in-

capable of delivering or accepting characters. These rou-

tines have no eﬀect if the ﬁle has already been closed. The

value returned is unspeciﬁed.

6.6.2. Input

(read) library procedure

(read port) library procedure

Read converts external representations of Scheme objects

into the objects themselves. That is, it is a parser for the

nonterminal hdatumi (see sections 7.1.2 and 6.3.2). Read

returns the next object parsable from the given input port,

updating port to point to the ﬁrst character past the end

of the external representation of the object.

If an end of ﬁle is encountered in the input before any char-

acters are found that can begin an object, then an end of

ﬁle object is returned. The port remains open, and fur-

ther attempts to read will also return an end of ﬁle object.

If an end of ﬁle is encountered after the beginning of an

object’s external representation, but the external represen-

tation is incomplete and therefore not parsable, an error is

signalled.

The port argument may be omitted, in which case it de-

faults to the value returned by current-input-port. It is

an error to read from a closed port.

(read-char) procedure

(read-char port) procedure

Returns the next character available from the input port,

updating the port to point to the following character. If

no more characters are available, an end of ﬁle object is

returned. Port may be omitted, in which case it defaults

to the value returned by current-input-port.

(peek-char) procedure

(peek-char port) procedure

Returns the next character available from the input port,

without updating the port to point to the following char-

acter. If no more characters are available, an end of ﬁle

object is returned. Port may be omitted, in which case it

defaults to the value returned by current-input-port.

Note: The value returned by a call to peek-char is the same as

the value that would have been returned by a call to read-char

with the same port. The only diﬀerence is that the very next call

to read-char or peek-char on that port will return the value

returned by the preceding call to peek-char. In particular, a

call to peek-char on an interactive port will hang waiting for

input whenever a call to read-char would have hung.

(eof-object? obj ) procedure

Returns #t if obj is an end of ﬁle object, otherwise returns

#f. The precise set of end of ﬁle objects will vary among

implementations, but in any case no end of ﬁle object will

ever be an object that can be read in using read.

(char-ready?) procedure

(char-ready? port) procedure

Returns #t if a character is ready on the input port and

returns #f otherwise. If char-ready returns #t then the

next read-char operation on the given port is guaranteed

not to hang. If the port is at end of ﬁle then char-ready?

returns #t. Port may be omitted, in which case it defaults

to the value returned by current-input-port.

Rationale: Char-ready? exists to make it possible for a pro-

gram to accept characters from interactive ports without getting

stuck waiting for input. Any input editors associated with such

ports must ensure that characters whose existence has been as-

serted by char-ready? cannot be rubbed out. If char-ready?

were to return #f at end of ﬁle, a port at end of ﬁle would

be indistinguishable from an interactive port that has no ready

characters.

6.6.3. Output

6. Standard procedures 37

(write obj ) library procedure

(write obj port) library procedure

Writes a written representation of obj to the given port.

Strings that appear in the written representation are en-

closed in doublequotes, and within those strings backslash

and doublequote characters are escaped by backslashes.

Character objects are written using the #\ notation. Write

returns an unspeciﬁed value. The port argument may be

omitted, in which case it defaults to the value returned by

current-output-port.

(display obj ) library procedure

(display obj port) library procedure

Writes a representation of obj to the given port. Strings

that appear in the written representation are not enclosed

in doublequotes, and no characters are escaped within

those strings. Character objects appear in the represen-

tation as if written by write-char instead of by write.

Display returns an unspeciﬁed value. The port argument

may be omitted, in which case it defaults to the value re-

turned by current-output-port.

Rationale: Write is intended for producing machine-readable

output and display is for producing human-readable output.

Implementations that allow “slashiﬁcation” within symbols will

probably want write but not display to slashify funny charac-

ters in symbols.

(newline) library procedure

(newline port) library procedure

Writes an end of line to port. Exactly how this is done

diﬀers from one operating system to another. Returns

an unspeciﬁed value. The port argument may be omit-

ted, in which case it defaults to the value returned by

current-output-port.

(write-char char) procedure

(write-char char port) procedure

Writes the character char (not an external representa-

tion of the character) to the given port and returns an

unspeciﬁed value. The port argument may be omit-

ted, in which case it defaults to the value returned by

current-output-port.

6.6.4. System interface

Questions of system interface generally fall outside of the

domain of this report. However, the following operations

are important enough to deserve description here.

(load ﬁlename) optional procedure

Filename should be a string naming an existing ﬁle con-

taining Scheme source code. The load procedure reads ex-

pressions and deﬁnitions from the ﬁle and evaluates them

sequentially. It is unspeciﬁed whether the results of the

expressions are printed. The load procedure does not

aﬀect the values returned by current-input-port and

current-output-port. Load returns an unspeciﬁed value.

Rationale: For portability, load must operate on source ﬁles.

Its operation on other kinds of ﬁles necessarily varies among

implementations.

(transcript-on ﬁlename) optional procedure

(transcript-off) optional procedure

Filename must be a string naming an output ﬁle to be cre-

ated. The eﬀect of transcript-on is to open the named

ﬁle for output, and to cause a transcript of subsequent

interaction between the user and the Scheme system to

be written to the ﬁle. The transcript is ended by a call

to transcript-off, which closes the transcript ﬁle. Only

one transcript may be in progress at any time, though some

implementations may relax this restriction. The values re-

turned by these procedures are unspeciﬁed.

