Revised Rep ort on the Algorithmic Language
Scheme
JONATHAN REES AND WILLIAM CLINGER Editors
H ABELSON R K DYBVIG C T HAYNES G J ROZAS
N I ADAMS IV D P FRIEDMAN E KOHLBECKER G J SUSSMAN
D H BARTLEY R HALSTEAD D OXLEY M WAND
G BROOKS C HANSON K M PITMAN
Dedicated to the Memory of ALGOL
CONTENTS SUMMARY
Introduction
The rep ort gives a de ning description of the program
Overview of Scheme
ming language Scheme Scheme is a statically scop ed and
Semantics
prop erly tail recursive dialect of the Lisp programming
Syntax
language invented by Guy Lewis Steele Jr and Gerald
Notation and terminology
Jay Sussman It was designed to have an exceptionally
Lexical conventions
clear and simple semantics and few di erent ways to form
Identi ers
expressions A wide variety of programming paradigms in
Whitespace and comments
cluding imp erative functional and message passing styles
Other notations
nd convenient expression in Scheme
Basic concepts
The introduction o ers a brief history of the language and
Variables and regions
of the rep ort
True and false
External representations
The rst three chapters present the fundamental ideas of
Expressions
the language and describ e the notational conventions used
Primitive expression types
for describing the language and for writing programs in the
Derived expression types
language
Program structure
Chapters and describ e the syntax and semantics of
Programs
expressions programs and de nitions
De nitions
Standard pro cedures
Chapter describ es Scheme s built in pro cedures which
Bo oleans
include all of the language s data manipulation and in
Equivalence predicates
put output primitives
Pairs and lists
Chapter provides a formal syntax for Scheme written in
Symbols
extended BNF along with a formal denotational semantics
Numbers
The rep ort concludes with an example of the use of the
Characters
language and an alphab etic index
Strings
Vectors
Control features
Input and output
Formal syntax and semantics
Formal syntax
Formal semantics
Derived expression types
Notes
Example
Bibliography and references
Alphab etic index of de nitions of concepts
keywords and pro cedures
Revised Scheme
INTRODUCTION
Programming languages should b e designed not by piling agenda for the workshop but was unable to attend the ses
feature on top of feature but by removing the weaknesses sions
and restrictions that make additional features app ear nec
Subsequent electronic mail discussions and committee work
essary Scheme demonstrates that a very small number of
completed the de nition of the language Gerry Sussman
rules for forming expressions with no restrictions on how
drafted the section on numbers Chris Hanson drafted the
they are comp osed su ce to form a practical and e cient
sections on characters and strings and Gary Bro oks and
programming language that is exible enough to supp ort
William Clinger drafted the sections on input and output
most of the ma jor programming paradigms in use to day
William Clinger recorded the decisions of the workshop and
compiled the pieces into a coherent do cument The Re
Scheme has in uenced the evolution of Lisp Scheme was
vised revised rep ort on Scheme was published at MIT
one of the rst programming languages to incorp orate rst
and Indiana University in the summer of Another
class pro cedures as in the lambda calculus thereby proving
round of revision in the spring of again accomplished
the usefulness of static scop e rules and blo ck structure in
almost entirely by electronic mail resulted in the present
a dynamically typed language Scheme was the rst ma jor
rep ort
dialect of Lisp to distinguish pro cedures from lambda ex
pressions and symbols to use a single lexical environment
We intend this rep ort to b elong to the entire Scheme com
for all variables and to evaluate the op erator p osition of
munity and so we grant p ermission to copy it in whole or in
a pro cedure call in the same way as an op erand p osition
part without fee In particular we encourage implementors
By relying entirely on pro cedure calls to express iteration
of Scheme to use this rep ort as a starting p oint for manuals
Scheme emphasized the fact that tail recursive pro cedure
and other do cumentation mo difying it as necessary
calls are essentially goto s that pass arguments Scheme
was the rst widely used programming language to em
Acknowledgements
brace rst class escap e pro cedures from which all known
sequential control structures can b e synthesized A few
We would like to thank the following p eople for their
of these innovations have recently b een incorp orated into
comments and criticisms Alan Bawden George Carrette
Common Lisp while others remain to b e adopted
Andy Cromarty Andy Freeman Richard Gabriel Yekta
Gursel Ken Haase Paul Hudak Richard Kelsey Chris
Lindblad Mark Meyer Jim Miller Jim Philbin John
Background
Ramsdell Guy Lewis Steele Jr Julie Sussman Perry Wa
The rst description of Scheme was written in A
gle Daniel Weise and Henry Wu We thank Carol Fes
revised rep ort app eared in which describ ed the
senden Daniel Friedman and Christopher Haynes for p er
evolution of the language as its MIT implementation was
mission to use text from the Scheme version reference
upgraded to supp ort an innovative compiler Three
manual We thank Texas Instruments Inc for p ermission
distinct pro jects b egan in and to use variants
to use text from the TI Scheme Language Reference Man
of Scheme for courses at MIT Yale and Indiana Univer
ual We gladly acknowledge the in uence of manuals for
sity An introductory computer science text
MIT Scheme T Scheme Common Lisp and Algol
b o ok using Scheme was published in
We also thank Betty Dexter for the extreme e ort she put
As might b e exp ected of a language used primarily for ed
into setting this rep ort in T X and Donald Knuth for de
E
ucation and research Scheme has always evolved rapidly
signing the program that caused her troubles
This was no problem when Scheme was used only within
The Arti cial Intelligence Lab oratory of the Massachusetts
MIT but as Scheme b ecame more widespread lo cal di
Institute of Technology and the Computer Science Depart
alects b egan to diverge until students and researchers o c
ment of Indiana University supp orted the preparation of
casionally found it di cult to understand co de written at
this rep ort Supp ort for the MIT work was provided in
other sites
part by the Advanced Research Pro jects Agency of the
Fifteen representatives of the ma jor implementations of
Department of Defense under O ce of Naval Research con
Scheme therefore met in Octob er to work toward
tract N C Supp ort for the Indiana Univer
a b etter and more widely accepted standard for Scheme
sity work was provided by NSF grants NCS and
Participating in this workshop were Hal Ab elson Nor
NCS
man Adams David Bartley Gary Bro oks William Clinger
Daniel Friedman Rob ert Halstead Chris Hanson Christo
pher Haynes Eugene Kohlb ecker Don Oxley Jonathan
Rees Guillermo Rozas Gerald Jay Sussman and Mitchell
Wand Kent Pitman made valuable contributions to the
Overview of Scheme
DESCRIPTION OF THE LANGUAGE
Overview of Scheme ML C and APL are three other languages that always
pass arguments by value This is distinct from the lazy
evaluation semantics of SASL or the call by name seman
Semantics
tics of Algol where an argument expression is not eval
This section gives an overview of Scheme s semantics A
uated unless its value is needed by the pro cedure
detailed informal semantics is the sub ject of chapters
through For reference purp oses section provides a
Syntax
formal semantics of Scheme
Scheme employs a parenthesized list Polish notation to de
Following Algol Scheme is a statically scop ed program
scrib e programs and other data The syntax of Scheme
ming language Each use of a variable is asso ciated with a
like that of most Lisp dialects provides for great expressive
lexically apparent binding of that variable
p ower largely due to its simplicity An imp ortant conse
Scheme has latent as opp osed to manifest types Types
quence of this simplicity is the susceptibility of Scheme
are asso ciated with values also called ob jects rather than
programs and data to uniform treatment by other Scheme
with variables Some authors refer to languages with
programs As with other Lisp dialects the read primitive
latent types as weakly typed or dynamically typed lan
parses its input that is it p erforms syntactic as well as
guages Other languages with latent types are APL
lexical decomp osition of what it reads
Snob ol and other dialects of Lisp Languages with mani
fest types sometimes referred to as strongly typed or stat
ically typed languages include Algol Pascal and C
Notation and terminology
All ob jects created in the course of a Scheme computation
Essential and non essential features
including pro cedures and continuations have unlimited ex
It is required that every implementation of Scheme supp ort
tent No Scheme ob ject is ever destroyed The reason that
features that are marked as b eing essential Features not
implementations of Scheme do not usually run out of
explicitly marked as essential are not essential Implemen
storage is that they are p ermitted to reclaim the storage
tations are free to omit non essential features of Scheme
o ccupied by an ob ject if they can prove that the ob ject
or to add extensions provided the extensions are not in
cannot p ossibly matter to any future computation Other
con ict with the language rep orted here
languages in which most ob jects have unlimited extent in
clude APL and other Lisp dialects
Error situations and unsp eci ed b ehavior
Implementations of Scheme are required to b e prop erly
When sp eaking of an error situation this rep ort uses the
tail recursive This allows the execution of an iterative
phrase an error is signalled to indicate that implemen
computation in constant space even if the iterative compu
tations must detect and rep ort the error If such wording
tation is describ ed by a syntactically recursive pro cedure
do es not app ear in the discussion of an error then imple
Thus with a tail recursive implementation iteration can b e
mentations are not required to detect or rep ort the error
expressed using the ordinary pro cedure call mechanics so
though they are encouraged to do so An error situation
that sp ecial iteration constructs are useful only as syntactic
that implementations are not required to detect is usually
sugar
referred to simply as an error
Scheme pro cedures are ob jects in their own right Pro
For example it is an error for a pro cedure to b e passed an
cedures can b e created dynamically stored in data struc
argument that the pro cedure is not explicitly sp eci ed to
tures returned as results of pro cedures and so on Other
handle even though such domain errors are seldom men
languages with these prop erties include Common Lisp and
tioned in this rep ort Implementations may extend a pro
ML
cedure s domain of de nition to include other arguments
One distinguishing feature of Scheme is that continuations
If the value of an expression is said to b e unsp eci ed
which in most other languages only op erate b ehind the
then the expression must evaluate to some ob ject without
scenes also have rst class status Continuations are
signalling an error but the value dep ends on the imple
useful for implementing a wide variety of advanced control
mentation this rep ort explicitly do es not say what value
constructs including non lo cal exits backtracking and
should b e returned
coroutines See section
Arguments to Scheme pro cedures are always passed by
Entry format
value which means that the actual argument expressions
are evaluated b efore the pro cedure gains control whether Chapters and are organized into entries Each entry de
the pro cedure needs the result of the evaluation or not scrib es one language feature or a group of related features
Revised Scheme
where a feature is either a syntactic construct or a built in Evaluation examples
pro cedure An entry b egins with one or more header lines
The symbol used in program examples should b e
of the form
read evaluates to For example
template essential category
if the feature is an essential feature or simply
means that the expression evaluates to the ob
template category
ject Or more precisely the expression given by the
sequence of characters evaluates in the initial
if the feature is not an essential feature
environment to an ob ject that may b e represented exter
If category is syntax the entry describ es an expression
nally by the sequence of characters See section
type and the header line gives the syntax of the expression
for a discussion of external representations of ob jects
type Comp onents of expressions are designated by syn
tactic variables which are written using angle brackets
Lexical conventions
for example hexpressioni hvariablei Syntactic variables
should b e understo o d to denote segments of program text
This section gives an informal account of some of the lexical
for example hexpressioni stands for any string of charac
conventions used in writing Scheme programs For a formal
ters which is a syntactically valid expression The notation
syntax of Scheme see section
hthing i
Upp er and lower case forms of a letter are never distin
guished except within character and string constants For
indicates zero or more o ccurrences of a hthingi and
example Foo is the same identi er as FOO and x AB is
hthing i hthing i
the same number as X ab
indicates one or more o ccurrences of a hthingi
Identi ers
If category is pro cedure then the entry describ es a pro
cedure and the header line gives a template for a call to the
Most identi ers allowed by other programming languages
pro cedure Argument names in the template are italicized
are also acceptable to Scheme The precise rules for form
Thus the header line
ing identi ers vary among implementations of Scheme but
in all implementations a sequence of letters digits and ex
vector ref vector k essential pro cedure
tended alphab etic characters that b egins with a character
indicates that the essential built in pro cedure vector ref
that cannot b egin a number is an identi er In addition
takes two arguments a vector vector and an exact non
and which can b egin numbers are identi ers Here are
negative integer k see b elow The header lines
some examples of identi ers
append list list essential pro cedure
lambda q
append list pro cedure
list vector soup
V a
indicate that in all implementations the append pro cedure
a kTMNs
must b e de ned to take two arguments and some imple
the word recursion has many meani ngs
mentations will extend it to take zero or more arguments
Extended alphab etic characters may b e used in identi ers
It is an error for an op eration to b e presented with an ar
exactly as if they were letters The following are extended
gument that it is not sp eci ed to handle For succinctness
alphab etic characters
we follow the convention that if an argument name is also
the name of a type then this implies a restriction on the
type of that argument to the pro cedure For example the
See section for a formal syntax of identi ers
header line for vector ref given ab ove dictates that rst
Identi ers have several uses within Scheme programs
argument to vector ref must b e a vector The following
naming conventions also imply type restrictions
Certain identi ers are reserved for use as syntactic
keywords see b elow
obj any ob ject
z z z complex real rational integer
j
Any identi er that is not a syntactic keyword may b e
x x x real rational integer
j
used as a variable see section
y y y real rational integer
j
q q q rational integer
j
When an identi er app ears as a literal or within a
n n n integer
j
literal see section it is b eing used to denote a
k k k exact non negative integer
j
symbol see section
Basic concepts
The following identi ers are syntactic keywords and delimited plus or minus sign by itself is also an identi
should not b e used as variables er A delimited p erio d not o ccurring within a num
b er or identi er is used in the notation for pairs sec
do or
tion and to indicate a rest parameter in a formal
and else quasiquote
parameter list section
begin if quote
case lambda set
Parentheses are used for grouping and to notate lists
cond let unquote
section
define let unquote splicing
delay letrec
The single quote character is used to indicate literal data
section
Some implementations allow all identi ers including syn
tactic keywords to b e used as variables This is a com
The backquote character is used to indicate almost
patible extension to the language but ambiguities in the
constant data section
language result when the restriction is relaxed and the
ways in which these ambiguities are resolved vary b etween
The character comma and the sequence comma at
implementations
sign are used in conjunction with backquote sec
tion
The characters and have no sp ecial prop erties they
are extended alphab etic characters By convention how
The double quote character is used to delimit strings
ever most predicate pro cedures those that return b o olean
section
values are named by identi ers that end in and most
data mutation pro cedures are named by identi ers that end
Backslash is used in the syntax for character constants
in
and as an escap e character within string constants
section
Whitespace and comments
Left and right square brackets and curly braces
are reserved for p ossible future extensions to the lan
Whitespace characters are spaces and newlines Imple
