perm filename LSPDOC[206,LSP] blob
sn#271252 filedate 1977-03-23 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00038 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00004 00002 .require "book.pub[let,jmc]" source_file
C00011 00003 .INSERT CONTENTS
C00015 00004 .ss Atoms.
C00017 00005 .ss List structures.
C00023 00006 .ss S-expressions.
C00032 00007 .ss The basic functions and predicates of LISP.
C00045 00008 .ss Conditional expressions.
C00052 00009 .ss Boolean expressions.
C00056 00010 .ss Recursive function definitions.
C00074 00011 .ss Lambda expressions and functions with functions as arguments.
C00086 00012 .ss Label.
C00089 00013 .ss Numerical computation.
C00096 00014 .s HOW TO WRITE RECURSIVE FUNCTION DEFINITIONS
C00102 00015 .ss Simple list recursion.
C00111 00016 .ss Simple S-expression recursion.
C00114 00017 .ss Other structural recursions.
C00117 00018 .ss Tree search recursion.
C00132 00019 .ss Game trees.
C00144 00020 .s COMPILING IN LISP
C00148 00021 .ss Some facts about the PDP-10.
C00154 00022 .ss Code produced by LISP compilers.
C00155 00023 .BEGIN VERBATIM SELECT 1
C00158 00024 Note that all three compilers produce the same first line of
C00161 00025 .ss Listings of LCOM0 and LCOM4
C00162 00026 .SKIP TO LINE 1
C00163 00027 .SKIP TO LINE 1
C00176 00028 .SKIP TO LINE 1
C00177 00029 .s COMPUTABILITY
C00202 00030 .s PROVING LISP PROGRAMS CORRECT
C00205 00031 .ss First order logic with conditional forms and lambda-expressions.
C00217 00032 .ss Conditional forms.
C00223 00033 .ss Lambda-expressions.
C00225 00034 .ss Algebraic axioms for S-expressions and lists.
C00229 00035 .ss Axiom schemas of induction.
C00232 00036 .ss Proofs by structural induction.
C00235 00037 .ss First Order Theory of Partial Functions
C00236 00038 .BACK
C00237 ENDMK
C⊗;
.require "book.pub[let,jmc]" source_file;
.font B "ms25";
.font C "grfx25"
.font D "grfx35"
.PORTION TITLEPAGE
.sname ← SSNAME ← NULL
.skip 20
.cb RECURSIVE PROGRAMMING IN LISP
.SKIP 4
.CB by John McCarthy
.skip 20
.cb Stanford University
.skip 5
.cb Copyright %Bc%* 1977 by John McCarthy
.skip 10
.once center
.TURN ON "{}"
This version printed at {time} on {date}.
.TURN OFF "{}∂[]"
.INSERT CONTENTS
.PORTION MAINPORTION
.sname ← SSNAME ← NULL
.s INTRODUCTION TO LISP
.ss Lists.
Symbolic information in LISP is expressed by S-expressions
and these are represented in the memory of the computer by list
structures. Before giving formal definitions, we shall give some
examples.
The most common form of S-expression is the list, and here
are some lists:
.NOFILL
The list
(A B C E)
%1has four elements.
The list
(A B (C D) E)
%1has four elements one of which is itself a list.
The list
(A)
%1has one element.
The list
((A B C D))
%1also has one element which itself is a list.
The list
()
%1has no elements; it is also written
NIL.
%1The list
(PLUS X Y)
%1has three elements and may be used to represent the expression
%2x + %2y.
%1The list
(PLUS (TIMES X Y) X 3)
%1has four elements and may be used to represent the expression
%2xy + %2x + 3.
%1The list
(EXIST X (ALL Y (IMPLIES (P X) (P Y))))
%1may be used to represent the logical expression
(∃%2x)(∀%2y).%2P(%2x)⊃%2P(%2y).
%1The list
(INTEGRAL 0 ∞ (TIMES (EXP (TIMES I X Y)) (F X)) X)
%1may be used to represent the expression
.TURN ON "↑↓[]&_∪";
%8I%60%5↑∞%2e%5ixy%2f(%2x)%2dx.
.TURN OFF
%1The list
((A B) (B A C D) (C B D E) (D B C E) (E C D F) (F E))
%1
.FILL
is used to represent the network of Figure 1 according to a scheme
whereby there is a sublist for each vertex consisting of the vertex
itself followed by the vertices to which it is connected.