38 Revised

Scheme

7. Formal syntax and semantics

This chapter provides formal descriptions of what has al-

ready been described informally in previous chapters of this

report.

7.1. Formal syntax

This section provides a formal syntax for Scheme written

in an extended BNF.

All spaces in the grammar are for legibility. Case is insignif-

icant; for example, #x1A and #X1a are equivalent. hemptyi

stands for the empty string.

The following extensions to BNF are used to make the de-

scription more concise: hthingi* means zero or more occur-

rences of hthingi; and hthingi

means at least one hthingi.

7.1.1. Lexical structure

This section describes how individual tokens (identiﬁers,

numbers, etc.) are formed from sequences of characters.

The following sections describe how expressions and pro-

grams are formed from sequences of tokens.

hIntertoken spacei may occur on either side of any token,

but not within a token.

Tokens which require implicit termination (identiﬁers,

numbers, characters, and dot) may be terminated by any

hdelimiteri, but not necessarily by anything else.

The following ﬁve characters are reserved for future exten-

sions to the language: [ ] { } |

htokeni −→ hidentiﬁeri | hbooleani | hnumberi

| hcharacteri | hstringi

| ( | ) | #( | ’ | ` | , | ,@ | .

hdelimiteri −→ hwhitespacei | ( | ) | " | ;

hwhitespacei −→ hspace or newlinei

hcommenti −→ ; hall subsequent characters up to a

line breaki

hatmospherei −→ hwhitespacei | hcommenti

hintertoken spacei −→ hatmospherei*

hidentiﬁeri −→ hinitiali hsubsequenti*

| hpeculiar identiﬁeri

hinitiali −→ hletteri | hspecial initiali

hletteri −→ a | b | c | ... | z

hspecial initiali −→ ! | $ | % | & | * | / | : | < | =

| > | ? | ^ | _ | ~

hsubsequenti −→ hinitiali | hdigiti

| hspecial subsequenti

hdigiti −→ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

hspecial subsequenti −→ + | - | . | @

hpeculiar identiﬁeri −→ + | - | ...

hsyntactic keywordi −→ hexpression keywordi

| else | => | define

| unquote | unquote-splicing

hexpression keywordi −→ quote | lambda | if

| set! | begin | cond | and | or | case

| let | let* | letrec | do | delay

| quasiquote

hvariablei −→ hany hidentiﬁeri that isn’t

also a hsyntactic keywordii

hbooleani −→ #t | #f

hcharacteri −→ #\ hany characteri

| #\ hcharacter namei

hcharacter namei −→ space | newline

hstringi −→ " hstring elementi* "

hstring elementi −→ hany character other than " or \i

| \" | \\

hnumberi −→ hnum 2i| hnum 8i

| hnum 10i| hnum 16i

The following rules for hnum Ri, hcomplex Ri, hreal Ri,

hureal Ri, huinteger Ri, and hpreﬁx Ri should be repli-

cated for R = 2, 8, 10, and 16. There are no rules for

hdecimal 2i, hdecimal 8i, and hdecimal 16i, which means

that numbers containing decimal points or exponents must

be in decimal radix.

hnum Ri −→ hpreﬁx Ri hcomplex Ri

hcomplex Ri −→ hreal Ri | hreal Ri @ hreal Ri

| hreal Ri + hureal Ri i | hreal Ri - hureal Ri i

| hreal Ri + i | hreal Ri - i

| + hureal Ri i | - hureal Ri i | + i | - i

hreal Ri −→ hsigni hureal Ri

hureal Ri −→ huinteger Ri

| huinteger Ri / huinteger Ri

| hdecimal Ri

hdecimal 10i −→ huinteger 10i hsuﬃxi

| . hdigit 10i

#* hsuﬃxi

| hdigit 10i

. hdigit 10i* #* hsuﬃxi

| hdigit 10i

. #* hsuﬃxi

huinteger Ri −→ hdigit Ri

hpreﬁx Ri −→ hradix Ri hexactnessi

| hexactnessi hradix Ri

hsuﬃxi −→ hemptyi

| hexponent markeri hsigni hdigit 10i

hexponent markeri −→ e | s | f | d | l

hsigni −→ hemptyi | + | -

hexactnessi −→ hemptyi | #i | #e

hradix 2i −→ #b

hradix 8i −→ #o

hradix 10i −→ hemptyi | #d

7. Formal syntax and semantics 39

hradix 16i −→ #x

hdigit 2i −→ 0 | 1

hdigit 8i −→ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7

hdigit 10i −→ hdigiti

hdigit 16i −→ hdigit 10i | a | b | c | d | e | f

7.1.2. External representations

hDatumi is what the read procedure (section 6.6.2) suc-

cessfully parses. Note that any string that parses as an

hexpressioni will also parse as a hdatumi.

hdatumi −→ hsimple datumi | hcompound datumi

hsimple datumi −→ hbooleani | hnumberi

| hcharacteri | hstringi | hsymboli

hsymboli −→ hidentiﬁeri

hcompound datumi −→ hlisti | hvectori

hlisti −→ (hdatumi*) | (hdatumi

. hdatumi)

| habbreviationi

habbreviationi −→ habbrev preﬁxi hdatumi

habbrev preﬁxi −→ ’ | ` | , | ,@

hvectori −→ #(hdatumi*)