guage
mentations typically provide additional whitespace char
acters such as tab or page break Whitespace is used for
Sharp sign is used for a variety of purp oses dep ending
improved readability and as necessary to separate tokens
on the character that immediately follows it
from each other a token b eing an indivisible lexical unit
such as an identi er or number but is otherwise insigni
t f These are the b o olean constants section
cant Whitespace may o ccur b etween any two tokens but
not within a token Whitespace may also o ccur inside a
This introduces a character constant section
string where it is signi cant
This introduces a vector constant section Vector
A semicolon indicates the start of a comment The
constants are terminated by
comment continues to the end of the line on which the
semicolon app ears Comments are invisible to Scheme but
e i l s b o d x These are used in the nota
the end of the line is visible as whitespace This prevents a
tion for numbers section
comment from app earing in the middle of an identi er or
number
Basic concepts
The FACT procedure computes the factorial
of a non negative integer
Variables and regions
define fact
lambda n
Any identi er that is not a syntactic keyword see sec
if n
tion may b e used as a variable A variable may name
Base case return
a lo cation where a value can b e stored A variable that
n fact n
do es so is said to b e bound to the lo cation The set of all
such bindings in e ect at some p oint in a program is known
Other notations
as the environment in e ect at that p oint The value stored
in the lo cation to which a variable is b ound is called the
For a description of the notations used for numbers see
variable s value By abuse of terminology the variable is
section
sometimes said to name the value or to b e b ound to the
These are used in numbers and may also o ccur any value This is not quite accurate but confusion rarely re
where in an identi er except as the rst character A sults from this practice
Revised Scheme
Certain expression types are used to create new lo cations External representations can also b e used for input and
and to bind variables to those lo cations The most funda output The pro cedure read section parses ex
mental of these binding constructs is the lambda expres ternal representations and the pro cedure write sec
sion b ecause all other binding constructs can b e explained tion generates them Together they provide an
in terms of lambda expressions The other binding con elegant and p owerful input output facility
structs are let let letrec and do expressions see sec
Note that the sequence of characters is not an
tions and
external representation of the integer even though it is an
Like Algol and Pascal and unlike most other dialects of
expression evaluating to the integer rather it is an exter
Lisp except for Common Lisp Scheme is a statically scop ed
nal representation of a three element list the elements of
language with blo ck structure To each place where a vari
which are the symbol and the integers and Scheme s
able is b ound in a program there corresp onds a region of the
syntax has the prop erty that any sequence of characters
program text within which the binding is e ective The re
which is an expression is also the external representation
gion is determined by the particular binding construct that
of some ob ject This can lead to confusion since it may
establishes the binding if the binding is established by a
not b e obvious out of context whether a given sequence of
lambda expression for example then its region is the en
characters is intended to denote data or program but it is
tire lambda expression Every reference to or assignment
also a source of p ower since it facilitates writing programs
of a variable refers to the binding of the variable that es
such as interpreters and compilers which treat programs as
tablished the innermost of the regions containing the use
data or vice versa
If there is no binding of the variable whose region contains
The syntax of external representations of various kinds of
the use then the use refers to the binding for the variable
ob jects accompanies the description of the primitives for
in the top level environment if any section if there is
manipulating the ob jects in the appropriate sections of
no binding for the identi er it is said to b e unbound
chapter
True and false
Expressions
Any Scheme value can b e used as a b o olean value for the
purp ose of a conditional test As explained in section
A Scheme expression is a construct that returns a value
all values count as true in such a test except for f and
such as a variable reference literal pro cedure call or con
the empty list which count as false This rep ort uses the
ditional
word true to refer to any Scheme value that counts as
true and the word false to refer to any Scheme value
Expression types are categorized as primitive or derived
that counts as false
Primitive expression types include variables and pro cedure
calls Derived expression types are not semantically primi
tive but can instead b e explained in terms of the primitive
External representations
constructs as in section They are redundant in the
strict sense of the word but they capture common pat
An imp ortant concept in Scheme and Lisp is that of the
terns of usage and are therefore provided as convenient
external representation of an ob ject as a sequence of char
abbreviations
acters For example an external representation of the inte
ger is the sequence of characters and an external
representation of a list consisting of the integers and
Primitive expression types
is the sequence of characters
Variable references
The external representation of an ob ject is not necessarily
unique The integer also has representations
and x c and the list in the previous paragraph also has
hvariablei essential syntax
the representations and
An expression consisting of a variable section is a
see section
variable reference The value of the variable reference is
Many ob jects have standard external representations but
the value stored in the lo cation to which the variable is
some such as pro cedures do not have standard represen
b ound It is an error to reference an unbound variable
tations although particular implementations may de ne
representations for them
define x
x
An external representation may b e written in a program to
obtain the corresp onding ob ject see quote section
Expressions
Literal expressions Note In many dialects of Lisp the empty combination
is a legitimate expression In Scheme combinations must have
quote hdatumi essential syntax
at least one sub expression so is not a syntactically valid
hdatumi essential syntax
expression
hconstanti essential syntax
Lambda expressions
quote hdatumi evaluates to hdatumi hDatumi may b e
any external representation of a Scheme ob ject see sec
lambda hformalsi hb o dyi essential syntax
tion This notation is used to include literal constants
Syntax hFormalsi should b e a formal arguments list as
in Scheme co de
describ ed b elow and hb o dyi should b e a sequence of one
quote a a
or more expressions
quote a b c a b c
Semantics A lambda expression evaluates to a pro cedure
quote
The environment in e ect when the lambda expression was
quote hdatumi may b e abbreviated as hdatumi The
evaluated is remembered as part of the pro cedure When
two notations are equivalent in all resp ects
the pro cedure is later called with some actual arguments
the environment in which the lambda expression was evalu
a a
ated will b e extended by binding the variables in the formal
a b c a b c
argument list to fresh lo cations the corresp onding actual
quote a quote a
argument values will b e stored in those lo cations and the
a quote a
expressions in the b o dy of the lambda expression will b e
evaluated sequentially in the extended environment The
Numeric constants string constants character constants
result of the last expression in the b o dy will b e returned
and b o olean constants evaluate to themselves they need
as the result of the pro cedure call
not b e quoted
lambda x x x a procedure
abc abc
lambda x x x
abc abc
define reverse subtract
lambda x y y x
t t
reverse subtract
t t
define foo
It is an error to alter a constant i e the value of a literal
let x
expression using a mutation pro cedure like set car or
lambda y x y
string set
foo
Pro cedure calls
hFormalsi should have one of the following forms
hvariable i The pro cedure takes a xed num
hop eratori hop erand i essential syntax
b er of arguments when the pro cedure is called the
A pro cedure call is written by simply enclosing in parenthe
arguments will b e stored in the bindings of the corre
ses expressions for the pro cedure to b e called and the ar
sp onding variables
guments to b e passed to it The op erator and op erand ex
pressions are evaluated in an indeterminate order and the
hvariablei The pro cedure takes any number of ar
resulting pro cedure is passed the resulting arguments
guments when the pro cedure is called the sequence
of actual arguments is converted into a newly allo
if f
cated list and the list is stored in the binding of the
hvariablei
A number of pro cedures are available as the values of vari
ables in the initial environment for example the addition
hvariable i hvariable i hvariable i If a
n n
and multiplication pro cedures in the ab ove examples are
space delimited p erio d precedes the last variable then
the values of the variables and New pro cedures are cre
the value stored in the binding of the last variable will
ated by evaluating lambda expressions see section
b e a newly allo cated list of the actual arguments left
over after all the other actual arguments have b een
Pro cedure calls are also called combinations
matched up against the other formal arguments
Note In contrast to other dialects of Lisp the order of
lambda x x
evaluation is unsp eci ed and the op erator expression and the
lambda x y z z
op erand expressions are always evaluated with the same evalu
ation rules
Revised Scheme
hclausei contains only the htesti and no hexpressionis then Conditionals
the value of the htesti is returned as the result If all htestis
if htesti hconsequenti halternatei essential syntax
evaluate to false values and there is no else clause then
if htesti hconsequenti syntax
the result of the conditional expression is unsp eci ed if
there is an else clause then its hexpressionis are evaluated
Syntax hTesti hconsequenti and halternatei may b e arbi
and the value of the last one is returned
trary expressions
Semantics An if expression is evaluated as follows rst
cond greater
htesti is evaluated If it yields a true value see section
less greater
cond greater
then hconsequenti is evaluated and its value is returned
less
Otherwise halternatei is evaluated and its value is returned
else equal equal
If htesti yields a false value and no halternatei is sp eci ed
then the result of the expression is unsp eci ed
Some implementations supp ort an alternative hclausei syn
if yes no yes
tax htesti hrecipienti where hrecipienti is an expres
if yes no no
sion If htesti evaluates to a true value then hrecipienti is
if
evaluated Its value must b e a pro cedure of one argument
this pro cedure is then invoked on the value of the htesti
cond assv b a b cadr
Assignments
else f
set hvariablei hexpressioni essential syntax
hExpressioni is evaluated and the resulting value is stored
case hkeyi hclause i hclause i syntax
in the lo cation to which hvariablei is b ound hVariablei
Syntax hKeyi may b e any expression Each hclausei
must b e b ound either in some region enclosing the set
should have the form
expression or at top level The result of the set expression
is unsp eci ed
hdatum i hexpression i hexpression i
define x
where each hdatumi is an external representation of some
x
ob ject All the hdatumis must b e distinct The last
set x unspeci ed
hclausei may b e an else clause which has the form
x
else hexpression i hexpression i
Derived expression types
Semantics A case expression is evaluated as follows
hKeyi is evaluated and its result is compared against each
For reference purp oses section gives rewrite rules that
hdatumi If the result of evaluating hkeyi is equivalent
will convert constructs describ ed in this section into the
in the sense of eqv see section to a hdatumi then
primitive constructs describ ed in the previous section
the expressions in the corresp onding hclausei are evaluated
from left to right and the result of the last expression in
Conditionals
the hclausei is returned as the result of the case expres
sion If the result of evaluating hkeyi is di erent from every
cond hclause i hclause i essential syntax
hdatumi then if there is an else clause its expressions are
Syntax Each hclausei should b e of the form
evaluated and the result of the last is the result of the case
expression otherwise the result of the case expression is
htesti hexpressioni
unsp eci ed
where htesti is any expression The last hclausei may b e
an else clause which has the form
case
else hexpression i hexpression i
prime
composite composite
Semantics A cond expression is evaluated by evaluating
case car c d
the htesti expressions of successive hclauseis in order until
a a
one of them evaluates to a true value see section
b b unspeci ed
When a htesti evaluates to a true value then the remaining
case car c d
hexpressionis in its hclausei are evaluated in order and the
a e i o u vowel
result of the last hexpressioni in the hclausei is returned
w y semivowel
as the result of the entire cond expression If the selected else consonant consonant
Expressions
and htest i syntax let x y
x y
The htesti expressions are evaluated from left to right and
the value of the rst expression that evaluates to a false
let x y
value see section is returned Any remaining expres
let foo lambda z x y z
sions are not evaluated If all the expressions evaluate to
x
foo
true values the value of the last expression is returned If
there are no expressions then t is returned
See also named let section
and t
and f
and c f g f g
let hbindingsi hb o dyi syntax
and t
Syntax hBindingsi should have the form
hvariable i hinit i
or htest i syntax
and hb o dyi should b e a sequence of one or more expres
sions
The htesti expressions are evaluated from left to right and
the value of the rst expression that evaluates to a true
Semantics Let is similar to let but the bindings are
value see section is returned Any remaining expres
p erformed sequentially from left to right and the region of
sions are not evaluated If all expressions evaluate to false
a binding indicated by hvariablei hiniti is that part of
values the value of the last expression is returned If there
the let expression to the right of the binding Thus the
are no expressions then f is returned
second binding is done in an environment in which the rst
binding is visible and so on
or t
or t
let x y x
or f f f f
y
or memq b a b c
b c
letrec hbindingsi hb o dyi essential syntax
Binding constructs
Syntax hBindingsi should have the form
hvariable i hinit i
The three binding constructs let let and letrec give
Scheme a blo ck structure like Algol The syntax of the
and hb o dyi should b e a sequence of one or more expres
three constructs is identical but they di er in the regions
sions
they establish for their variable bindings In a let ex
pression the initial values are computed b efore any of the Semantics The hvariableis are b ound to fresh lo cations
variables b ecome b ound in a let expression the bind holding unde ned values the hinitis are evaluated in the
ings and evaluations are p erformed sequentially while in resulting environment in some unsp eci ed order each
a letrec expression the bindings are in e ect while their
hvariablei is assigned to the result of the corresp onding
initial values are b eing computed thus allowing mutually hiniti the hb o dyi is evaluated in the resulting environment
recursive de nitions and the value of the last expression in hb o dyi is returned
Each binding of a hvariablei has the entire letrec expres
sion as its region making it p ossible to de ne mutually
let hbindingsi hb o dyi essential syntax
recursive pro cedures
Syntax hBindingsi should have the form
letrec even
lambda n
hvariable i hinit i
if zero n
where each hiniti is an expression and hb o dyi should b e a
t
sequence of one or more expressions
odd n
odd
Semantics The hinitis are evaluated in the current envi
lambda n
ronment in some unsp eci ed order the hvariableis are
if zero n
b ound to fresh lo cations holding the results the hb o dyi is
f
evaluated in the extended environment and the value of
even n
the last expression of hb o dyi is returned Each binding of
even
t a hvariablei has hb o dyi as its region
Revised Scheme
One restriction on letrec is very imp ortant it must b e The region of the binding of a hvariablei consists of the
p ossible to evaluate each hiniti without referring to the entire do expression except for the hinitis
value of any hvariablei If this restriction is violated then
A hstepi may b e omitted in which case the e ect is the
the e ect is unde ned and an error may b e signalled dur
same as if hvariablei hiniti hvariablei had b een written
ing evaluation of the hinitis The restriction is necessary
instead of hvariablei hiniti
b ecause Scheme passes arguments by value rather than by
name In the most common uses of letrec all the hinitis
do vec make vector
are lambda expressions and the restriction is satis ed au