.begin verbatim;
C
B E
A F
D
Figure 1
.END
The elements of a list are surrounded by parentheses and
separated by spaces. A list may have any number of terms and any of
these terms may themselves be lists. In this case, the spaces
surrounding a sublist may be omitted, and extra spaces between
elements of a list are allowed. Thus the lists
(A B(C D) E)
%1and
(A B (C D) E)
%1are regarded as the same.
.ss Atoms.
The expressions A, B, X, Y, 3, PLUS, %1and ALL%1 occurring in the
above lists are called atoms. In general, an atom is expressed by a
sequence of capital letters, digits, and special characters with
certain exclusions. The exclusions are <space>, <carriage return>,
and the other non-printing characters, and also the parentheses,
brackets, semi-colon, and comma. Numbers are expressed as signed
decimal or octal numbers, the exact convention depending on the
implementation. Floating point numbers are written with decimal
points and, when appropriate, an exponent notation depending on the
implementation. The reader should consult the programmer's manual
for the LISP implementation he intends to use.
Some examples of atoms are
.NOFILL
THE-LAST-TRUMP
A307B
345
3.14159,
.FILL
%1and from these we can form lists like
((345 3.14159 -47) A307B THE-LAST-TRUMP -45.21).
%1
.ss List structures.
Lists are represented in the memory of the computer by list
structures. A list structure is a collection of memory words each
of which is divided into two parts, and each part is capable of
containing an address in memory. The two parts are called are called
the a-part and the d-part. There is one computer word for each
element of the list, and the a-part of the word contains the address
of the list or atom representing the element, and the d-part contains
the address of the word representing the next element of the list.
If the list element is itself a list, then, of course, the address of
the first word of its list structure is given in the a-part of the
word representing that element. A diagram shows this more clearly
than words, and the list structure corresponding to the list (PLUS
(TIMES X Y) X 3)%1 which may represent the expression %2xy + %2x + 3 is
shown in figure 2.
.begin verbatim; select c; preface 0;
⊂αααπααα⊃ ⊂αααπααα⊃ ⊂αααπααα⊃ ⊂αααπααα⊃
ααααα→~ ~ εαααα→~ ~ εαααα→~ ~ εαααα→~ ~ εααααααα⊃
%απα∀ααα$ %απα∀ααα$ %απα∀ααα$ %απα∀ααα$ ~
↓ ~ ~ ↓ ~
PLUS ~ ~ 3 ~
~ ⊂αααπααα⊃ ~ ⊂αααπααα⊃ ⊂αααπααα⊃ ~
%α→~ ~ εααβα→~ ~ εαααα→~ ~ ~ ~
%απα∀ααα$ ~ %απα∀ααα$ %απα∀απα$ ~
↓ ~ ~ ↓ ~ ~
TIMES %απαα$ Y %ααπα$
↓ ↓
X NIL
Figure 2.
.END
Atoms are represented by the addresses of their property
lists which are list structures of a special kind depending on the
implementation. (In some implementations, the first word of a
property list is in a special area of memory, in others the first word
is distinguished by sign, in still others it has a special a-part.
For basic LISP programming, it is enough to know that atoms are
distinguishable from other list structures by a predicate called
%3at%1.)
The last word of a list cannot have the address of a next
word in its d-part since there isn't any next word, so it has the
address of a special atom called NIL%1.
A program referrs to a list by the address of its first
element. According to this convention, we see that the a-part of a
list word is the list element and the d-part is a pointer to a
sublist formed by deleting the first element. Thus the first word of
the list structure of figure 2 contains a pointer to the list
structure representing the atom PLUS%1, while its d-part points to the
list ((TIMES X Y) X 3)%1. The second word contains the list structure
representing (TIMES X Y)%1 in its a-part and the list structure
representing (X 3)%1 in its d-part. The last word points to the atom
3%1 in its a-part and has a pointer to the atom NIL%1 in its d-part.
This is consistent with the convention that NIL%1 represents the null
list.
.ss S-expressions.
When we examine the way list structures represent lists we
see a curious asymmetry. Namely, the a-part of a list word can
contain an atom or a list, but the d-part can contain only a list or
the special atom NIL%1. This restriction is quite unnatural from the
computing point of view, and we shall allow arbitrary atoms to
inhabit the d-parts of words, but then we must generalize the way
list structures are expressed as character strings. To do this, we
introduce the notion of S-expression.