7.1.3. Expressions

hexpressioni −→ hvariablei

| hliterali

| hprocedure calli

| hlambda expressioni

| hconditionali

| hassignmenti

| hderived expressioni

| hmacro usei

| hmacro blocki

hliterali −→ hquotationi | hself-evaluatingi

hself-evaluatingi −→ hbooleani | hnumberi

| hcharacteri | hstringi

hquotationi −→ ’hdatumi | (quote hdatumi)

hprocedure calli −→ (hoperatori hoperandi*)

hoperatori −→ hexpressioni

hoperandi −→ hexpressioni

hlambda expressioni −→ (lambda hformalsi hbodyi)

hformalsi −→ (hvariablei*) | hvariablei

| (hvariablei

. hvariablei)

hbodyi −→ hdeﬁnitioni* hsequencei

hsequencei −→ hcommandi* hexpressioni

hcommandi −→ hexpressioni

hconditionali −→ (if htesti hconsequenti halternatei)

htesti −→ hexpressioni

hconsequenti −→ hexpressioni

halternatei −→ hexpressioni | hemptyi

hassignmenti −→ (set! hvariablei hexpressioni)

hderived expressioni −→

(cond hcond clausei

)

| (cond hcond clausei* (else hsequencei))

| (case hexpressioni

hcase clausei

)

| (case hexpressioni

hcase clausei*

(else hsequencei))

| (and htesti*)

| (or htesti*)

| (let (hbinding speci*) hbodyi)

| (let hvariablei (hbinding speci*) hbodyi)

| (let* (hbinding speci*) hbodyi)

| (letrec (hbinding speci*) hbodyi)

| (begin hsequencei)

| (do (hiteration speci*)

(htesti hdo resulti)

hcommandi*)

| (delay hexpressioni)

| hquasiquotationi

hcond clausei −→ (htesti hsequencei)

| (htesti)

| (htesti => hrecipienti)

hrecipienti −→ hexpressioni

hcase clausei −→ ((hdatumi*) hsequencei)

hbinding speci −→ (hvariablei hexpressioni)

hiteration speci −→ (hvariablei hiniti hstepi)

| (hvariablei hiniti)

hiniti −→ hexpressioni

hstepi −→ hexpressioni

hdo resulti −→ hsequencei | hemptyi

hmacro usei −→ (hkeywordi hdatumi*)

hkeywordi −→ hidentiﬁeri

hmacro blocki −→

(let-syntax (hsyntax speci*) hbodyi)

| (letrec-syntax (hsyntax speci*) hbodyi)

hsyntax speci −→ (hkeywordi htransformer speci)

7.1.4. Quasiquotations

The following grammar for quasiquote expressions is not

context-free. It is presented as a recipe for generating an

inﬁnite number of production rules. Imagine a copy of the

following rules for D = 1, 2, 3, . . .. D keeps track of the

nesting depth.

hquasiquotationi −→ hquasiquotation 1i

hqq template 0i −→ hexpressioni

40 Revised

Scheme

hquasiquotation Di −→ `hqq template Di

| (quasiquote hqq template Di)

hqq template Di −→ hsimple datumi

| hlist qq template Di

| hvector qq template Di

| hunquotation Di

hlist qq template Di −→ (hqq template or splice Di*)

| (hqq template or splice Di

. hqq template Di)

| ’hqq template Di

| hquasiquotation D + 1i

hvector qq template Di −→ #(hqq template or splice Di*)

hunquotation Di −→ ,hqq template D − 1i

| (unquote hqq template D − 1i)

hqq template or splice Di −→ hqq template Di

| hsplicing unquotation Di

hsplicing unquotation Di −→ ,@hqq template D − 1i

| (unquote-splicing hqq template D − 1i)

In hquasiquotationis, a hlist qq template Di can some-

times be confused with either an hunquotation Di or

a hsplicing unquotation Di. The interpretation as an

hunquotationi or hsplicing unquotation Di takes prece-

dence.

7.1.5. Transformers

htransformer speci −→

(syntax-rules (hidentiﬁeri*) hsyntax rulei*)

hsyntax rulei −→ (hpatterni htemplatei)

hpatterni −→ hpattern identiﬁeri

| (hpatterni*)

| (hpatterni

. hpatterni)

| (hpatterni* hpatterni hellipsisi)

| #(hpatterni*)

| #(hpatterni* hpatterni hellipsisi)

| hpattern datumi

hpattern datumi −→ hstringi

| hcharacteri

| hbooleani

| hnumberi

htemplatei −→ hpattern identiﬁeri

| (htemplate elementi*)

| (htemplate elementi

. htemplatei)

| #(htemplate elementi*)

| htemplate datumi

htemplate elementi −→ htemplatei

| htemplatei hellipsisi

htemplate datumi −→ hpattern datumi

hpattern identiﬁeri −→ hany identiﬁer except ...i

hellipsisi −→ hthe identiﬁer ...i

7.1.6. Programs and deﬁnitions

hprogrami −→ hcommand or deﬁnitioni*

hcommand or deﬁnitioni −→ hcommandi

| hdeﬁnitioni

| hsyntax deﬁnitioni

| (begin hcommand or deﬁnitioni

)

hdeﬁnitioni −→ (define hvariablei hexpressioni)

| (define (hvariablei hdef formalsi) hbodyi)

| (begin hdeﬁnitioni*)

hdef formalsi −→ hvariablei*

| hvariablei* . hvariablei

hsyntax deﬁnitioni −→

(define-syntax hkeywordi htransformer speci)