i i
tomatically
i vec
vector set vec i i
Sequencing
let x
do x x cdr x
begin hexpression i hexpression i essential syntax
sum sum car x
null x sum
The hexpressionis are evaluated sequentially from left to
right and the value of the last hexpressioni is returned
This expression type is used to sequence side e ects such
let hvariablei hbindingsi hb o dyi syntax
as input and output
Some implementations of Scheme p ermit a variant on the
begin set x
syntax of let called named let which provides a more
x
general lo oping construct than do and may also b e used
to express recursions
begin display plus equals
display unspeci ed
Named let has the same syntax and semantics as ordi
and prints plus equals
nary let except that hvariablei is b ound within hb o dyi to
a pro cedure whose formal arguments are the b ound vari
Note uses the keyword sequence instead of begin
ables and whose b o dy is hb o dyi Thus the execution of
hb o dyi may b e rep eated by invoking the pro cedure named
Iteration
by hvariablei
do hvariable i hinit i hstep i syntax
let loop numbers
nonneg
htesti hexpressioni
neg
hcommandi
cond null numbers list nonneg neg
car numbers
Do is an iteration construct It sp eci es a set of variables
loop cdr numbers
to b e b ound how they are to b e initialized at the start
cons car numbers nonneg
and how they are to b e up dated on each iteration When a
neg
termination condition is met the lo op exits with a sp eci ed
car numbers
result value
loop cdr numbers
nonneg
Do expressions are evaluated as follows The hiniti ex
cons car numbers neg
pressions are evaluated in some unsp eci ed order the
hvariableis are b ound to fresh lo cations the results of
the hiniti expressions are stored in the bindings of the
hvariableis and then the iteration phase b egins
Delayed evaluation
Each iteration b egins by evaluating htesti if the result is
false see section then the hcommandi expressions are
delay hexpressioni syntax
evaluated in order for e ect the hstepi expressions are eval
The delay construct is used together with the pro cedure
uated in some unsp eci ed order the hvariableis are b ound
force to implement lazy evaluation or call by need delay
to fresh lo cations the results of the hstepis are stored in the
hexpressioni returns an ob ject called a promise which at
bindings of the hvariableis and the next iteration b egins
some p oint in the future may b e asked by the force pro
If htesti evaluates to a true value then the hexpressionis
cedure to evaluate hexpressioni and deliver the resulting
are evaluated from left to right and the value of the last
value
hexpressioni is returned as the value of the do expression
See the description of force section for a complete
If no hexpressionis are present then the value of the do
description of delay
expression is unsp eci ed
Program structure
Program structure Quasiquotation
quasiquote htemplatei syntax
Programs
htemplatei syntax
A Scheme program consists of a sequence of expressions
Backquote or quasiquote expressions are useful for
and de nitions Expressions are describ ed in chapter
constructing a list or vector structure when most but not
de nitions are the sub ject of the rest of the present chapter
all of the desired structure is known in advance If no
Programs are typically stored in les or entered inter
commas app ear within the htemplatei the result of evalu
actively to a running Scheme system although other
ating htemplatei is equivalent to the result of evaluating
paradigms are p ossible questions of user interface lie out
htemplatei If a comma app ears within the htemplatei
side the scop e of this rep ort Indeed Scheme would still b e
however the expression following the comma is evaluated
useful as a notation for expressing computational metho ds
unquoted and its result is inserted into the structure
even in the absence of a mechanical implementation
instead of the comma and the expression If a comma ap
p ears followed immediately by an at sign then the fol
De nitions o ccurring at the top level of a program are inter
lowing expression must evaluate to a list the op ening and
preted declaratively They cause bindings to b e created in
closing parentheses of the list are then stripp ed away and
the top level environment Expressions o ccurring at the top
the elements of the list are inserted in place of the comma
level of a program are interpreted imp eratively they are
at sign expression sequence
executed in order when the program is invoked or loaded
and typically p erform some kind of initialization
list list
let name a list name name
list a quote a
De nitions
a map abs b
a b
De nitions are valid in some but not all contexts where
foo cdr c car cons
expressions are allowed They are valid only at the top
foo cons
level of a hprogrami and in some implementations at the
sqrt map sqrt
b eginning of a hb o dyi
A de nition should have one of the following forms
Quasiquote forms may b e nested Substitutions are made
only for unquoted comp onents app earing at the same nest
define hvariablei hexpressioni
ing level as the outermost backquote The nesting level in
This syntax is essential
creases by one inside each successive quasiquotation and
decreases by one inside each unquotation
define hvariablei hformalsi hb o dyi
a b foo d e f
This syntax is not essential hFormalsi should b e ei
a b foo d e f
ther a sequence of zero or more variables or a sequence
of one or more variables followed by a space delimited let name x
name y
p erio d and another variable as in a lambda expres
a b name name d e
sion This form is equivalent to
a b x y d e
define hvariablei
lambda hformalsi hb o dyi
The notations htemplatei and quasiquote htemplatei
are identical in all resp ects hexpressioni is identical to
define hvariablei hformali hb o dyi
unquote hexpressioni and hexpressioni is identical to
This syntax is not essential hFormali should b e a
unquote splicing hexpressioni The external syntax
single variable This form is equivalent to
generated by write for two element lists whose car is one
of these symbols may vary b etween implementations
define hvariablei
lambda hformali hb o dyi
quasiquote list unquote
list
Top level de nitions
quasiquote list unquote
list
At the top level of a program a de nition
i e quasiquote list unquote
define hvariablei hexpressioni
Unpredictable b ehavior can result if any of the symbols
has essentially the same e ect as the assignment expres
quasiquote unquote or unquote splicing app ear in
sion
p ositions within a htemplatei otherwise than as describ ed
ab ove
set hvariablei hexpressioni
Revised Scheme
if hvariablei is b ound If hvariablei is not b ound however Bo oleans
then the de nition will bind hvariablei to a new lo cation
The standard b o olean ob jects for true and false are written
b efore p erforming the assignment whereas it would b e an
as t and f What really matters though are the ob jects
error to p erform a set on an unbound variable
that the Scheme conditional expressions if cond and
or do treat as true or false The phrase a true value
define add
or sometimes just true means any ob ject treated as
lambda x x
add true by the conditional expressions and the phrase a false
define first car
value or false means any ob ject treated as false by the
first
conditional expressions
Of all the standard Scheme values only f and the empty
All Scheme implementations must supp ort top level de ni
list count as false in conditional expressions Everything
tions
else including t pairs symbols numbers strings vec
Some implementations of Scheme use an initial environ
tors and pro cedures counts as true
ment in which all p ossible variables are b ound to lo cations
The empty list counts as false for compatibility with exist
most of which contain unde ned values Top level de ni
ing programs and implementations that assume this to b e
tions in such an implementation are truly equivalent to
the case
assignments
Programmers accustomed to other dialects of Lisp should
b eware that Scheme distinguishes false and the empty list
Internal de nitions
from the symbol nil
Some implementations of Scheme p ermit de nitions to o c
Bo olean constants evaluate to themselves so they don t
cur at the b eginning of a hb o dyi that is the b o dy of a
need to b e quoted in programs
lambda let let letrec or define expression Such
t t
de nitions are known as internal de nitions as opp osed to
f f
the top level de nitions describ ed ab ove The variable de
f f
ned by an internal de nition is lo cal to the hb o dyi That
is hvariablei is b ound rather than assigned and the region
of the binding is the entire hb o dyi For example
not obj essential pro cedure
let x
Not returns t if obj is false and returns f otherwise
define foo lambda y bar x y
not t f
define bar lambda a b a b a
not f
foo x
not list f
not f t
A hb o dyi containing internal de nitions can always b e con
not t
verted into a completely equivalent letrec expression For
not list t
example the let expression in the ab ove example is equiv
alent to
boolean obj essential pro cedure
let x
letrec foo lambda y bar x y
Boolean returns t if obj is either t or f and returns
bar lambda a b a b a
f otherwise
foo x
boolean f t
boolean f
Standard pro cedures
nil variable
This chapter describ es Scheme s built in pro cedures The
t variable
initial or top level Scheme environment starts out with
a number of variables b ound to lo cations containing useful
Some implementations provide variables nil and t whose
values most of which are primitive pro cedures that manip
values in the initial environment are f and t resp ectively
ulate data For example the variable abs is b ound to a
t t
lo cation initially containing a pro cedure of one argument
nil f
that computes the absolute value of a number and the
nil nil
variable is b ound to a pro cedure that computes sums
Standard pro cedures
Equivalence predicates prove that obj and obj are op erationally equivalent If it
can t it always errs on the conservative side and returns
A predicate is a pro cedure that always returns a b o olean
f
value t or f An equivalence predicate is the compu
The only situation in which it might fail to prove is when
tational analogue of a mathematical equivalence relation
obj and obj are op erationally equivalent pro cedures that
it is symmetric re exive and transitive Of the equiva
were created at di erent times In general op erational
lence predicates describ ed in this section eq is the nest
equivalence of pro cedures is uncomputable but it is guar
or most discriminating and equal is the coarsest Eqv
anteed that eqv can recognize a pro cedure created at a
is slightly less discriminating than eq
given time by a given lambda expression as b eing itself
Two ob jects are operationally equivalent if and only if there
This is useful for applications in which pro cedures are b e
is no way that they can b e distinguished using Scheme
ing used to implement ob jects with lo cal state
primitives other than eqv or eq or those like memq and
eqv a a t
assv whose meaning is de ned explicitly in terms of eqv
eqv a b f
or eq It is guaranteed that ob jects maintain their op era
eqv t
tional identity despite b eing named by variables or fetched
eqv t
from or stored into data structures
eqv t
This de nition can b e interpreted in the following ways for
eqv t
various kinds of ob jects
eqv cons cons f
eqv lambda
The two b o olean values t and f are op erationally
lambda f
distinct b ecause they b ehave in opp osite ways in con
eqv f nil f
ditionals
let p lambda x x
eqv p p t
Two symbols are op erationally equivalent if they print
the same way
The following examples illustrate cases in which eqv is
p ermitted to fail to prove op erational equivalence dep end
Two numbers are op erationally equivalent if they are
ing on the implementation In every case it will return
numerically equal see section and are either
either t or f but which one it returns is implementation
b oth exact or b oth inexact see section
dep endent Compare with the last example in the previ
ous set
Two characters are op erationally equivalent if they are
the same character according to char section
eqv lambda x x
lambda x x unspeci ed
Two structured mutable ob jects pairs vectors or
eqv lambda x x
strings are op erationally equivalent if they have op
lambda y y unspeci ed
erationally equivalent values in corresp onding p osi
The next set of examples shows the use of eqv with pro
tions and applying a mutation pro cedure to one
cedures that have lo cal state Gen counter must return a
causes the other to change as well A mutation pro
distinct pro cedure every time since each pro cedure has its
cedure is one like set car which alters a data struc
own internal counter Gen loser however returns equiv
ture For example two pairs are not op erationally
alent pro cedures each time since the lo cal state do es not
equivalent if a set car op eration on one do es not
a ect the value or side e ects of the pro cedures
change the car eld of the other
define gen counter
There is only one empty list and it is op erationally
lambda
equivalent to itself All empty vectors are op era
let n
tionally equivalent to each other All empty strings
lambda set n n n
are op erationally equivalent to each other Whether
let g gen counter
there is more than one empty vector or string is
eqv g g t
implementation dependent
eqv gen counter gen counter
f
Two pro cedures are op erationally equivalent if when
define gen loser
called on op erationally equivalent arguments they re
lambda
turn the same value and p erform the same side e ects
let n
lambda set n n
let g gen loser
eqv obj obj essential pro cedure
eqv g g t
The eqv pro cedure implements an approximation to the
eqv gen loser gen loser
unspeci ed relation of op erational equivalence It returns t if it can
Revised Scheme
Note It will usually b e p ossible to implement eq much more Ob jects of distinct types are never op erationally equiva
e ciently than eqv for example as a simple p ointer compari lent except that false and the empty list are p ermitted to
son instead of as some more complicated op eration One reason b e identical and the character type need not b e disjoint
is that it may not b e p ossible to compute eqv of two numbers from other types
in constant time whereas eq implemented as p ointer compar
eqv f unspeci ed
ison will always nish in constant time Eq may b e used like
eqv A unspeci ed
eqv in applications using pro cedures to implement ob jects with
state since it ob eys the same constraints as eqv
Since it is an error to mo dify constant ob jects those re
turned by literal expressions implementations are p er
mitted though not required to share structure b etween
equal obj obj essential pro cedure
constants where appropriate Thus the value of eqv on
Equal recursively compares the contents of pairs vectors
constants is sometimes implementation dependent
and strings applying eqv on other ob jects such as num
let x a
b ers and symbols A rule of thumb is that ob jects are
eqv x x t
generally equal if they print the same Equal may fail
eqv a a unspeci ed
to terminate if its arguments are circular data structures
eqv a a unspeci ed
eqv b cdr a b unspeci ed
equal a a t
equal a a t
Note The ab ove de nition of eqv allows implementations
equal a b c
latitude in their treatment of pro cedures and literals imple
a b c t
mentations are free to either detect or fail to detect that two
equal abc abc t
pro cedures or two literals are op erationally equivalent to each
equal t
other and can decide whether or not to merge representations
equal make vector a
of equivalent ob jects by using the same p ointer or bit pattern
make vector a t
equal lambda x x
to represent b oth
lambda y y unspeci ed
eq obj obj essential pro cedure
Pairs and lists
Eq is similar to eqv except that in some cases it is capable
of discerning distinctions ner than those detectable by
A pair sometimes called a dotted pair is a record structure
eqv
with two elds called the car and cdr elds for historical
reasons Pairs are created by the pro cedure cons The
Eq and eqv are guaranteed to have the same b ehavior
car and cdr elds are accessed by the pro cedures car and
on symbols b o oleans the empty list pairs and non empty
cdr The car and cdr elds are assigned by the pro cedures
strings and vectors Eq s b ehavior on numbers and charac
set car and set cdr
ters is implementation dependent but it will always return
either true or false and will return true only when eqv
Pairs are used primarily to represent lists A list can b e
would also return true Eq may also b ehave di erently
de ned recursively as either the empty list or a pair whose
from eqv on empty vectors and empty strings
cdr is a list The ob jects in the car elds of successive
pairs of a list are the elements of the list For example
eq a a t
a two element list is a pair whose car is the rst element
eq a a unspeci ed
and whose cdr is a pair whose car is the second element
eq list a list a f
and whose cdr is the empty list The length of a list is the
eq a a unspeci ed
eq unspeci ed
number of elements which is the same as the number of
eq t
pairs
eq unspeci ed
The empty list is a sp ecial ob ject of its own type it is not
eq A A unspeci ed
a pair it has no elements and its length is zero
eq car car t
let n
The most general notation external representation for
eq n n unspeci ed
Scheme pairs is the dotted notation c c where c
let x a
is the value of the car eld and c is the value of the cdr
eq x x t