An S-expression is either an atom or a pair of S-expressions
separated by "_._" and surrounded by parentheses. In BNF, we can
write
<S-expression> ::= <atom> | (<S-expression> . <S-expression>).
Examples of S-expressions are
.NOFILL
A
(A . B)
(A . (B . A))
(PLUS (X . (Y . NIL)))
(3 . 3.4)
.FILL
%1The spaces around the . may be omitted when this will not cause
confusion. The only possible confusion is of the dot separator with
a decimal point in numbers. Thus, in the above cases, we may write
(A.B), (A.(B.A))%1, and (PLUS.(X.(Y.NIL)))%1, but if we wrote (3.3.4)%1
confusion would result.
In the memory of a computer, an S-expression is represented
by the address of a word whose a-part contains the first element of
the pair and whose d-part contains the second element of the pair.
Thus, the S-expressions (A.B), (A.(B.A))%1, and (PLUS.(X.(Y.NIL))) %1
are represented by the list structures of figure 3.
.begin verbatim; select c; preface 0;
⊂αααπααα⊃ ⊂αααπααα⊃ ⊂αααπααα⊃
~ ~ ~ ~ ~ εαααα→~ ~ ~
%απα∀απα$ %απα∀ααα$ %απα∀απα$
↓ ↓ ~ ↓ ~
A B ~ B ~
~ ~
~ ~
%ααααααααπαααααααα$
↓
A
⊂αααπααα⊃ ⊂αααπααα⊃ ⊂αααπααα⊃
~ ~ εαααα→~ ~ εαααα→~ ~ ~
%απα∀ααα$ %απα∀ααα$ %απα∀απα$
↓ ↓ ↓ ↓
PLUS X Y NIL
Figure 3.
.END
Note that the list (PLUS X Y)%1 and the S-expression
(PLUS . (X . (Y . NIL)))%1 are represented in memory by the same list
structure. The simplest way to treat this is to regard S-expressions
as primary and lists as abbreviations for certain S-expressions,
namely those that never have any atom but NIL as the second part of
a pair. In giving input to LISP, either the list form or the
S-expression form may be used for lists. On output, LISP will print
a list structure as a list as far as it can, otherwise as an
S-expression. Thus, some list structures will be printed in a mixed
notation, e.g. ((A . B) (C . D) (3))%1.
In general, the list (%2a b . . . z)%1 may be regarded
as an abbreviation of the S-expression (%2a . (%2b . (%2c .
(... (%2z . NIL) ...)))%1.
.SKIP 2
.GROUP
Exercises
1. If we represent sums and products as indicated above and
.APART
use (MINUS X), (QUOTIENT X Y)%1, and (POWER X Y)%1 as representations
of -%2x%1, %2x/%2y%1, and %2x%5y%1 respectively, then
a. What do the lists
(QUOTIENT 2 (POWER (PLUS X (MINUS Y)) 3))
%1and
(PLUS -2 (MINUS 2) (TIMES X (POWER Y 3.3)))
%1represent?
b. How are the expressions %2xyz+3(%2u+%2v)%5-3%1 and
(%2xy-%2yx)/(%2xy+%2yx)%1 to be represented?
2. In the above mentioned graph notation, what graph is
represented by the list
((A D E F)(B D E F)(C D E F)(D A B C)(E A B C)(F A B C))?
%13. Write the list ((PLUS (TIMES X Y) X 3)%1 as an S-expression.
This is sometimes referred to as "dot-notation".
4. Write the following S-expressions in list notation to
whatever extent is possible:
.NOFILL
a. (A . NIL)
b. (A . B)
c. ((A . NIL) . B)
d. ((A . B) . ((C . D) . NIL).
.FILL
5. Prove that the number of "shapes" of S-expressions
with %2n%1 atoms is %2(2n - 2)!/(n!(n - 1)!)%1. (The two shapes
of S-expressions with three atoms are (A.(B.C))%1 and ((A.B).C)%1).
6. The above result can also be obtained by writing %2S_=_A_+_S%3x%2S%1
as an "equation" satisfied by the set of S-expressions with the
interpretation that an S-expression is either an atom or a pair of
S-expressions. The next step is to regard the equation as the
quadratic %2S%52%2_-_S_+_A = 0%1, solve it by the quadratic formula
choosing the minus sign for the square root. Then the radical is
regarded as the 1/2 power and expanded by the binomial theorem. Manipulating
the binomial coefficients yields the above formula as the coefficient
of %2A%5n%1 in the expansion. Why does this somewhat ill-motivated
and irregular procedure give the right answer?