7.2. Formal semantics

This section provides a formal denotational semantics for

the primitive expressions of Scheme and selected built-in

procedures. The concepts and notation used here are de-

scribed in [29]; the notation is summarized below:

h . . . i sequence formation

s ↓ k kth member of the sequence s (1-based)

#s length of sequence s

s § t concatenation of sequences s and t

s † k drop the ﬁrst k members of sequence s

t → a, b McCarthy conditional “if t then a else b”

ρ[x/i] substitution “ρ with x for i”

x in D injection of x into domain D

x | D projection of x to domain D

The reason that expression continuations take sequences

of values instead of single values is to simplify the formal

treatment of procedure calls and multiple return values.

The boolean ﬂag associated with pairs, vectors, and strings

will be true for mutable objects and false for immutable

objects.

The order of evaluation within a call is unspeciﬁed. We

mimic that here by applying arbitrary permutations per-

mute and unpermute, which must be inverses, to the argu-

ments in a call before and after they are evaluated. This is

not quite right since it suggests, incorrectly, that the order

of evaluation is constant throughout a program (for any

given number of arguments), but it is a closer approxima-

tion to the intended semantics than a left-to-right evalua-

tion would be.

The storage allocator new is implementation-dependent,

but it must obey the following axiom: if new σ

∈

L, then

σ (new σ | L) ↓ 2 = false.

The deﬁnition of K is omitted because an accurate deﬁni-

tion of K would complicate the semantics without being

very interesting.

If P is a program in which all variables are deﬁned before

being referenced or assigned, then the meaning of P is

E[[((lambda (I*) P’) hundeﬁnedi . . . )]]

7. Formal syntax and semantics 41

where I* is the sequence of variables deﬁned in P, P

is the

sequence of expressions obtained by replacing every deﬁni-

tion in P by an assignment, hundeﬁnedi is an expression

that evaluates to undeﬁned, and E is the semantic function

that assigns meaning to expressions.

7.2.1. Abstract syntax

∈

Con constants, including quotations

∈

Ide identiﬁers (variables)

∈

Exp expressions

∈

Com = Exp commands

Exp −→ K | I | (E

E*)

| (lambda (I*) Γ* E

)

| (lambda (I* . I) Γ* E

)

| (lambda I Γ* E

)

| (if E

) | (if E

)

| (set! I E)

7.2.2. Domain equations

∈

L locations

∈

N natural numbers

T = {false, true} booleans

Q symbols

H characters

R numbers

= L × L × T pairs

= L* × T vectors

= L* × T strings

M = {false, true, null, undeﬁned, unspeciﬁed}

miscellaneous

∈

F = L × (E* → K → C) procedure values



∈

E = Q + H + R + E

+ E

+ M + F

expressed values

∈

S = L → (E × T) stores

∈

U = Ide → L environments

∈

C = S → A command continuations

∈

K = E* → C expression continuations

A answers

X errors

7.2.3. Semantic functions

K : Con → E

E : Exp → U → K → C

E* : Exp* → U → K → C

C : Com* → U → C → C

Deﬁnition of K deliberately omitted.

E[[K]] = λρκ . send (K[[K]]) κ

E[[I]] = λρκ . hold (lookup ρ I)

(single(λ .  = undeﬁned →

wrong “undeﬁned variable”,

send  κ))

E[[(E

E*)]] =

λρκ . E*(permute(hE

i § E*))

(λ* . ((λ* . applicate (* ↓ 1) (* † 1) κ)

(unpermute *)))

E[[(lambda (I*) Γ* E

)]] =

λρκ . λσ .

new σ

∈

L →

send (hnew σ | L,

λ*κ

. #* = #I* →

tievals(λα* . (λρ

. C[[Γ*]]ρ

(E[[E

]]ρ

))

(extends ρ I* α*))

*,

wrong “wrong number of arguments”i

in E)

(update (new σ | L) unspeciﬁed σ),

wrong “out of memory” σ

E[[(lambda (I* . I) Γ* E

)]] =

λρκ . λσ .

new σ

∈

L →

send (hnew σ | L,

λ*κ

. #* ≥ #I* →

tievalsrest

(λα* . (λρ

. C[[Γ*]]ρ

(E[[E

]]ρ

))

(extends ρ (I* § hIi) α*))

*

(#I*),

wrong “too few arguments”i in E)

(update (new σ | L) unspeciﬁed σ),

wrong “out of memory” σ

E[[(lambda I Γ* E

)]] = E[[(lambda (. I) Γ* E

)]]

E[[(if E

)]] =

λρκ . E[[E

]] ρ (single (λ . truish  → E[[E

]]ρκ,

E[[E

]]ρκ))

E[[(if E

)]] =

λρκ . E[[E

]] ρ (single (λ . truish  → E[[E

]]ρκ,

send unspeciﬁed κ))

Here and elsewhere, any expressed value other than undeﬁned

may be used in place of unspeciﬁed.