eld For example is a pair whose car is and
let x
whose cdr is Note that is the external repre
eq x x t
sentation of a pair not an expression that evaluates to a
let p lambda x x
eq p p t pair
Standard pro cedures
A more streamlined notation can b e used for lists the pair obj essential pro cedure
elements of the list are simply enclosed in parentheses and
Pair returns t if obj is a pair and otherwise returns f
separated by spaces The empty list is written For
pair a b t
example
pair a b c t
a b c d e
pair f
pair a b f
and
a b c d e
cons obj obj essential pro cedure
are b oth representations of the same list of symbols
Returns a newly allo cated pair whose car is obj and whose
A chain of pairs not ending in the empty list is called an
cdr is obj The pair is guaranteed to b e di erent in the
improper list Note that an improp er list is not a list
sense of eqv from every existing ob ject
The list and dotted notations can b e combined to represent
cons a a
improp er lists
cons a b c d a b c d
cons a b c a b c
a b c d
cons a a
is equivalent to
cons a b c a b c
a b c d
car pair essential pro cedure
Whether a given pair is a list dep ends up on what is stored
in the cdr eld When the set cdr pro cedure is used an
Returns the contents of the car eld of pair Note that it
ob ject can b e a list one moment and not the next
is an error to take the car of the empty list
car a b c a
define x list a b c
car a b c d a
define y x
car
y a b c
car error
set cdr x unspeci ed
x a
eqv x y t
cdr pair essential pro cedure
y a
Returns the contents of the cdr eld of pair Note that it
It is often convenient to sp eak of a homogeneous list of ob
is an error to take the cdr of the empty list
jects of some particular data type as for example
is a list of integers To b e more precise supp ose D is some
cdr a b c d b c d
data type Any predicate de nes a data type consisting
cdr
of those ob jects of which the predicate is true Then
cdr error
The empty list is a list of D
set car pair obj essential pro cedure
If list is a list of D then any pair whose cdr is list and
Stores obj in the car eld of pair The value returned by
whose car is an element of the data type D is also a
set car is unsp eci ed
list of D
There are no other lists of D
set cdr pair obj essential pro cedure
Within literal expressions and representations of ob
Stores obj in the cdr eld of pair The value returned by
jects read by the read pro cedure the forms hdatumi
set cdr is unsp eci ed
hdatumi hdatumi and hdatumi denote two ele
ment lists whose rst elements are the symbols quote
quasiquote unquote and unquote splicing resp ec
caar pair essential pro cedure
tively The second element in each case is hdatumi This
cadr pair essential pro cedure
convention is supp orted so that arbitrary Scheme pro
grams may b e represented as lists That is according
cdddar pair essential pro cedure
to Scheme s grammar every hexpressioni is also a hdatumi
cddddr pair essential pro cedure
see section Among other things this p ermits the
These pro cedures are comp ositions of car and cdr where use of the read pro cedure to parse Scheme programs See
for example caddr could b e de ned by section
Revised Scheme
define caddr lambda x car cdr cdr x define list tail
lambda x k
Arbitrary comp ositions up to four deep are provided
if zero k
There are twenty eight of these pro cedures in all
x
list tail cdr x k
null obj essential pro cedure
list ref list k pro cedure
Returns t if obj is the empty list otherwise returns f
Returns the k th element of list This is the same as the
In implementations in which the empty list is the same as
car of list tail list k
f null will return t if obj is f
list ref a b c d c
list obj essential pro cedure
last pair list pro cedure
Returns a list of its arguments
Returns the last pair in the nonempty p ossibly improp er
list a c a c
list list Last pair could b e de ned by
list
define last pair
lambda x
length list essential pro cedure
if pair cdr x
last pair cdr x
Returns the length of list
x
length a b c
length a b c d e
memq obj list essential pro cedure
length
memv obj list essential pro cedure
member obj list essential pro cedure
append list list essential pro cedure
These pro cedures return the rst sublist of list whose car
append list pro cedure
is obj If obj do es not o ccur in list f n b not the empty
list is returned Memq uses eq to compare obj with the
Returns a list consisting of the elements of the rst list
elements of list while memv uses eqv and member uses
followed by the elements of the other list s
equal
append x y x y
memq a a b c a b c
append a b c d a b c d
memq b a b c b c
append a b c a b c
memq a b c d f
memq list a b a c f
The resulting list is always newly allo cated except that
member list a
it shares structure with the last list argument The last
b a c a c
argument may actually b e any ob ject an improp er list
memq unspeci ed
results if it is not a prop er list
memv
append a b c d a b c d
append a a
assq obj alist essential pro cedure
assv obj alist essential pro cedure
assoc obj alist essential pro cedure
reverse list pro cedure
Alist for asso ciation list must b e a list of pairs These
Returns a newly allo cated list consisting of the elements of
pro cedures nd the rst pair in alist whose car eld is obj
list in reverse order
and returns that pair If no pair in alist has obj as its car
reverse a b c c b a
f is returned Assq uses eq to compare obj with the car
reverse a b c d e f
elds of the pairs in alist while assv uses eqv and assoc
e f d b c a
uses equal
define e a b c
assq a e a
list tail list k pro cedure
assq b e b
Returns the sublist of list obtained by omitting the rst k
assq d e f
assq list a a b c elements List tail could b e de ned by
Standard pro cedures
f its name contains alphab etic characters then the string
assoc list a a b c
returned will contain characters in the implementation s
a
preferred standard case some implementations will prefer
assq
upp er case others lower case If the symbol was returned
unspeci ed
by string symbol the case of characters in the string
assv
returned will b e the same as the case in the string that
was passed to string symbol It is an error to apply
Note Although they are ordinarily used as predicates memq
mutation pro cedures like string set to strings returned
memv member assq assv and assoc do not have question
by this pro cedure
marks in their names b ecause they return useful values rather
The following examples assume that the implementation s
than just t or f
standard case is lower case
symbol string flying fish
Symbols
flying fish
symbol string Martin martin
Symbols are ob jects whose usefulness rests on the fact that
symbol string
two symbols are identical in the sense of eqv if and only
string symbol Malvina
if their names are sp elled the same way This is exactly the
Malvina
prop erty needed to represent identi ers in programs and
so most implementations of Scheme use them internally for
that purp ose Symbols are useful for many other applica
string symbol string essential pro cedure
tions for instance they may b e used the way enumerated
Returns the symbol whose name is string This pro cedure
values are used in Pascal
can create symbols with names containing sp ecial charac
The rules for writing a symbol are exactly the same as the
ters or letters in the non standard case but it is usually
rules for writing an identi er see sections and
a bad idea to create such symbols b ecause in some imple
It is guaranteed that any symbol that has b een returned as
mentations of Scheme they cannot b e read as themselves
part of a literal expression or read using the read pro ce
See symbol string
dure and subsequently written out using the write pro ce
The following examples assume that the implementation s
dure will read back in as the identical symbol in the sense
standard case is lower case
of eqv The string symbol pro cedure however can
eq mISSISSIppi mississippi
create symbols for which this write read invariance may
t
not hold b ecause their names contain sp ecial characters or
string symbol mISSISSIppi
letters in the non standard case
the symbol with name mISSISSIppi
Note Some implementations of Scheme have a feature known
eq bitBlt string symbol bitBlt
as slashi cation in order to guarantee write read invariance
f
for all symbols but historically the most imp ortant use of this
eq JollyWog
feature has b een to comp ensate for the lack of a string data
string symbol
type
symbol string JollyWog
t
Some implementations also have uninterned symbols which
string K Harper M D
defeat write read invariance even in implementations with
symbol string
slashi cation and also generate exceptions to the rule that two
string symbol K Harper M D
symbols are the same if and only if their names are sp elled the
t
same
Numbers
symbol obj essential pro cedure
Numerical computation has traditionally b een neglected by
Returns t if obj is a symbol otherwise returns f
the Lisp community Until Common Lisp there has b een
symbol foo t
no carefully thought out strategy for organizing numeri
symbol car a b t
cal computation and with the exception of the MacLisp
symbol bar f
system there has b een little e ort to execute numeri
cal co de e ciently We applaud the excellent work of the
symbol string symbol essential pro cedure
Common Lisp committee and we accept many of their rec
ommendations In some ways we simplify and generalize Returns the name of symbol as a string If the symbol was
their prop osals in a manner consistent with the purp oses part of an ob ject returned as the value of a literal expres
of Scheme sion section or by a call to the read pro cedure and
Revised Scheme
Scheme s numerical op erations treat numbers as abstract Not all implementations of Scheme must provide the whole
data as indep endent of their representation as is p ossible tower but they must implement a coherent subset consis
Thus the casual user should b e able to write simple pro tent with b oth the purp oses of the implementation and the
grams without having to know that the implementation spirit of the Scheme language
may use xed p oint oating p oint and p erhaps other rep
resentations for his data Unfortunately this illusion of
Exactness
uniformity can b e sustained only approximately the im
Numbers are either exact or inexact A number is exact
plementation of numbers will leak out of its abstraction
if it was derived from exact numbers using only exact
whenever the user must b e in control of precision or accu
op erations A number is inexact if it mo dels a quantity
racy or when he must construct esp ecially e cient com
e g a measurement known only approximately if it was
putations Thus the language must also provide escap e
derived using inexact ingredients or if it was derived us
mechanisms so that a sophisticated programmer can exer
ing inexact op erations Thus inexactness is a contagious
cise more control over the execution of his co de and the
prop erty of a number Some op erations such as the square
representation of his data when necessary
ro ot of non square numbers must b e inexact b ecause of
It is imp ortant to distinguish b etween the abstract num
the nite precision of our representations Other op era
b ers their machine representations and their written rep
tions are inexact b ecause of implementation requirements
resentations We will use mathematical terms number
We emphasize that exactness is indep endent of the p osi
complex real rational and integer for prop erties of
tion of the number on the tower It is p erfectly p ossible to
the abstract numbers the names fixnum bignum ratnum
have an inexact integer or an exact real may
and flonum for machine representations and the names
b e an exact rational or it may b e an inexact rational
int fix flo sci rat polar and rect for input output
approximation to pi dep ending on the application
formats
Op erationally it is the system s resp onsibility to combine
exact numbers using exact metho ds such as in nite pre
cision integer and rational arithmetic where p ossible An
Numbers
implementation may not b e able to do this if it do es not
A Scheme system provides data of type number which is
use in nite precision integers and rationals but if a num
the most general numerical type supp orted by that sys
b er b ecomes inexact for implementation reasons there is
tem Number is likely to b e a complicated union type im
likely to b e an imp ortant error condition such as integer
plemented in terms of fixnums bignums flonums and so
over ow to b e rep orted Arithmetic on inexact numbers
forth but this should not b e apparent to a naive user
is not so constrained The system may use oating p oint
What the user should see is that the usual op erations
and other ill b ehaved representation strategies for inexact
on numbers pro duce the mathematically exp ected results
numbers This is not to say that implementors need not
within the limits of the implementation Thus if the user
use the b est known algorithms for inexact computations
divides the exact number by the exact number he
only that approximate metho ds of high quality are allowed
should get something like or the exact fraction
In a system that cannot explicitly distinguish exact from
If he adds that result to itself and the implementation is
inexact numbers the system must do its b est to main
go o d enough he should get an exact
tain precision Scheme systems must not burden users
with numerical op erations describ ed in terms of hardware
Mathematically numbers may b e arranged into a tower of
and op erating system dep endent representations such as
subtypes with pro jections and injections relating adjacent
fixnum and flonum however b ecause these representation
levels of the tower
issues are hardly ever germane to the user s problems
We highly recommend that the IEEE bit and bit
number
oating p oint standards b e adopted for implementations
complex
that use oating p oint representations internally To min
real
imize loss of precision we adopt the following rules If an
rational
implementation uses several di erent sizes of oating p oint
integer
formats the results of any op eration with a oating p oint
result must b e expressed in the largest format used to ex
We imp ose a uniform rule of downward co ercion a num press any of the oating p oint arguments to that op eration
b er of one type is also of a lower type if the injection up It is desirable but not required for p otentially irrational
of the pro jection down of a number leaves the number op erations such as sqrt when applied to exact arguments
unchanged Since this tower is a genuine mathematical to pro duce exact answers whenever p ossible for example
structure Scheme provides predicates and pro cedures to the square ro ot of an exact ought to b e an exact If
access the tower an exact number or an inexact number represented as
Standard pro cedures
a fixnum a bignum or a ratnum is op erated up on so as number obj essential pro cedure
to pro duce an inexact result as by sqrt and if the re complex obj essential pro cedure
sult is represented as a flonum then the largest available real obj essential pro cedure
flonum format must b e used but if the result is expressed rational obj essential pro cedure
as a ratnum then the rational approximation must have at integer obj essential pro cedure
least as much precision as the largest available flonum
These numerical type predicates can b e applied to any kind
of argument including non numbers They return true if
the ob ject is of the named type In general if a type pred
Number syntax
icate is true of a number then all higher type predicates
Scheme allows the traditional ways of writing numerical
are also true of that number Not every system supp orts
constants though any particular implementation may sup
all of these types for example it is entirely p ossible to
p ort only some of them These syntaxes are intended to
have a Scheme system that has only integers Nonethe
b e purely notational any kind of number may b e written
less every implementation of Scheme must have all of these
in any form that the user deems convenient Of course
predicates
writing as a limited precision decimal fraction will not
express the number exactly but this approximate form of
zero z essential pro cedure
expression may b e just what the user wants to see
positive x essential pro cedure
The syntax of numbers is describ ed formally in sec
negative x essential pro cedure
tion See section for many examples of rep
odd n essential pro cedure
resentations of numbers
even n essential pro cedure
exact z essential pro cedure
A numerical constant may b e represented in binary o ctal
inexact z essential pro cedure
decimal or hex by the use of a radix pre x The radix
These numerical predicates test a number for a particular
pre xes are b binary o o ctal d decimal and x
prop erty returning t or f
hex With no radix pre x a number is assumed to b e
expressed in decimal
z z essential pro cedure
A numerical constant may b e sp eci ed to b e either exact
x x essential pro cedure
or inexact by a pre x The pre xes are e for exact and
x x essential pro cedure
i for inexact An exactness pre x may app ear b efore or
x x essential pro cedure
after any radix pre x that is used If the representation of
x x essential pro cedure
a numerical constant has no exactness pre x the constant
Some implementations allow these pro cedures to take many
may b e assumed to b e exact or inexact at the discretion of
arguments to facilitate range checks These pro cedures