The last two exercises are unconnected with subsequent
material in these notes.
.ss The basic functions and predicates of LISP.
The main form of LISP program is the LISP function, and
LISP functions are built up from basic LISP functions, predicates,
variables and constants.
An expression with variables in it whose value depends on the values
of the variables is called a %2form%1. LISP functions are constructed
from LISP %2forms%1 which are written in two languages - %2publication
language%1 and %2internal language%1. Publication language is easier
for people to read and write, but publication language programs are
not S-expressions, and therefore it is not easy to write LISP
programs to generate, interpret or compile other LISP programs
when these programs are in publication language. The internal
language programs are S-expressions and are harder for a person
to read and write, but it is easier for a person to write a program
to manipulate %2object programs%1 when the object programs are
in internal language.
Besides all that, most LISP implementations accept only internal
language programs rather than some form of publication language.
Just as numerical computer programs are based on the four
arithmetic operations of addition subtraction, multiplication, and
division, and the arthmetic predicates of equality and comparison, so
symbolic computation has its basic predicates and functions. LISP
has three basic functions and two predicates. Each will be explained
first in its publication form, and then the internal form will be
given.
First, we have two functions %3a%1 and %3d%1. %3a %2x%1 is
the a-part of the S-expression x%1. Thus, %3a(A . B) = A%1,
and %3a((A . B) . A) = (A . B)%1. %3d %2x%1 is the d-part of
the S-expression %2x%1. Thus %3d(A . B) = B%1, and
%3d((A . B) . A) = A%1. %3a %2x%1 and %3d %2x%1 are undefined in
basic LISP when %2x%1 is an atom, but they may have a meaning in some
implementations. The internal forms of %3a %2x%1 and %3d %2x%1 are
(CAR X)%1 and (CDR X)%1 respectively. The names CAR%1 and CDR%1
stood for "contents of the address part of register" and "contents
of the decrement part of register" in a 1957 precursor of LISP
projected for the IBM 704 computer. The names have persisted for
lack of a clearly preferable alternative.
A certain ambiguity arises when we want to use S-expression
constants in the internal language. Namely, the S-expression A%1
could either stand for itself or stand for a variable A%1 whose
value is wanted. Even (CAR X)%1 is ambiguous since sometime the
S-expression (CAR X)%1 itself must be referred to and not the
result of evaluating it. Internal language avoids the ambiguity
by using (QUOTE %2e)%1 to stand for the S-exression %2e%1.
Thus the publication language form %3a (A B)%1 corresponds
to the internal language form (CAR (QUOTE (A.B)))%1 and both
have the value A%1 obtained by taking the %3a%1-part of the
S-expression (A.B)%1.
Since lists are a particular kind of S-expression, the
meanings of %3a%1 and %3d%1 for lists are also determined by the above
definitions. Thus, we have
%3a(A B C D) =%3 A
%1and
%3d(A B C D) = (B C D),
%1and we see that, in general, %3a %2x%1 is the first element of the list
%2x%1, and %3d %2x%1 is the rest of the list, deleting that first element.
Notice that for S-expressions, the definitions of %3a %2x%1 and
%3d %2x%1 are symmetrical, while for lists the symmetry is lost. This is
because of the unsymmetrical nature of the convention that defines
lists in terms of S-expressions.
Notice, further, our convention of writing %3a%1 and %3d%1__
without brackets surrounding the argument. Also, we use lower case
letters for our function names and for variables, reserving the upper
case letters for the S-expressions themselves.
The third function %2cons[%2x, %2y]%1 forms the S-expression whose
a-part and d-part are %2x%1 and %2y%1 respectively. Thus
%2cons[(A.B), A] = ((A.B).A).
%1We see that for lists, %2cons%1 is a prefixing operation.
Namely, %2cons[%2x, %2y]%1 is the list obtained by putting the element %2x%1__
onto the front of the list %2y%1. Thus
%2cons[A, (B C D E)] = (A B C D E).
%1If we want %2cons[%2x, %2y]%1 to be a list, then %2y%1 must be a list
(possibly the null list NIL%1), and %2x%1 must be a list or an atom. In
the case of S-expressions in general, there is no restriction on the
arguments of %2cons%1. Usually, we shall write %2x . y%1 instead of
%2cons[%2x, %2y]%1 since this is briefer.