E[[(set! I E)]] =

λρκ . E[[E]] ρ (single(λ . assign (lookup ρ I)



(send unspeciﬁed κ)))

E*[[ ]] = λρκ . κh i

E*[[E

E*]] =

λρκ . E[[E

]] ρ (single(λ

. E*[[E*]] ρ (λ* . κ (h

i § *))))

C[[ ]] = λρθ . θ

C[[Γ

Γ*]] = λρθ . E[[Γ

]] ρ (λ* . C[[Γ*]]ρθ)

42 Revised

Scheme

7.2.4. Auxiliary functions

lookup : U → Ide → L

lookup = λρI . ρI

extends : U → Ide* → L* → U

extends =

λρI*α* . #I* = 0 → ρ,

extends (ρ[(α* ↓ 1)/(I* ↓ 1)]) (I* † 1) (α* † 1)

wrong : X → C [implementation-dependent]

send : E → K → C

send = λκ . κhi

single : (E → C) → K

single =

λψ* . #* = 1 → ψ(* ↓ 1),

wrong “wrong number of return values”

new : S → (L + {error}) [implementation-dependent]

hold : L → K → C

hold = λακσ . send (σα ↓ 1)κσ

assign : L → E → C → C

assign = λαθσ . θ(update ασ)

update : L → E → S → S

update = λασ . σ[h, truei/α]

tievals : (L* → C) → E* → C

tievals =

λψ*σ . #* = 0 → ψh iσ,

new σ

∈

L → tievals (λα* . ψ(hnew σ | Li § α*))

(* † 1)

(update(new σ | L)(* ↓ 1)σ),

wrong “out of memory”σ

tievalsrest : (L* → C) → E* → N → C

tievalsrest =

λψ*ν . list (dropﬁrst *ν)

(single(λ . tievals ψ ((takeﬁrst *ν) § hi)))

dropﬁrst = λln . n = 0 → l, dropﬁrst (l † 1)(n − 1)

takeﬁrst = λln . n = 0 → h i, hl ↓ 1i § (takeﬁrst (l † 1)(n − 1))

truish : E → T

truish = λ .  = false → false, true

permute : Exp* → Exp* [implementation-dependent]

unpermute : E* → E* [inverse of permute]

applicate : E → E* → K → C

applicate =

λ*κ . 

∈

F → ( | F ↓ 2)*κ, wrong “bad procedure”

onearg : (E → K → C) → (E* → K → C)

onearg =

λζ*κ . #* = 1 → ζ(* ↓ 1)κ,

wrong “wrong number of arguments”

twoarg : (E → E → K → C) → (E* → K → C)

twoarg =

λζ*κ . #* = 2 → ζ(* ↓ 1)(* ↓ 2)κ,

wrong “wrong number of arguments”

list : E* → K → C

list =

λ*κ . #* = 0 → send null κ,

list (* † 1)(single(λ . consh* ↓ 1, iκ))

cons : E* → K → C

cons =

twoarg (λ



κσ . new σ

∈

L →

(λσ

. new σ

∈

L →

send (hnew σ | L, new σ

| L, truei

in E)

(update(new σ

| L)

wrong “out of memory”σ

)

(update(new σ | L)

σ),

wrong “out of memory”σ)

less : E* → K → C

less =

twoarg (λ



κ . (

∈

R ∧ 

∈

R) →

send (

| R < 

| R → true, false)κ,

wrong “non-numeric argument to <”)

add : E* → K → C

add =

twoarg (λ



κ . (

∈

R ∧ 

∈

R) →

send ((

| R + 

| R) in E)κ,

wrong “non-numeric argument to +”)

car : E* → K → C

car =

onearg (λκ . 

∈

→ hold ( | E

↓ 1)κ,

wrong “non-pair argument to car”)

cdr : E* → K → C [similar to car]

setcar : E* → K → C

setcar =

twoarg (λ



κ . 

∈

→

(

| E

↓ 3) → assign (

| E

↓ 1)



(send unspeciﬁed κ),

wrong “immutable argument to set-car!”,

wrong “non-pair argument to set-car!”)

eqv : E* → K → C

eqv =

twoarg (λ



κ . (

∈

M ∧ 

∈

M) →

send (

| M = 

| M → true, false)κ,

(

∈

Q ∧ 

∈

Q) →

send (

| Q = 

| Q → true, false)κ,

(

∈

H ∧ 

∈

H) →

send (

| H = 

| H → true, false)κ,

(

∈

R ∧ 

∈

R) →

send (

| R = 

| R → true, false)κ,

(

∈

∧ 

∈

) →

send ((λp

. ((p

↓ 1) = (p

↓ 1)∧

↓ 2) = (p

↓ 2)) → true,

false)

(

| E

)

(

| E

))

κ,

7. Formal syntax and semantics 43

(

∈

∧ 

∈

) → . . . ,

(

∈

∧ 

∈

) → . . . ,

(

∈

F ∧ 

∈

F) →

send ((

| F ↓ 1) = (

| F ↓ 1) → true, false)

κ,

send false κ)

apply : E* → K → C

apply =

twoarg (λ



κ . 