the implementation except that integers expressed without
return t if their arguments are resp ectively numeri
decimal p oints and without use of exp onential notation are
cally equal monotonically increasing monotonically de
assumed exact
creasing monotonically nondecreasing or monotonically
In systems with b oth single and double precision flonums
nonincreasing Warning on inexact numbers the equal
we may want to sp ecify which size we want to use to rep
ity tests will give unreliable results and the other numer
resent a constant internally For example we may want
ical comparisons will b e useful only heuristically when in
a constant that has the value of pi rounded to the single
doubt consult a numerical analyst
precision length or we might want a long number that has
the value In either case we are sp ecifying an explicit
max x x essential pro cedure
way to represent an inexact number For this purp ose we
max x x pro cedure
may express a number with a pre x that indicates short or
min x x essential pro cedure
long flonum representation
min x x pro cedure
S
These pro cedures return the maximum or minimum of their
Round to short
arguments
L
Extend to long
z z essential pro cedure
z pro cedure
Numerical op erations
z z essential pro cedure
z pro cedure
The reader is referred to section for a summary of
the naming conventions used to sp ecify restrictions on the These pro cedures return the sum or pro duct of their argu
types of arguments to numerical routines ments
Revised Scheme
numerator q pro cedure
denominator q pro cedure
These pro cedures return the numerator or denominator of
their argument
numerator
denominator
z z essential pro cedure
z z pro cedure
z z essential pro cedure
gcd n pro cedure
z z pro cedure
lcm n pro cedure
With two or more arguments these pro cedures return the
These pro cedures return the greatest common divisor or
di erence or quotient of their arguments asso ciating to the
least common multiple of their arguments The result is
left With one argument however they return the additive
always non negative
or multiplicative inverse of their argument
gcd
gcd
lcm
lcm
floor x pro cedure
abs z essential pro cedure
ceiling x pro cedure
truncate x pro cedure
Abs returns the magnitude of its argument
round x pro cedure
abs
rationalize x y pro cedure
abs i
rationalize x pro cedure
quotient n n essential pro cedure
These pro cedures create integers and rationals Their re
remainder n n essential pro cedure
sults are exact if and only if their arguments are exact
modulo n n pro cedure
Floor returns the largest integer not larger than x
These are intended to implement number theoretic inte
Ceiling returns the smallest integer not smaller than x
ger division For p ositive integers n and n if n and n
Truncate returns the integer of maximal absolute value
are integers such that n n n n and n n
not larger than the absolute value of x Round returns the
then
closest integer to x rounding to even when x is halfway
b etween two integers With two arguments rationalize
quotient n n n
pro duces the rational number with smallest denominator remainder n n n
modulo n n n
di ering from x by no more than y With one argument
rationalize pro duces the b est rational approximation to
For all integers n and n with n not equal to
x preserving all of the precision in its representation
n n quotient n n
Note Round rounds to even for consistency with the rounding
remainder n n
mo des required by the IEEE oating p oint standard
t
The value returned by quotient always has the sign of the
pro duct of its arguments Remainder and modulo di er on
exp z pro cedure
negative arguments the remainder always has the sign of
log z pro cedure
the dividend the modulo always has the sign of the divisor
sin z pro cedure
cos z pro cedure
modulo
tan z pro cedure
remainder
asin z pro cedure
modulo
acos z pro cedure
remainder
atan z pro cedure
atan y x pro cedure
modulo
These pro cedures are part of every implementation that
remainder
supp orts real numbers they compute the usual transcen
dental functions Log computes the natural logarithm of
modulo
remainder z not the base logarithm Asin acos and atan
Standard pro cedures
compute arcsine sin arccosine cos and arctan Note Magnitude is the same as abs but abs must b e present
gent tan resp ectively The two argument variant of in all implementations whereas magnitude will only b e present
atan computes angle make rectangular x y see in implementations that supp ort complex numbers
b elow even in implementations that don t supp ort com
plex numbers
exact inexact z pro cedure
In general the mathematical functions log arcsine arcco
inexact exact z pro cedure
sine and arctangent are multiply de ned For nonzero real
Exact inexact returns an inexact representation of z
x the value of log x is de ned to b e the one whose imagi
which is a fairly harmless thing to do Inexact exact
nary part lies in the range exclusive to inclusive
returns an exact representation of z Since the law of
log is unde ned The value of log z when z is complex is
garbage in garbage out remains in force inexact
de ned according to the formula
exact should not b e used casually
log z log magnitude z i angle z
With log de ned this way the values of sin z cos z
Numerical input and output
and tan z are according to the following formulae
p
sin z i log iz z
number string number format pro cedure
p
z cos z i log z i
The conventions used to pro duce the printed representation
p
of a number can b e sp eci ed by a format as describ ed
z tan z i log iz
in section The pro cedure number string takes a
number and a format and returns as a string the printed
The ab ove sp eci cation follows which in turn fol
representation of the given number in the given format
lows refer to these sources for more detailed discussion
This pro cedure will mostly b e used by sophisticated users
of branch cuts b oundary conditions and implementation
and in system programs In general a naive user will need
of these functions
to know nothing ab out the formats b ecause the system
printer will have reasonable default formats for all types of
sqrt z pro cedure
numbers
Returns the principal square ro ot of z The result will have
either p ositive real part or zero real part and non negative
string number string exactness radix pro cedure
imaginary part
The system reader will construct reasonable default numer
ical types for numbers expressed in each of the formats it
expt z z pro cedure
recognizes A user who needs control of the co ercion from
strings to numbers will use string number Exactness
Returns z raised to the p ower z
must b e a symbol either E for exact or I for inexact
z log z z
2 1 2
e z
Radix must also b e a symbol B for binary O for o ctal D for
is de ned to b e equal to
decimal and X for hexadecimal Returns a number of the
maximally precise representation expressed by the given
string It is an error if string do es not express a number
make rectangular x x pro cedure
according to the grammar in section
make polar x x pro cedure
real part z pro cedure
Formats
imag part z pro cedure
magnitude z pro cedure
A format is a list b eginning with a format descriptor which
angle z pro cedure
is a symbol such as sci Following the descriptor are pa
rameters used by that descriptor such as the number of
These pro cedures are part of every implementation that
signi cant digits to b e used Default values are supplied
supp orts complex numbers Supp ose x x x and x
for any parameters that are omitted Mo di ers may app ear
are real numbers and z is a complex number such that
after the parameters such as the radix and exactness for
ix
z x x i x e
mats describ ed b elow which themselves take parameters
Then make rectangular and make polar return z
Details of particular formats such as sci and fix are given
real part returns x imag part returns x magnitude
in section
returns x and angle returns x In the case of angle
whose value is not uniquely determined by the preceding For example the format sci exactness s sp ec
rule the value returned will b e the one in the range i es that a number is to b e expressed in Fortran scien
exclusive to inclusive ti c format with signi cant places and two places after
Revised Scheme
the radix p oint and that its exactness pre x is to b e sup
pressed
In the following examples the comment shows the format
rat n format
that was used to pro duce the output shown
Express as a rational fraction n sp eci es the largest de
int
nominator to b e used in constructing a rational approxi
int
mation to the number b eing expressed If n is omitted it
rat
defaults to in nity
fix
fix
flo
fix n format
flo
Express with a xed radix p oint n sp eci es the number of
e sci
places to the right of the radix p oint n defaults to the size
e e e sci
of a single precision flonum If there are not enough signif
i rect int int
icant places then insigni cant digits will b e agged For
polar fix
flo
example an inexact represented internally as
a digit flonum would b e printed with a fix format
A format may sp ecify that a number should b e expressed
as
in a particular radix The radix pre x may also b e sup
pressed For example one may express a complex number
in p olar form with the magnitude in o ctal and the angle in
decimal as follows
flo n format
o d polar flo radix o
Express with a oating radix p oint n sp eci es the total
flo radix d
number of places to b e displayed n defaults to the size of
o polar flo radix o
a single precision flonum If the number is out of range
flo radix d s
it is converted to sci flo h expresses n in oating
A format may sp ecify that a number should b e expressed
p oint format heuristically for human consumption
with an explicit exactness pre x exactness e or it
may force the exactness to b e suppressed exactness
sci n m format
s For example the following are ways to express an
inexact value for pi
Express in exp onential notation n sp eci es the total num
b er of places to b e displayed n defaults to the size of a
i rat exactness e
single precision flonum m sp eci es the number of places
rat exactness s
to the right of the radix p oint m defaults to n sci
i fix exactness e
h do es heuristic expression
An attempt to pro duce more digits than are available in
the internal machine representation of a number will b e
rect r i format
marked with a lling the extra digits This is not a
statement that the implementation knows or keeps track
Express as a rectangular form complex number r and i
of the signi cance of a number just that the machine will
are formats for the real and imaginary parts resp ectively
ag attempts to pro duce digits of a number that has
They default to heur
only digits of machine representation
flo exactness s
polar m a format
Express as a p olar form complex number m and a are
Details of formats
formats for the magnitude and angle resp ectively m and
a default to heur The format descriptors are
heur format int format
Express heuristically using the minimum number of digits Express as an integer The radix p oint is implicit If
required to get an expression that when co erced back to there are not enough signi cant places then insigni cant
a number pro duces the original machine representation digits will b e agged For example an inexact integer
Exact numbers are expressed as int or rat Inexact represented internally as a digit flonum
numbers are expressed as flo h or sci h dep ending would b e printed as
Standard pro cedures
on their range Complex numbers are expressed in rect implementations that supp ort the notation for input do
This is the normal default of the system printer not have to supp ort it for output
Some of the pro cedures that op erate on characters ignore
the di erence b etween upp er case and lower case The pro
The following mo di ers may b e added to a numerical for
cedures that ignore case have the su x ci for case in
mat sp eci cation
sensitive If the op eration is a predicate then the ci
su x precedes the at the end of the name
exactness s format
This controls the expression of the exactness pre x of a
char obj essential pro cedure
number s must b e a symbol either E or S indicating
Returns t if obj is a character otherwise returns f
whether the exactness is to b e expressed or suppressed re
sp ectively If no exactness mo di er is sp eci ed for a format
char char char essential pro cedure
then the exactness is by default suppressed
char char char essential pro cedure
char char char essential pro cedure
char char char essential pro cedure
radix r s format
char char char essential pro cedure
This forces a number to b e expressed in the radix r r may
These pro cedures imp ose a total ordering on the set of
b e the symbol B binary O o ctal D decimal or X hex
characters It is guaranteed that under this ordering
s must b e a symbol either E or S indicating whether the
radix pre x is to b e expressed or suppressed resp ectively
The upp er case characters are in order For example
s defaults to E expressed If no radix mo di er is sp eci ed
char A B returns t
then the default is decimal and the pre x is suppressed
The lower case characters are in order For example
char a b returns t
Characters
The digits are in order For example char
returns t
Characters are ob jects that represent printed characters
such as letters and digits There is no requirement that the
Either all the digits precede all the upp er case letters
data type of characters b e disjoint from other data types
or vice versa
implementations are encouraged to have a separate char
acter data type but may choose to represent characters as
Either all the digits precede all the lower case letters
integers strings or some other type
or vice versa
Characters are written using the notation hcharacteri or Some implementations may generalize these pro cedures to
take more than two arguments as with the corresp onding
hcharacter namei For example
numeric predicates
a lower case letter
char ci char char pro cedure
A upp er case letter
char ci char char pro cedure
left parenthesis
char ci char char pro cedure
the space character
char ci char char pro cedure
space the preferred way to write a space
char ci char char pro cedure
newline the newline character
These pro cedures are similar to char et cetera but they
Case is signi cant in hcharacteri but not in hcharacter treat upp er case and lower case letters as the same For
namei If hcharacteri in hcharacteri is alphab etic then
example char ci A a returns t Some imple
the character following hcharacteri must b e a delimiter
mentations may generalize these pro cedures to take more
character such as a space or parenthesis This rule resolves
than two arguments as with the corresp onding numeric
the ambiguous case where for example the sequence of
predicates
characters space could b e taken to b e either a repre
sentation of the space character or a representation of the
char alphabetic char pro cedure
character s followed by a representation of the symbol
char numeric char pro cedure
pace
char whitespace char pro cedure
Characters written in the notation are self evaluating These pro cedures return t if their arguments are alpha
That is they do not have to b e quoted in programs The b etic numeric or whitespace characters resp ectively oth
notation is not an essential part of Scheme however Even erwise they return f The following remarks which are
Revised Scheme
sp eci c to the ASCI I character set are intended only as A backslash can b e written inside a string only by escaping
a guide The alphab etic characters are the upp er and it with another backslash Scheme do es not sp ecify the
lower case letters The numeric characters are the ten dec e ect of a backslash within a string that is not followed by
imal digits The whitespace characters are space tab line a doublequote or backslash
feed form feed and carriage return
A string may continue from one line to the next but this is
usually a bad idea b ecause the exact e ect may vary from
one computer system to another
char upper case letter pro cedure
The length of a string is the number of characters that
char lower case letter pro cedure
it contains This number is a non negative integer that
Letter must b e an alphab etic character These pro cedures
is xed when the string is created The valid indexes of
return t if their arguments are upp er case or lower case
a string are the exact non negative integers less than the
characters resp ectively otherwise they return f
length of the string The rst character of a string has
index the second has index and so on
In phrases such as the characters of string b eginning with
char integer char essential pro cedure
index start and ending with index end it is understo o d
integer char n essential pro cedure
that the index start is inclusive and the index end is ex
Given a character char integer returns an integer rep
clusive Thus if start and end are the same index a null
resentation of the character Given an integer that is the
substring is referred to and if start is zero and end is the
image of a character under char integer integer
length of string then the entire string is referred to
char returns a character These pro cedures implement or
Some of the pro cedures that op erate on strings ignore the
der isomorphisms b etween the set of characters under the
di erence b etween upp er and lower case The versions that
char ordering and some subset of the integers under
ignore case have the su x ci for case insensitive If
the ordering That is if
the op eration is a predicate then the ci su x precedes