∈

F → valueslist h

i(λ* . applicate 

*κ),

wrong “bad procedure argument to apply”)

valueslist : E* → K → C

valueslist =

onearg (λκ . 

∈

→

cdrhi

(λ* . valueslist

*

(λ* . carhi(single(λ . κ(hi § *))))),

 = null → κh i,

wrong “non-list argument to values-list”)

cwcc : E* → K → C [call-with-current-continuation]

cwcc =

onearg (λκ . 

∈

F →

(λσ . new σ

∈

L →

applicate 

hhnew σ | L, λ*κ

. κ*i in Ei

(update (new σ | L)

unspeciﬁed

σ),

wrong “out of memory” σ),

wrong “bad procedure argument”)

values : E* → K → C

values = λ*κ . κ*

cwv : E* → K → C [call-with-values]

cwv =

twoarg (λ



κ . applicate 

h i(λ* . applicate 

*))

7.3. Derived expression types

This section gives macro deﬁnitions for the derived expres-

sion types in terms of the primitive expression types (lit-

eral, variable, call, lambda, if, set!). See section 6.4 for

a possible deﬁnition of delay.

(define-syntax cond

(syntax-rules (else =>)

((cond (else result1 result2 ...))

(begin result1 result2 ...))

((cond (test => result))

(let ((temp test))

(if temp (result temp))))

((cond (test => result) clause1 clause2 ...)

(let ((temp test))

(if temp

(result temp)

(cond clause1 clause2 ...))))

((cond (test)) test)

((cond (test) clause1 clause2 ...)

(let ((temp test))

(if temp

temp

(cond clause1 clause2 ...))))

((cond (test result1 result2 ...))

(if test (begin result1 result2 ...)))

((cond (test result1 result2 ...)

clause1 clause2 ...)

(if test

(begin result1 result2 ...)

(cond clause1 clause2 ...)))))

(define-syntax case

(syntax-rules (else)

((case (key ...)

clauses ...)

(let ((atom-key (key ...)))

(case atom-key clauses ...)))

((case key

(else result1 result2 ...))

(begin result1 result2 ...))

((case key

((atoms ...) result1 result2 ...))

(if (memv key ’(atoms ...))

(begin result1 result2 ...)))

((case key

((atoms ...) result1 result2 ...)

clause clauses ...)

(if (memv key ’(atoms ...))

(begin result1 result2 ...)

(case key clause clauses ...)))))

(define-syntax and

(syntax-rules ()

((and) #t)

((and test) test)

((and test1 test2 ...)

(if test1 (and test2 ...) #f))))

(define-syntax or

(syntax-rules ()

((or) #f)

((or test) test)

((or test1 test2 ...)

(let ((x test1))

(if x x (or test2 ...))))))

(define-syntax let

(syntax-rules ()

((let ((name val) ...) body1 body2 ...)

((lambda (name ...) body1 body2 ...)

val ...))

((let tag ((name val) ...) body1 body2 ...)

((letrec ((tag (lambda (name ...)

body1 body2 ...)))

tag)

44 Revised

Scheme

val ...))))

(define-syntax let*

(syntax-rules ()

((let* () body1 body2 ...)

(let () body1 body2 ...))

((let* ((name1 val1) (name2 val2) ...)

body1 body2 ...)

(let ((name1 val1))

(let* ((name2 val2) ...)

body1 body2 ...)))))

The following letrec macro uses the symbol <undefined>

in place of an expression which returns something that

when stored in a location makes it an error to try to ob-

tain the value stored in the location (no such expression is

deﬁned in Scheme). A trick is used to generate the tempo-

rary names needed to avoid specifying the order in which

the values are evaluated. This could also be accomplished

by using an auxiliary macro.

(define-syntax letrec

(syntax-rules ()

((letrec ((var1 init1) ...) body ...)

(letrec "generate temp names"

(var1 ...)

()

((var1 init1) ...)

body ...))

((letrec "generate temp names"

()

(temp1 ...)

((var1 init1) ...)

body ...)

(let ((var1 <undefined>) ...)

(let ((temp1 init1) ...)

(set! var1 temp1)

...

body ...)))

((letrec "generate temp names"

(x y ...)

(temp ...)

((var1 init1) ...)

body ...)

(letrec "generate temp names"

(y ...)

(newtemp temp ...)

((var1 init1) ...)

body ...))))

(define-syntax begin

(syntax-rules ()

((begin exp ...)

((lambda () exp ...)))))

The following alternative expansion for begin does not

make use of the ability to write more than one expression

in the body of a lambda expression. In any case, note that

these rules apply only if the body of the begin contains no

deﬁnitions.

(define-syntax begin

(syntax-rules ()

((begin exp)

exp)

((begin exp1 exp2 ...)

(let ((x exp1))

(begin exp2 ...)))))

The following deﬁnition of do uses a trick to expand the

variable clauses. As with letrec above, an auxiliary macro

would also work. The expression (if #f #f) is used to

obtain an unspeciﬁc value.

(define-syntax do

(syntax-rules ()

((do ((var init step ...) ...)

(test expr ...)

command ...)

(letrec

((loop

(lambda (var ...)

(if test

(begin

(if #f #f)

expr ...)

(begin

command

...

(loop (do "step" var step ...)

...))))))

(loop init ...)))

((do "step" x)

((do "step" x y)

y)))

Notes 45

NOTES

Language changes

This section enumerates the changes that have been made

to Scheme since the “Revised

report” [6] was published.

• The report is now a superset of the IEEE standard

for Scheme [13]: implementations that conform to the

report will also conform to the standard. This required

the following changes:

– The empty list is now required to count as true.

– The classiﬁcation of features as essential or

inessential has been removed. There are now

three classes of built-in procedures: primitive, li-

brary, and optional. The optional procedures are

load, with-input-from-file, with-output-

to-file, transcript-on, transcript-off, and

interaction-environment, and - and / with

more than two arguments. None of these are in

the IEEE standard.

– Programs are allowed to redeﬁne built-in proce-

dures. Doing so will not change the behavior of

other built-in procedures.

• Port has been added to the list of disjoint types.

• The macro appendix has been removed. High-level

macros are now part of the main body of the report.

The rewrite rules for derived expressions have been

replaced with macro deﬁnitions. There are no reserved

identiﬁers.

• Syntax-rules now allows vector patterns.

• Multiple-value returns, eval, and dynamic-wind have

been added.

• The calls that are required to be implemented in a

properly tail-recursive fashion are deﬁned explicitly.

• ‘@’ can be used within identiﬁers. ‘|’ is reserved for

possible future extensions.

ADDITIONAL MATERIAL

The Internet Scheme Repository at

http://www.cs.indiana.edu/scheme-repository/

contains an extensive Scheme bibliography, as well as pa-

pers, programs, implementations, and other material re-

lated to Scheme.

EXAMPLE

Integrate-system integrates the system

= f

, y

, . . . , y

), k = 1, . . . , n

of diﬀerential equations with the method of Runge-Kutta.

The parameter system-derivative is a function that

takes a system state (a vector of values for the state vari-

ables y

, . . . , y

) and produces a system derivative (the val-

ues y

, . . . , y

). The parameter initial-state provides

an initial system state, and h is an initial guess for the

length of the integration step.

The value returned by integrate-system is an inﬁnite

stream of system states.

(define integrate-system

(lambda (system-derivative initial-state h)

(let ((next (runge-kutta-4 system-derivative h)))

(letrec ((states

(cons initial-state

(delay (map-streams next

states)))))

states))))

Runge-Kutta-4 takes a function, f, that produces a system

derivative from a system state. Runge-Kutta-4 produces

a function that takes a system state and produces a new

system state.

(define runge-kutta-4

(lambda (f h)

(let ((*h (scale-vector h))

(*2 (scale-vector 2))

(*1/2 (scale-vector (/ 1 2)))

(*1/6 (scale-vector (/ 1 6))))

(lambda (y)

;; y is a system state

(let* ((k0 (*h (f y)))

(k1 (*h (f (add-vectors y (*1/2 k0)))))

(k2 (*h (f (add-vectors y (*1/2 k1)))))

(k3 (*h (f (add-vectors y k2)))))

(add-vectors y

(*1/6 (add-vectors k0

(*2 k1)

(*2 k2)

k3))))))))

(define elementwise

(lambda (f)

(lambda vectors

(generate-vector

(vector-length (car vectors))

(lambda (i)

(apply f

(map (lambda (v) (vector-ref v i))

vectors)))))))

(define generate-vector

(lambda (size proc)

46 Revised

Scheme

(let ((ans (make-vector size)))

(letrec ((loop

(lambda (i)

(cond ((= i size) ans)

(else

(vector-set! ans i (proc i))

(loop (+ i 1)))))))

(loop 0)))))

(define add-vectors (elementwise +))

(define scale-vector

(lambda (s)

(elementwise (lambda (x) (* x s)))))

Map-streams is analogous to map: it applies its ﬁrst argu-

ment (a procedure) to all the elements of its second argu-

ment (a stream).

(define map-streams

(lambda (f s)

(cons (f (head s))

(delay (map-streams f (tail s))))))

Inﬁnite streams are implemented as pairs whose car holds

the ﬁrst element of the stream and whose cdr holds a

promise to deliver the rest of the stream.

(define head car)

(define tail

(lambda (stream) (force (cdr stream))))

The following illustrates the use of integrate-system in

integrating the system

= −i

−

= v

which models a damped oscillator.

(define damped-oscillator

(lambda (R L C)

(lambda (state)

(let ((Vc (vector-ref state 0))

(Il (vector-ref state 1)))

(vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))

(/ Vc L))))))

(define the-states

(integrate-system

(damped-oscillator 10000 1000 .001)

’#(1 0)

.01))

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48 Revised

Scheme

ALPHABETIC INDEX OF DEFINITIONS OF CONCEPTS,

KEYWORDS, AND PROCEDURES

The principal entry for each term, procedure, or keyword is

listed ﬁrst, separated from the other entries by a semicolon.