the at the end of the name
char a b t and x y t
and x and y are in the range of char integer then
string obj essential pro cedure
Returns t if obj is a string otherwise returns f
char integer a
char integer b t
make string k pro cedure
char integer char x
make string k char pro cedure
integer char y t
k must b e a non negative integer and char must b e a char
acter Make string returns a newly allo cated string of
length k If char is given then all elements of the string
char upcase char pro cedure
are initialized to char otherwise the contents of the string
char downcase char pro cedure
are unsp eci ed
These pro cedures return a character char such that
char ci char char In addition if char is alpha
string length string essential pro cedure
b etic then the result of char upcase is upp er case and
Returns the number of characters in the given string
the result of char downcase is lower case
string ref string k essential pro cedure
k must b e a valid index of string String ref returns
Strings
character k of string using zero origin indexing
Strings are sequences of characters In some implemen
tations of Scheme they are immutable other implementa
string set string k char pro cedure
tions provide destructive pro cedures such as string set
k must b e a valid index of string String set stores char
that alter string ob jects
in element k of string and returns an unsp eci ed value
Strings are written as sequences of characters enclosed
within doublequotes A doublequote can b e written
string string string essential pro cedure
inside a string only by escaping it with a backslash as
string ci string string pro cedure
in
Returns t if the two strings are the same length and con
The word recursion has many meanings tain the same characters in the same p ositions otherwise
Standard pro cedures
returns f String ci treats upp er and lower case let Vectors
ters as though they were the same character but string
Vectors are heterogenous mutable structures whose ele
treats upp er and lower case as distinct characters
ments are indexed by integers
The length of a vector is the number of elements that it
string string string essential pro cedure
contains This number is a non negative integer that is
string string string essential pro cedure
xed when the vector is created The valid indexes of a
string string string essential pro cedure
vector are the exact non negative integers less than the
string string string essential pro cedure
length of the vector The rst element in a vector is indexed
string ci string string pro cedure
by zero and the last element is indexed by one less than
string ci string string pro cedure
the length of the vector
string ci string string pro cedure
string ci string string pro cedure
Vectors are written using the notation obj For
example a vector of length containing the number zero
These pro cedures are the lexicographic extensions to
in element the list in element and the
strings of the corresp onding orderings on characters For
string Anna in element can b e written as following
example string is the lexicographic ordering on strings
induced by the ordering char on characters If two
Anna
strings di er in length but are the same up to the length
Note that this is the external representation of a vector not
of the shorter string the shorter string is considered to b e
an expression evaluating to a vector Like list constants
lexicographically less than the longer string
vector constants must b e quoted
Implementations may generalize these and the string
Anna
and string ci pro cedures to take more than two argu
Anna
ments as with the corresp onding numeric predicates
substring string start end essential pro cedure
vector obj essential pro cedure
String must b e a string and start and end must b e exact
Returns t if obj is a vector otherwise returns f
integers satisfying
start end string length string
make vector k essential pro cedure
make vector k l l pro cedure
Substring returns a newly allo cated string formed from
the characters of string b eginning with index start inclu
Returns a newly allo cated vector of k elements If a second
sive and ending with index end exclusive
argument is given then each element is initialized to l l
Otherwise the initial contents of each element is unsp eci
string append string string essential pro cedure
ed
string append string pro cedure
Returns a new string whose characters form the concate
vector obj essential pro cedure
nation of the given strings
Returns a newly allo cated vector whose elements contain
the given arguments Analogous to list
string list string essential pro cedure
vector a b c a b c
list string chars essential pro cedure
String list returns a newly allo cated list of the char
vector length vector essential pro cedure
acters that make up the given string List string re
turns a string formed from the characters in the list chars
Returns the number of elements in vector
String list and list string are inverses so far as
equal is concerned Implementations that provide de
vector ref vector k essential pro cedure
structive op erations on strings should ensure that the re
k must b e a valid index of vector Vector ref returns the
sult of list string is newly allo cated
contents of element k of vector
vector ref
string copy string pro cedure
Returns a newly allo cated copy of the given string
vector set vector k obj essential pro cedure
string fill string char pro cedure
k must b e a valid index of vector Vector set stores obj
in element k of vector The value returned by vector set
Stores char in every element of the given string and returns
is unsp eci ed
an unsp eci ed value
Revised Scheme
let vec vector Anna map proc list essential pro cedure
vector set vec Sue Sue
map proc list list pro cedure
vec
The list s must b e lists and proc must b e a pro cedure taking
Sue Sue Anna
as many arguments as there are lists If more than one list
is given then they must all b e the same length Map applies
vector list vector essential pro cedure
proc element wise to the elements of the list s and returns
list vector list essential pro cedure
a list of the results The order in which proc is applied to
the elements of the list s is unsp eci ed
Vector list returns a newly created list of the ob jects
contained in the elements of vector List vector returns
map cadr a b d e g h
a newly created vector initialized to the elements of the list
b e h
list
map lambda n expt n n
vector list dah dah didah
dah dah didah
list vector dididit dah
dididit dah
map
let count
vector fill vector l l pro cedure
map lambda ignored
Stores l l in every element of vector The value returned
set count count
by vector fill is unsp eci ed
count
a b c unspeci ed
Control features
for each proc list essential pro cedure
This chapter describ es various primitive pro cedures which
for each proc list list pro cedure
control the ow of program execution in sp ecial ways The
procedure predicate is also describ ed here
The arguments to for each are like the arguments to map
but for each calls proc for its side e ects rather than for
its values Unlike map for each is guaranteed to call proc
procedure obj essential pro cedure
on the elements of the list s in order from the rst element to
Returns t if obj is a pro cedure otherwise returns f
the last and the value returned by for each is unsp eci ed
procedure car t
let v make vector
procedure car f
for each lambda i
procedure lambda x x x
vector set v i i i
t
procedure lambda x x x
v
f
call with current continuati on procedure
t
force promise pro cedure
Forces the value of promise see delay section If
apply proc args essential pro cedure
no value has b een computed for the promise then a value
apply proc arg args pro cedure
is computed and returned The value of the promise is
Proc must b e a pro cedure and args must b e a list The
cached or memoized so that if it is forced a second
rst essential form calls proc with the elements of args as
time the previously computed value is returned without
the actual arguments The second form is a generalization
any recomputation
of the rst that calls proc with the elements of the list
force delay
append list arg args as the actual arguments
let p delay
apply list list force p force p
define compose
lambda f g define a stream
lambda args letrec next
f apply g args lambda n
cons n delay next n
compose sqrt next
Standard pro cedures
eqv delay unspeci ed define head car
pair delay cons unspeci ed define tail
lambda stream force cdr stream
Some implementations will implement implicit forc
ing where the value of a promise is forced by primi
head tail tail a stream
tive pro cedures like cdr and
delay
Force and delay are mainly intended for programs writ
ten in functional style The following examples should not
b e considered to illustrate go o d programming style but
call with current continuation proc
they illustrate the prop erty that the value of a promise is
essential pro cedure
computed at most once
Proc must b e a pro cedure of one argument The pro cedure
define count
call with current continuation packages up the cur
define p delay begin set count count
rent continuation see the rationale b elow as an escap e
x
pro cedure and passes it as an argument to proc The
define x
escap e pro cedure is a Scheme pro cedure of one argument
count
that if it is later passed a value will ignore whatever con
p a promise
tinuation is in e ect at that later time and will give the
force p
value instead to the continuation that was in e ect when
p a promise stil l
the escap e pro cedure was created
count
force p
The escap e pro cedure created by
count
call with current continuation has unlimited extent
just like any other pro cedure in Scheme It may b e stored
Here is a p ossible implementation of delay and force We
in variables or data structures and may b e called as many
de ne the expression
times as desired
delay hexpressioni
The following examples show only the most common
uses of call with current continuation If all real
to have the same meaning as the pro cedure call
programs were as simple as these examples there
make promise lambda hexpressioni
would b e no need for a pro cedure with the p ower of
call with current continuation
where make promise is de ned as follows
call with current continuat ion
define make promise
lambda exit
lambda proc
for each lambda x
let already run f result f
if negative x
lambda
exit x
cond not already run
set result proc
t
set already run t
result
define list length
Promises are implemented here as pro cedures of no argu
lambda obj
call with current continuati on
ments and force simply calls its argument
lambda return
define force
letrec r
lambda object
lambda obj
object
cond null obj
pair obj
Various extensions to this semantics of delay and force
r cdr obj
are supp orted in some implementations
else return f
r obj
Calling force on an ob ject that is not a promise may
simply return the ob ject
list length
It may b e the case that there is no means by which
list length a b c f
a promise can b e op erationally distinguished from its
forced value That is expressions like the following
Rationale A
may evaluate to either t or to f dep ending on the
common use of call with current continuat ion is for struc
implementation tured non lo cal exits from lo ops or pro cedure b o dies but in
Revised Scheme
fact call with current continuation is extremely useful for automatically and the value yielded by the pro cedure is
implementing a wide variety of advanced control structures
returned If the pro cedure do es not return then Scheme
will not close the p ort unless it can prove that the p ort will
Whenever a Scheme expression is evaluated there is a contin
never again b e used for a read or write op eration
uation wanting the result of the expression The continuation
represents an entire default future for the computation If
Rationale Because Scheme s escap e pro cedures have un
the expression is evaluated at top level for example then the
limited extent it is p ossible to escap e from the current con
continuation will take the result print it on the screen prompt
tinuation but later to escap e back in If implementations
for the next input evaluate it and so on forever Most of the
were p ermitted to close the p ort on any escap e from the
time the continuation includes actions sp eci ed by user co de
current continuation then it would b e imp ossible to write
as in a continuation that will take the result multiply it by
p ortable co de using b oth call with current continuati on
the value stored in a lo cal variable add seven and give the
and call with input file or call with output file
answer to the top level continuation to b e printed Normally
these ubiquitous continuations are hidden b ehind the scenes and
programmers don t think much ab out them On rare o ccasions
input port obj essential pro cedure
however a programmer may need to deal with continuations ex
output port obj essential pro cedure
plicitly Call with current continuatio n allows Scheme pro
grammers to do that by creating a pro cedure that acts just like
Returns t if obj is an input p ort or output p ort resp ec
the current continuation
tively otherwise returns f
Most programming languages incorp orate one or more sp ecial
purp ose escap e constructs with names like exit return or
current input port essential pro cedure
even goto In however Peter Landin invented a
current output port essential pro cedure
general purp ose escap e op erator called the J op erator John
Reynolds describ ed a simpler but equally p owerful con
Returns the current default input or output p ort
struct in The catch sp ecial form describ ed by Sussman
and Steele in the rep ort on Scheme is exactly the same as
Reynolds s construct though its name came from a less general
with input from file string thunk pro cedure
construct in MacLisp Several Scheme implementors noticed
with output to file string thunk pro cedure
that the full p ower of the catch construct could b e provided by
Thunk must b e a pro cedure of no arguments and
a pro cedure instead of by a sp ecial syntactic construct and the
string must b e a string naming a le For
name call with current continuation was coined in
with input from file the le must already exist for
This name is descriptive but opinions di er on the merits of
with output to file the e ect is unsp eci ed if the le
such a long name and some p eople use the name call cc in
already exists The le is op ened for input or out
stead
put an input or output p ort connected to it is made
the default value returned by current input port or
current output port and the thunk is called with no ar
Input and output
guments When the thunk returns the p ort is closed and
Ports
the previous default is restored With input from file
and with output to file return the value yielded by
Ports represent input and output devices To Scheme an
thunk If an escap e pro cedure is used to escap e from the
input device is a Scheme ob ject that can deliver characters
continuation of these pro cedures their b ehavior is imple
up on command while an output device is a Scheme ob ject
mentation dep endent
that can accept characters
open input file lename pro cedure
call with input file string proc
essential pro cedure
Takes a string naming an existing le and returns an input
call with output file string proc
p ort capable of delivering characters from the le If the
essential pro cedure
le cannot b e op ened an error is signalled
Proc should b e a pro cedure of one argument and
string should b e a string naming a le For
open output file lename pro cedure
call with input file the le must already exist for
call with output file the e ect is unsp eci ed if the Takes a string naming an output le to b e created and
le already exists These pro cedures call proc with one ar returns an output p ort capable of writing characters to a
gument the p ort obtained by op ening the named le for new le by that name If the le cannot b e op ened an
input or output If the le cannot b e op ened an error is error is signalled If a le with the given name already
signalled If the pro cedure returns then the p ort is closed exists the e ect is unsp eci ed
Standard pro cedures
close input port port pro cedure b e indistinguishable from an interactive p ort that has no ready
close output port port pro cedure characters
Closes the le asso ciated with port rendering the port in
eof object obj essential pro cedure
capable of delivering or accepting characters These rou
tines have no e ect if the le has already b een closed The
Returns t if obj is an end of le ob ject otherwise returns
value returned is unsp eci ed
f The precise set of end of le ob jects will vary among
implementations but in any case no end of le ob ject will
ever b e a character or an ob ject that can b e read in using
Input
read
read essential pro cedure
Output
read port essential pro cedure
Read converts written representations of Scheme ob jects
write obj essential pro cedure
into the ob jects themselves That is it is a parser for the
write obj port essential pro cedure
nonterminal hdatumi see section Read returns the
Writes a representation of obj to the given port Strings
next ob ject parsable from the given input port up dating
that app ear in the written representation are enclosed in
port to p oint to the rst character past the end of the
doublequotes and within those strings backslash and dou
written representation of the ob ject
blequote characters are escap ed by backslashes Write re
turns an unsp eci ed value The port argument may b e
If an end of le is encountered in the input b efore any char
omitted in which case it defaults to the value returned by
acters are found that can b egin an ob ject then an end of
current output port
le ob ject is returned The p ort remains op en and fur
ther attempts to read will also return an end of le ob ject
If an end of le is encountered after the b eginning of an
display obj essential pro cedure
ob ject s written representation but the written represen
display obj port essential pro cedure
tation is incomplete and therefore not parsable an error is
Writes a representation of obj to the given port Strings
signalled
that app ear in the written representation are not enclosed
The port argument may b e omitted in which case it de
in doublequotes and no characters are escap ed within
faults to the value returned by current input port It is
those strings In those implementations that have a dis
an error to read from a closed p ort
tinct character type character ob jects app ear in the repre
sentation as if written by write char instead of by write
Display returns an unsp eci ed value The port argument
read char essential pro cedure
may b e omitted in which case it defaults to the value re
read char port essential pro cedure
turned by current output port
Returns the next character available from the input port
Rationale Write is intended for pro ducing machine readable
up dating the port to p oint to the following character If
output and display is for pro ducing human readable output
no more characters are available an end of le ob ject is
Implementations that allow slashi cation within symbols will
returned Port may b e omitted in which case it defaults
probably want write but not display to slashify funny charac
to the value returned by current input port
ters in symbols
char ready pro cedure
newline essential pro cedure
char ready port pro cedure
newline port essential pro cedure
Writes an end of line to port Exactly how this is done
Returns t if a character is ready on the input port and
di ers from one op erating system to another Returns
returns f otherwise If char ready returns t then the
an unsp eci ed value The port argument may b e omit
next read char op eration on the given port is guaranteed
ted in which case it defaults to the value returned by
not to hang If the port is at end of le then char ready
current output port
returns t Port may b e omitted in which case it defaults
to the value returned by current input port
write char char essential pro cedure
Rationale Char ready exists to make it p ossible for a pro
write char char port essential pro cedure
gram to accept characters from interactive p orts without getting
stuck waiting for input Any input editors asso ciated with such
Writes the character char not a written representation of
p orts must ensure that characters whose existence has b een as
the character to the given port and returns an unsp eci ed
serted by char ready cannot b e rubb ed out If char ready
value The port argument may b e omitted in which case it
were to return f at end of le a p ort at end of le would
defaults to the value returned by current output port
Revised Scheme
Formal syntax and semantics User interface
Questions of user interface generally fall outside of the do
This chapter provides formal descriptions of what has al
main of this rep ort However the following op erations are
ready b een describ ed informally in previous chapters of this
imp ortant enough to deserve description here
rep ort
load lename essential pro cedure
Formal syntax
Filename should b e a string naming an existing le con
This section provides a formal syntax for Scheme written
taining Scheme source co de The load pro cedure reads ex
in an extended BNF The syntax for the entire language
pressions and de nitions from the le and evaluates them
including features which are not essential is given here
sequentially It is unsp eci ed whether the results of the
All spaces in the grammar are for legibility Case is insignif
expressions are printed The load pro cedure do es not
icant for example x A and X a are equivalent hemptyi
a ect the values returned by current input port and
stands for the empty string
current output port Load returns an unsp eci ed value
The following extensions to BNF are used to make the de
Note For p ortability load must op erate on source les Its
scription more concise hthingi means zero or more o ccur
op eration on other kinds of les necessarily varies among im
rences of hthingi and hthingi means at least one hthingi
plementations
Lexical structure
transcript on lename pro cedure
This section describ es how individual tokens identi ers
transcript off pro cedure
numbers etc are formed from sequences of characters
Filename must b e a string naming an output le to b e cre
The following sections describ e how expressions and pro
ated The e ect of transcript on is to op en the named
grams are formed from sequences of tokens
le for output and to cause a transcript of subsequent
hIntertoken spacei may o ccur on either side of any token
interaction b etween the user and the Scheme system to
but not within a token
b e written to the le The transcript is ended by a call
to transcript off which closes the transcript le Only
Tokens which require implicit termination identi ers
one transcript may b e in progress at any time though some
numbers characters and dot may b e terminated by any
implementations may relax this restriction The values re
hdelimiteri but not necessarily by anything else
turned by these pro cedures are unsp eci ed
htokeni hidenti eri j hb o oleani j hnumberi
j hcharacteri j hstringi
j j j j j j j j
hdelimiteri hwhitespacei j j j j
hwhitespacei hspace or newlinei
hcommenti hall subsequent characters up to a
line breaki
hatmospherei hwhitespacei j hcommenti
hintertoken spacei hatmospherei
hidenti eri hinitiali hsubsequenti
j hp eculiar identi eri
hinitiali hletteri j hsp ecial initiali
hletteri a j b j c j j z
hsp ecial initiali j j j j j j j j
j j j j j
hsubsequenti hinitiali j hdigiti
j hsp ecial subsequenti
hdigiti j j j j j j j j j
hsp ecial subsequenti j j
hp eculiar identi eri j
hsyntactic keywordi hexpression keywordi
j else j j define
j unquote j unquote splicing
hexpression keywordi quote j lambda j if
Formal syntax and semantics
j set j begin j cond j and j or j case hsimple datumi hb o oleani j hnumberi
j let j let j letrec j do j delay j hcharacteri j hstringi j hsymboli
j quasiquote hsymboli hidenti eri
hcomp ound datumi hlisti j hvectori
hvariablei h any hidenti eri which isn t hlisti hdatumi j hdatumi hdatumi
also a hsyntactic keywordii j habbreviationi
habbreviationi habbrev pre xi hdatumi
hb o oleani t j f habbrev pre xi j j j
hcharacteri hany characteri hvectori hdatumi
j hcharacter namei
hcharacter namei space j newline
Expressions
hexpressioni hvariablei
hstringi hstring elementi
j hliterali
hstring elementi hany character other than or i
j hpro cedure calli
j j
j hlambda expressioni
j hconditionali
hnumberi hreali j hreali hureali i
j hassignmenti
j hreali hureali i j hreali hreali
j hderived expressioni
hreali hsigni hureali
hureali hureal i j hureal i j hureal i j hureal i
hliterali hquotationi j hself evaluatingi
hself evaluatingi hb o oleani j hnumberi
The following rules for hureal R i huinteger R i and hpre x
j hcharacteri j hstringi
R i should b e replicated for R and
hquotationi hdatumi j quote hdatumi
hureal R i hpre x R i huinteger R i hsu xi
hpro cedure calli hop eratori hop erandi
j hpre x R i huinteger R i huinteger R i hsu xi
hop eratori hexpressioni
j hpre x R i hdigit R i hsu xi
hop erandi hexpressioni
j hpre x R i hdigit R i hdigit R i hsu xi
j hpre x R i hdigit R i hsu xi
hlambda expressioni lambda hformalsi hb o dyi
hpre x R i hradix R i hexactnessi hprecisioni
hformalsi hvariablei j hvariablei
j hradix R i hprecisioni hexactnessi
j hvariablei hvariablei
j hexactnessi hradix R i hprecisioni
hb o dyi hde nitioni hsequencei
j hexactnessi hprecisioni hradix R i
hsequencei hcommandi hexpressioni
j hprecisioni hradix R i hexactnessi
hcommandi hexpressioni
j hprecisioni hexactnessi hradix R i
huinteger R i hdigit R i
hconditionali if htesti hconsequenti halternatei
htesti hexpressioni
hsigni hemptyi j j
hconsequenti hexpressioni
hsu xi hemptyi j e hsigni hdigiti
halternatei hexpressioni j hemptyi
hexactnessi hemptyi j i j e
hprecisioni hemptyi j s j l
hassignmenti set hvariablei hexpressioni
hradix i b
hradix i o
hderived expressioni
hradix i hemptyi j d
cond hcond clausei
hradix i x
j cond hcond clausei else hsequencei
hdigit i j
j case hexpressioni
hdigit i j j j j j j j
hcase clausei
hdigit i hdigiti
j case hexpressioni
hdigit i hdigiti j a j b j c j d j e j f
hcase clausei
else hsequencei
External representations
j and htesti
j or htesti
hDatumi is what the read pro cedure section suc
j let hbinding sp eci hb o dyi
cessfully parses Note that any string which parses as an
j let hvariablei hbinding sp eci hb o dyi
hexpressioni will also parse as a hdatumi
j let hbinding sp eci hb o dyi
hdatumi hsimple datumi j hcomp ound datumi j letrec hbinding sp eci hb o dyi
Revised Scheme
j begin hsequencei hcommand or de nitioni hcommandi j hde nitioni
j do hiteration sp eci hde nitioni define hvariablei hexpressioni
htesti hsequencei j define hvariablei hdef formalsi hb o dyi
hcommandi hdef formalsi hvariablei
j delay hexpressioni j hvariablei hvariablei
j hquasiquotationi
Formal semantics
hcond clausei htesti hsequencei
j else hsequencei
This section provides a formal denotational semantics for
j htesti
the primitive expressions of Scheme and selected built in
j htesti hrecipienti
pro cedures The concepts and notation used here are de
hrecipienti hexpressioni
scrib ed in the notation is summarized b elow
hcase clausei hdatumi hsequencei
h i sequence formation
j else hsequencei
s k k th member of the sequence s based
s length of sequence s
hbinding sp eci hvariablei hexpressioni
s x t concatenation of sequences s and t
hiteration sp eci hvariablei hiniti hstepi
s y k drop the rst k members of sequence s
j hvariablei hiniti
t a b McCarthy conditional if t then a else b
hiniti hexpressioni
x i substitution with x for i
hstepi hexpressioni
x in D injection of x into domain D
x j D pro jection of x to domain D
Quasiquotations
To avoid sp ecial treatment for a top level environment the
The following grammar for quasiquote expressions is not
semantics assumes that environments assign lo cations to all
context free It is presented as a recip e for generating an
variables
in nite number of pro duction rules Imagine a copy of the
The reason that expression continuations take sequences
following rules for D D keeps track of the
of values instead of single values is to simplify the formal
nesting depth
treatment of pro cedure calls and to make it easy to add
multiple return values
hquasiquotationi hquasiquotation i
htemplate i hexpressioni
The order of evaluation within a call is unsp eci ed We
hquasiquotation D i htemplate D i
mimic that here by applying arbitrary p ermutations per
j quasiquote htemplate D i
mute and unpermute which must b e inverses to the ar
htemplate D i hsimple datumi
guments in a call b efore and after they are evaluated
j hlist template D i
This still requires that the order of evaluation b e con
j hvector template D i
stant throughout a program for any given number of ar
j hunquotation D i
guments but it is a closer approximation to the intended
hlist template D i htemplate or splice D i
semantics than a left to right evaluation would b e
j htemplate or splice D i htemplate D i
The storage allo cator new is implementation dependent
j hquasiquotation D i
2
but it must ob ey the following axiom if new L then
j hquasiquotation D i
new j L false
hvector template D i htemplate or splice D i
hunquotation D i htemplate D i
The semantics in this section was translated by machine
j unquote htemplate D i
from an executable version of the semantics written in
htemplate or splice D i htemplate D i
Scheme itself
j hsplicing unquotation D i
hsplicing unquotation D i htemplate D i
Abstract syntax
j unquote splicing htemplate D i
2
K Con constants including quotations
In hquasiquotationis a hlist template D i can sometimes b e
2
I Ide identi ers variables
confused with an hunquotation D i or hsplicing unquotation
2
E Exp expressions
D i The interpretation as an hunquotationi or hsplicing
2
Com Exp commands
unquotation D i takes precedence
Exp K j I j E E
Programs and de nitions
j lambda I E
hprogrami hcommand or de nitioni j lambda I I E
Formal syntax and semantics
j lambda I E
update new j L unspeci ed
j if E E E j if E E
wrong out of memory
j set I E
E lambda I . I E
2
Domain equations
new L
send hnew j L
2
L lo cations
0
I
2
N natural numbers
tievalsrest
0 0 0 0
T ffalse trueg b o oleans
C E E
0
Q symbols
extends I x hI i
H characters
I
R numbers
wrong to o few arguments i in E
E L L pairs
p
E L vectors
v
update new j L unspeci ed
E L strings
s
wrong out of memory
M ffalse true nul l unde ned unspeci edg
miscellaneous
E lambda I E E lambda . I E
2
F L E K C pro cedure values
E if E E E
2
E Q H R E E E M F
p v s
E E single truish E E
expressed values
E E
2
S L E T stores
2
U Ide L environments
E if E E
E E single truish E E 2
C S A command continuations
send unspeci ed
2
K E C expression continuations
A answers
Here and elsewhere any expressed value other than unde ned
X errors
may b e used in place of unspeci ed
E set I E
Semantic functions
E E single assign lookup I
K Con E
E Exp U K C
send unspeci ed
E Exp U K C
E h i
C Com U C C
E E E
De nition of K delib erately omitted
E E single E E h i x
E K send K K
C
E I hold lookup I
C E C
single unde ned
wrong unde ned variable
Auxiliary functions
send
lookup U Ide L
E E E
lookup I I
E permute hE i x E
extends U Ide L U
applicate y
extends
unpermute
I I
extends I I y y
E lambda I E
wrong X C implementation dependent
2
new L
send hnew j L
send E K C
0
I
send h i
0 0 0 0
tievals C E E
single E C K
extends I
single
wrong wrong number of arguments i
wrong wrong number of return values
in E
Revised Scheme
less E K C new S L ferror g implementation dependent
less
hold L K C
2 2
twoarg R R
hold send
send j R j R true false
wrong non numeric argument to
assign L E C C
assign update
add E K C
add
update L E S S
2 2
twoarg R R
update h true i
send j R j R in E
wrong non numeric argument to
tievals L C E C
tievals
car E K C
h i
car
2
2
new L tievals hnew j Li x
onearg E hold j E
p p
y
wrong non pair argument to car
update new j L
cdr E K C similar to car
wrong out of memory
setcar E K C
tievalsrest L C E N C
setcar
tievalsrest
2
twoarg E assign j E
p p
list drop rst
single tievals take rst x h i
send unspeci ed
wrong non pair argument to set car
drop rst l n n l drop rst l y n
eqv E K C
take rst l n n h i hl i x take rst l y n
eqv
2 2
twoarg M M
truish E T
send j M j M true false
truish false nul l false true
2 2
Q Q
send j Q j Q true false
permute Exp Exp implementation dependent
2 2
H H
unpermute E E inverse of permute
send j H j H true false
2 2
R R
applicate E E K C
send j R j R true false
applicate
2 2
E E
p p
2
F j F wrong bad pro cedure
send p p p p
p p true
onearg E K C E K C
false
onearg
j E
p
j E
p
wrong wrong number of arguments
2 2
E E
v v
twoarg E E K C E K C
2 2
E E
s s
twoarg
2 2
F F
send j F j F true false
wrong wrong number of arguments
list E K C
send false
list
apply E K C
send nul l
apply
list y single cons h i
2
twoarg F valueslist h i applicate
wrong bad pro cedure argument to apply
cons E K C
cons
valueslist E K C
2
twoarg new L
valueslist
0 0
2
new L
2
onearg E
p
0
send hnew j L new j Li in E
cdr h i
valueslist
0 0
update new j L
0
wrong out of memory
car h i single h i x
update new j L
nul l h i
wrong out of memory
wrong non list argument to values list
Formal syntax and semantics
thunk lambda hsequencei cwcc E K C call with current continu ation
cwcc
2
elsethunk lambda onearg F
2
cond hmemvi key d thunk new L
applicate
0
else elsethunk hhnew j L i in Ei
where hmemvi is an expression evaluating to the memv pro
update new j L
cedure
unspeci ed
and t
wrong out of memory
and htesti htesti
wrong bad pro cedure argument
and htest i htest i
let x htest i
Derived expression types
thunk lambda and htest i
if x thunk x
This section gives rewrite rules for the derived expression
types By the application of these rules any expression
or f
can b e reduced to a semantically equivalent expression in
or htesti htesti
which only the primitive expression types literal variable
or htest i htest i
let x htest i
call lambda if set o ccur
thunk lambda or htest i
cond htesti hsequencei
if x x thunk
hclause i
if htesti
let hvariable i hinit i
begin hsequencei
hb o dyi
cond hclause i
lambda hvariable i hb o dyi hinit i
cond htesti
let hb o dyi
hclause i
lambda hb o dyi
or htesti cond hclause i
let hvariable i hinit i
hvariable i hinit i
cond htesti hrecipienti
hclause i
hb o dyi
let test result htesti
let hvariable i hinit i
thunk lambda hrecipienti
let hvariable i hinit i
thunk lambda cond hclause i
if test result
hb o dyi
thunk test result
thunk
letrec hvariable i hinit i
cond else hsequencei
hb o dyi
begin hsequencei
let hvariable i hunde nedi
cond
let htemp i hinit i
hsome expression returning an unsp eci ed valuei
set hvariable i htemp i
case hkeyi
d hsequencei
hb o dyi
let key hkeyi
where htemp i htemp i are variables distinct from
thunk lambda hsequencei