! 5

’ 8; 26

* 22

+ 22; 5, 42

, 13; 26

,@ 13

- 22; 5

-> 5

... 5; 14

/ 22

; 5

< 21; 42

<= 21

= 21; 22

=> 10

> 21

>= 21

? 4

‘ 13

abs 22; 24

acos 23

and 11; 43

angle 24

append 27

apply 32; 8, 43

asin 23

assoc 27

assq 27

assv 27

atan 23

#b 21; 38

backquote 13

begin 12; 16, 44

binding 6

binding construct 6

boolean? 25; 6

bound 6

caar 26

cadr 26

call 9

call by need 13

call-with-current-continuation 33; 8, 34, 43

call-with-input-file 35

call-with-output-file 35

call-with-values 34; 8, 43

call/cc 34

car 26; 42

case 10; 43

catch 34

cdddar 26

cddddr 26

cdr 26

ceiling 23

char->integer 29

char-alphabetic? 29

char-ci<=? 29

char-ci<? 29

char-ci=? 29

char-ci>=? 29

char-ci>? 29

char-downcase 29

char-lower-case? 29

char-numeric? 29

char-ready? 36

char-upcase 29

char-upper-case? 29

char-whitespace? 29

char<=? 29

char<? 29

char=? 29

char>=? 29

char>? 29

char? 29; 6

close-input-port 36

close-output-port 36

combination 9

comma 13

comment 5; 38

complex? 21; 19

cond 10; 15, 43

cons 26

constant 7

continuation 33

cos 23

current-input-port 35

current-output-port 35

#d 21

define 16; 14

define-syntax 17

deﬁnition 16

delay 13; 32

denominator 23

display 37

do 12; 44

dotted pair 25

dynamic-wind 34; 33

#e 21; 38

Index 49

else 10

empty list 25; 6, 26

eof-object? 36

eq? 18; 10

equal? 19

equivalence predicate 17

eqv? 17; 7, 10, 42

error 4

escape procedure 33

eval 35; 8

even? 22

exact 17

exact->inexact 24

exact? 21

exactness 19

exp 23

expt 24

#f 25

false 6; 25

floor 23

for-each 32

force 32; 13

gcd 23

hygienic 13

#i 21; 38

identiﬁer 5; 6, 28, 38

if 10; 41

imag-part 24

immutable 7

implementation restriction 4; 20

improper list 26

inexact 17

inexact->exact 24; 20

inexact? 21

initial environment 17

input-port? 35

integer->char 29

integer? 21; 19

interaction-environment 35

internal deﬁnition 16

keyword 13; 38

lambda 9; 16, 41

lazy evaluation 13

lcm 23

length 27; 20

let 11; 12, 15, 16, 43

let* 11; 16, 44

let-syntax 14; 16

letrec 11; 16, 44

letrec-syntax 14; 16

library 3

library procedure 17

list 27

list->string 30

list->vector 31

list-ref 27

list-tail 27

list? 26

load 37

location 7

log 23

macro 13

macro keyword 13

macro transformer 13

macro use 13

magnitude 24

make-polar 24

make-rectangular 24

make-string 30

make-vector 31

map 32

max 22

member 27

memq 27

memv 27

min 22

modulo 22

mutable 7

negative? 22

newline 37

nil 25

not 25

null-environment 35

null? 26

number 19

number->string 24

number? 21; 6, 19

numerator 23

numerical types 19

#o 21; 38

object 3

odd? 22

open-input-file 36

open-output-file 36

optional 3

or 11; 43

output-port? 35

pair 25

pair? 26; 6

peek-char 36

port 35

port? 6

50 Revised

Scheme

positive? 22

predicate 17

procedure call 9

procedure? 31; 6

promise 13; 32

proper tail recursion 7

quasiquote 13; 26

quote 8; 26

quotient 22

rational? 21; 19

rationalize 23

read 36; 26, 39

read-char 36

real-part 24

real? 21; 19

referentially transparent 13

region 6; 10, 11, 12

remainder 22

reverse 27

round 23

scheme-report-environment 35

set! 10; 16, 41

set-car! 26

set-cdr! 26

setcar 42

simplest rational 23

sin 23

sqrt 24

string 30

string->list 30

string->number 24

string->symbol 28

string-append 30

string-ci<=? 30

string-ci<? 30

string-ci=? 30

string-ci>=? 30

string-ci>? 30

string-copy 30

string-fill! 31

string-length 30; 20

string-ref 30

string-set! 30; 28

string<=? 30

string<? 30

string=? 30

string>=? 30

string>? 30

string? 30; 6

substring 30

symbol->string 28; 7

symbol? 28; 6

syntactic keyword 6; 5, 13, 38

syntax deﬁnition 17

syntax-rules 14; 17

#t 25

tail call 7

tan 23

token 38

top level environment 17; 6

transcript-off 37

transcript-on 37

true 6; 10, 25

truncate 23

type 6

unbound 6; 8, 16

unquote 13; 26

unquote-splicing 13; 26

unspeciﬁed 4

valid indexes 30; 31

values 34; 9

variable 6; 5, 8, 38

vector 31

vector->list 31

vector-fill! 31

vector-length 31; 20

vector-ref 31

vector-set! 31

vector? 31; 6

whitespace 5

with-input-from-file 35

with-output-to-file 35

write 37; 13

write-char 37

#x 21; 39

zero? 22