hvariable i which do not free o ccur in the original
hiniti expressions and hunde nedi is an expression which
cond hmemvi key d thunk
returns something which when stored in a lo cation makes
it an error to try to obtain the value stored in the lo cation
No such expression is de ned but one is assumed to ex
case hkeyi
ist for the purp oses of this rewrite rule The second let
d hsequencei
expression in the expansion is not strictly necessary but it
serves to preserve the prop erty that the hiniti expressions
else f f
are evaluated in an arbitrary order let key hkeyi
Revised Scheme
NOTES
begin hsequencei
lambda hsequencei
Language changes
The following alternative expansion for begin do es not
This section enumerates the changes that have b een made
make use of the ability to write more than one expres
to Scheme since the Revised revised rep ort was pub
sion in the b o dy of a lambda expression In any case note
lished
that these rules apply only if hsequencei contains no de
The character circum ex is now an extended al
nitions
phab etic character
begin hexpressioni hexpressioni
begin hcommandi hsequencei
The ob jects returned by literal expressions are p ermit
lambda ignore thunk thunk
ted to b e immutable
hcommandi
lambda begin hsequencei
The list to which a rest argument b ecomes b ound must
b e newly allo cated
do hvariable i hinit i hstep i
Do variables are up dated by rebinding rather than by
htesti hsequencei
assignment
hcommand i
letrec hlo opi
New expression type delay
lambda hvariable i
Quasiquote backquote has b een improved in several
if htesti
ways vectors are allowed nesting is allowed and an
begin hsequencei
external syntax for quasiquote expressions analogous
begin hcommand i
to that for quote has b een de ned
hlo opi hstep i
The semantics of de nitions of the form define
hlo opi hinit i
hvariablei hformalsi hb o dyi no longer involves an
where hlo opi is any variable which is distinct from
implicit rec or letrec
hvariable i and which do es not o ccur free in the do
The curried de nition syntax has b een removed
expression
The b o olean constants are now written t and f in
let hvariable i hvariable i hinit i
stead of true and false
hb o dyi
letrec hvariable i lambda hvariable i
The syntax null for the empty list has b een re
hb o dyi
moved
hvariable i
hinit i
New pro cedures boolean procedure and force
delay hexpressioni
The value of eq on numbers and characters is now
hmake promisei lambda hexpressioni
unsp eci ed
where hmake promisei is an expression evaluating to some
Eq and eqv now explicitly p ermit op erationally
pro cedure which b ehaves appropriately with resp ect to the
equivalent pro cedures to b e identi ed
force pro cedure see section
Eqv distinguishes exact numbers from inexact ones
even if they are equal according to
List string and vector indexes must b e exact integers
Atan now admits either one or two arguments
Expression types removed named lambda rec
sequence
Pro cedures
removed append string null substring fill
substring move left substring move right
object hash object unhash
Redundant pro cedure names removed
and
Example
Keywords as variable names EXAMPLE
Some implementations allow arbitrary syntactic keywords
Integrate system integrates the system
to b e used as variable names instead of reserving them
0
as this rep ort would have it But this creates ambiguities y f y y y k n
k n
k
in the interpretation of expressions for example in the
of di erential equations with the metho d of Runge Kutta
following it s not clear whether the expression if
The parameter system derivative is a function that
should b e treated as a pro cedure call or as a conditional
takes a system state a vector of values for the state vari
define if list
ables y y and pro duces a system derivative the val
n
if or
0 0
ues y y The parameter initial state provides
n
an initial system state and h is an initial guess for the
These ambiguities are usually resolved in some consistent
length of the integration step
way within any given implementation but no particular
treatment stands out as b eing clearly sup erior to any other
The value returned by integrate system is an in nite
so these situations were excluded for the purp oses of this
stream of system states
rep ort
define integrate system
lambda system derivative initial state h
Macros
let next runge kutta system derivative h
letrec states
Scheme do es not have any standard facility for de ning
cons initial state
new kinds of expressions
delay map streams next
The ability to alter the syntax of the language creates nu
states
merous problems All current implementations of Scheme
states
have macro facilities that solve those problems to one de
Runge Kutta takes a function f that pro duces a system
gree or another but the solutions are quite di erent and
derivative from a system state Runge Kutta pro duces
it isn t clear at this time which solution is b est or indeed
a function that takes a system state and pro duces a new
whether any of the solutions are truly adequate Rather
system state
than standardize we are encouraging implementations to
continue to exp eriment with di erent solutions
define runge kutta
lambda f h
The main problems with traditional macros are They
let h scale vector h
must b e de ned to the system b efore any co de using them
scale vector
is loaded this is a common source of obscure bugs They
scale vector
are usually global macros can b e made to follow lexical
scale vector
scop e rules but many p eople nd the resulting scop e rules
lambda y
confusing Unless they are written very carefully macros
y is a system state
are vulnerable to inadvertant capture of free variables to
let k h f y
get around this for example macros may have to generate
k h f add vectors y k
co de in which pro cedure values app ear as quoted constants
k h f add vectors y k
k h f add vectors y k
There is a similar problem with syntactic keywords if the
add vectors y
keywords of sp ecial forms are not reserved If keywords are
add vectors k
reserved then either macros introduce new reserved words
k
invalidating old co de or else sp ecial forms de ned by the
k
programmer do not have the same status as sp ecial forms
k
de ned by the system
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
Revised Scheme
BIBLIOGRAPHY AND REFERENCES let ans make vector size
letrec loop
Harold Ab elson and Gerald Jay Sussman with Julie
lambda i
cond i size ans
Sussman Structure and Interpretation of Computer
else
Programs MIT Press Cambridge
vector set ans i proc i
David H Bartley and John C Jensen The imple
loop i
loop
mentation of PC Scheme In Proceedings of the
ACM Conference on Lisp and Functional Program
define add vectors elementwise
ming pages
define scale vector
John Batali Edmund Go o dhue Chris Hanson Howie
lambda s
Shrob e Richard M Stallman and Gerald Jay Suss
elementwise lambda x x s
man The Scheme architecture system and chip
In Proceedings Conference on Advanced Research in
Map streams is analogous to map it applies its rst argu
VLSI pages Paul Pen eld Jr editor Artech
ment a pro cedure to all the elements of its second argu
House Washington Street Dedham MA
ment a stream
William Clinger editor The revised revised rep ort on
define map streams
Scheme or an uncommon Lisp MIT Arti cial Intel
lambda f s
ligence Memo August Also published as
cons f head s
delay map streams f tail s
Computer Science Department Technical Rep ort
Indiana University June
In nite streams are implemented as pairs whose car holds
the rst element of the stream and whose cdr holds a
William Clinger The Scheme compiler An exer
promise to deliver the rest of the stream
cise in denotational semantics In Conference Record
of the ACM Symposium on Lisp and Functional
define head car
Programming pages
define tail
lambda stream force cdr stream
R Kent Dybvig Daniel P Friedman and Christopher
T Haynes Expansion passing style Beyond conven
tional macros In Proceedings of the ACM Con
The following illustrates the use of integrate system in
ference on Lisp and Functional Programming pages
integrating the system
dv v
C C
C i
L
dt R
Michael A Eisenberg Bo chser an integrated Scheme
programming system MIT Lab oratory for Computer
di
L
v L
C
Science Technical Rep ort Octob er
dt
which mo dels a damp ed oscillator
Marc Feeley Deux appro ches a l implantation
du language Scheme M Sc thesis D epartement
define damped oscillator
d Informatique et de Recherche Op erationelle Univer
lambda R L C
sity of Montreal May
lambda state
let Vc vector ref state
Matthias Felleisen Daniel P Friedman Eugene
Il vector ref state
Kohlb ecker and Bruce Duba Reasoning with con
vector Vc R C Il C
tinuations In Proceedings of the Symposium on Logic
Vc L
in Computer Science pages IEEE Computer
So ciety Press Washigton DC
define the states
integrate system
Carol Fessenden William Clinger Daniel P Fried
damped oscillator
man and Christopher Haynes Scheme version
reference manual Indiana University Computer
Science Technical Rep ort February Su
p erceded by
Daniel P Friedman and Matthias Felleisen The Little
LISPer Science Research Asso ciates second edition
Bibliography
Daniel P Friedman Christopher T Haynes and Eu Steven S Muchnick and Uwe F Pleban A semantic
gene Kohlb ecker Programming with continuations In comparison of Lisp and Scheme In Conference Record
Program Transformation and Programming Environ of the Lisp Conference pages The Lisp
ments pages P Pepper editor Springer Conference
Verlag
Peter Naur et al Revised rep ort on the algorith
D Friedman C Haynes E Kohlb ecker and
mic language Algol Communications of the ACM
M Wand Scheme interim reference manual In
January
diana University Computer Science Technical Rep ort
Paul Pen eld Jr Principal values and branch cuts
January
in complex APL In APL Conference Proceed
Daniel P Friedman and Christopher T Haynes Con
ings pages ACM SIGAPL San Francisco
straining control In Proceedings of the Twelfth Annual
September Pro ceedings published as APL Quote
Symposium on Principles of Programming Languages
Quad ACM September
pages ACM January
Kent M Pitman Exceptional situations in Lisp MIT
Christopher T Haynes Daniel P Friedman and
Arti cial Intelligence Lab oratory Working Paper
Mitchell Wand Continuations and coroutines In Con
February
ference Record of the ACM Symposium on Lisp
Kent M Pitman The revised MacLisp manual satur
and Functional Programming pages
day evening edition MIT Arti cial Intelligence Lab
Christopher T Haynes Logic continuations In Pro
oratory Technical Rep ort May
ceedings of the Third International Conference on
Kent M Pitman Sp ecial forms in Lisp In Conference
Logic Programming pages Springer Verlag
Record of the Lisp Conference pages
July
The Lisp Conference
Christopher T Haynes and Daniel P Friedman En
Jonathan A Rees and Norman I Adams IV T A
gines build pro cess abstracions In Conference Record
dialect of Lisp or lambda The ultimate software to ol
of the ACM Symposium on Lisp and Functional
In Conference Record of the ACM Symposium on
Programming pages
Lisp and Functional Programming pages
Peter Henderson Functional Geometry In Confer
ence Record of the ACM Symposium on Lisp and
Jonathan A Rees Norman I Adams IV and James
Functional Programming pages
R Meehan The T manual fourth edition Yale Uni
versity Computer Science Department January
Eugene Edmund Kohlb ecker Jr Syntactic Extensions
in the Programming Language Lisp PhD thesis Indi
John Reynolds De nitional interpreters for higher or
ana University August
der programming languages In ACM Conference Pro
ceedings pages ACM
David Kranz Richard Kelsey Jonathan Rees Paul
Hudak James Philbin and Norman Adams Orbit
Guillermo J Rozas Liar an Algol like compiler for
An optimizing compiler for Scheme In Proceedings of
Scheme S B thesis MIT Department of Electrical
the SIGPLAN Symposium on Compiler Construc
Engineering and Computer Science January
tion pages ACM June Pro ceedings
Brian C Smith Re ection and semantics in a pro
published as SIGPLAN Notices July
cedural language MIT Lab oratory for Computer Sci
Peter Landin A corresp ondence b etween Algol and
ence Technical Rep ort January
Church s lambda notation Part I Communications of
Amitabh Srivastava Don Oxley and Aditya Srivas
the ACM February
tava An other integration of logic and functional pro
Drew McDermott An e cient environment allo cation
gramming In Proceedings of the Symposium on Logic
scheme in an interpreter for a lexically scop ed lisp
Programming pages IEEE
In Conference Record of the Lisp Conference
Richard M Stallman Phantom stacks if you lo ok
pages The Lisp Conference P O Box
to o hard they aren t there MIT Arti cial Intelligence
Redwoo d Estates CA Pro ceedings reprinted by
Memo July
ACM
MIT Department of Electrical Engineering and Com Guy Lewis Steele Jr and Gerald Jay Sussman
puter Science Scheme manual seventh edition Lambda the ultimate imp erative MIT Arti cial In
September telligence Memo March
Revised Scheme
Guy Lewis Steele Jr Lambda the ultimate declara Texas Instruments Inc TI Scheme Language Refer
tive MIT Arti cial Intelligence Memo November ence Manual Preliminary version November
Mitchell Wand Continuation based program transfor
mation strategies Journal of the ACM
Guy Lewis Steele Jr Debunking the exp ensive pro
cedure call myth or pro cedure call implementa
tions considered harmful or lambda the ultimate
Mitchell Wand Continuation based multiprocessing
GOTO In ACM Conference Proceedings pages
In Conference Record of the Lisp Conference
ACM
pages The Lisp Conference
Guy Lewis Steele Jr Macaroni is b etter than
spaghetti In Proceedings of the Symposium on Arti
cial Intel ligence and Programming Languages pages
These pro ceedings were published as a sp ecial
joint issue of SIGPLAN Notices and SIGART
Newsletter August
Guy Lewis Steele Jr Rabbit a compiler for Scheme
MIT Arti cial Intelligence Lab oratory Technical Re
p ort May
Guy Lewis Steele Jr An overview of Common Lisp
In Conference Record of the ACM Symposium on
Lisp and Functional Programming pages
Guy Lewis Steele Jr Common Lisp The Language
Digital Press Burlington MA
Guy Lewis Steele Jr and Gerald Jay Sussman The
revised rep ort on Scheme a dialect of Lisp MIT Ar
ti cial Intelligence Memo January
Guy Lewis Steele Jr and Gerald Jay Sussman The
art of the interpreter or the mo dularity complex
parts zero one and two MIT Arti cial Intelligence
Memo May
Guy Lewis Steele Jr and Gerald Jay Sussman De
sign of a Lisp based pro cessor Communications of the
ACM November
Guy Lewis Steele Jr and Gerald Jay Sussman The
dream of a lifetime a lazy variable extent mechanism
In Conference Record of the Lisp Conference
pages The Lisp Conference
Gerald Jay Sussman and Guy Lewis Steele Jr
Scheme an interpreter for extended lambda calcu
lus MIT Arti cial Intelligence Memo December
Gerald Jay Sussman Jack Holloway Guy Lewis
Steele Jr and Alan Bell Scheme Lisp on a chip
IEEE Computer July
Joseph E Stoy Denotational Semantics The Scott
Strachey Approach to Programming Language Theory
MIT Press Cambridge
Index
ALPHABETIC INDEX OF DEFINITIONS OF CONCEPTS
KEYWORDS AND PROCEDURES
The principal entry for each term pro cedure or keyword is cddddr
listed rst separated from the other entries by a semicolon cdr
ceiling
char integer
char alphabetic
char ci
char ci
char ci
char ci
char ci
char downcase
char lower case
char numeric
char ready
char ready
char upcase
char upper case
char whitespace
char
abs
char
acos
char
and
char
angle
char
append
char
apply
close input port
asin
close output port
assoc
combination
assq
comma
assv
comment
at sign
complex
atan
cond
cons
b
cos
backquote
current input port
begin
current output port
binding
binding construct
d
boolean
define
b ound
delay
denominator
caar
display
caddr
do
cadr
dotted pair
call
call by need
e
call with current continuation
else
call with input file
empty list
call with output file
eof object
call cc
eq
car
case equal
catch equivalence predicate
cdddar eqv
Revised Scheme
error list ref
escap e pro cedure list tail
essential load
even log
exact inexact
macros
exact
magnitude
exactness
make polar
exactness
make promise
exp
make rectangular
expt
make string
f make vector
false map
fix map streams
flo max
floor member
foo memq
for each memv
force min
modulo
gcd
mutation pro cedure
gen counter
gen loser negative
newline
heur
nil
not
i
null
identi er
number
if
number string
imag part
number
improp er list
numerator
inexact exact
inexact
o
initial environment
odd
input port
open input file
int
open output file
integer char
op erationally equivalent
integer
or
integrate system
output port
internal de nition
pair
keyword
pair
polar
l
positive
lambda
predicate
lambda expression
pro cedure call
last pair
procedure
lazy evaluation
promise
lcm
length
quasiquote
let
quote
let
quotient
letrec
list radix
list string rat
list vector rational
Index
rationalize unbound
read unquote
unquote splicing
read char
real part
valid indexes
real
values list
rect
variable
region
vector
remainder
vector list
return
vector fill
reverse
vector length
round
vector ref
runge kutta
vector set
vector
s
sci
whitespace
sequence
with input from file
set
with output to file
set car
write
set cdr
write char
sin
sqrt
x
string list
string number
zero
string symbol
string append
string ci
string ci
string ci
string ci
string ci
string copy
string fill
string length
string ref
string set
string
string
string
string
string
string
substring
symbol string
symbol
syntactic keyword
t
t
tan
token
top level environment
transcript off
transcript on
true
truncate