The internal language form of %2cons[x,y]%1 is (CONS X Y)%1. The name
"CONS" comes from "construct" referring to the way %2cons[x,y]%1 is constructed
from the free storage list.
The first predicate is %3at %2x%1. %3at %2x%1 is true if the
S-expression %2x%1 is atomic and false otherwise.
The internal form is (ATOM X)%1.
The second predicate is equality of atomic symbols written
%2x %3eq %2y%1. Equality of S-expressions is tested by a program based on
%3eq%1. Actually %3eq%1 does a bit more than testing equality of atoms.
Namely, %2x %3eq %2y%1 is true if and only if the addresses of the first
words of the S-expressions %2x%1 and %2y%1 are equal. Therefore, if
%2x %3eq %2y%1, then %2x%1 and %2y%1 are certainly the same S-expression since
they are represented by the same structure in memory. The converse
is false because there is no guarantee in general that the same
S-expression is not represented by two different list structures. In
the particular case where at least one of the S-expressions is known
to be an atom, this guarantee can be given, because LISP represents
atoms uniquely in memory.
The internal form is (EQ X Y)%1.
The above are all the basic functions of LISP; all other LISP
functions can be built from them using recursive conditional
expressions as will shortly be explained. However, the use of certain
abbreviations makes LISP programs easier to write and understand.
%3n %2x%1 is an abbreviation for %2x %3eq NIL%1. It rates a special
notation because NIL%1 plays the same role among lists that zero
plays among numbers, and list variables often have to be tested to
see if their value is NIL%1.
Its internal form is (NULL X)%1.
The notation %2list[%2x1, x2, . . . , xn]%1 is used to denote the
composition of %2cons%1's that forms a list from its elements. Thus
%2list[x, y, z] = cons[x, cons[y, cons[z, NIL%2]]]%1
and forms a list of three elements out of the quantities %2x, y, %1 and
%2z%1. We often write <%2x y . . . z>%1 instead of %2list[%2x, y, .
. . , z]%1 when this will not cause confusion. The experienced
implementer of programming languages will expect that since %2list%1 has a
variable number of arguments, its implementation will pose problems. He
will be right. The internal form of %2<x y ... z>%1 is (LIST X Y ...
Z)%1.
Compositions of %3a%1 and %3d%1 are written by concatenating the
letters %3a%1 and %3d%1. Thus, it is easily seen that %3ad %2x%1 denotes
the second element of the list %2x%1, and %3add %2x%1 denotes the third
element of that list. The internal forms of these functions are CADR%1
and CADDR%1 respectively.
Besides the basic functions of LISP, there will be user-defined
functions. We haven't given the mechanism function definition yet, but
suppose a function %2subst%1 taking three arguments has been defined.
It may be used in forms like %2subst[x,y,z]%1 having internal form
(SUBST X Y Z)%1.
As in other programming languages and in mathematics generally,
new forms can be constructed by applying the functions and predicates
to other forms and not just to variables and constants. Thus we have
forms like %2subst[x,y,%3a z%2] . subst[x,y,%3d %2z]%1 involving
a user defined function %2subst%1. Its internal form is
(CONS (SUBST X Y (CAR Z)) (SUBST X Y (CDR Z)))%1.
.ss Conditional expressions.
When the form that represents a function depends on whether a
certain predicate is true of the arguments, conditional expressions.
are used. The conditional expression (more properly %2conditional form%1)
%3if %2p %3then %2a %3else %2b
%1is evaluated as follows: First %2p%1 is evaluated and determined to be
true or false. If %2p%1 is true, then %2a%1 is evaluated and its value
is the value of the expression. If %2p%1 is false, then %2b%1 is
evaluated and its value is the value of the expression. Note that if
%2p%1 is true, %2b%1 is never evaluated, and if %2p%1 is false, then %2a%1 is
never evaluated. The importance of this will be explained later. A
familiar function that can be written conveniently using conditional
expressions is the absolute value of a real number. We have
|%2x| = %3if %2x<0 %3then -%2x %3else %2x.
%1A more general form of conditional expression is
%3if %2p %3then %2a %3else if %2q %3then %2b . . . %3else if %2s %3then %2d %3else %2e.
%1There can be any number of terms; the value is determined by
evaluating the propositional terms %2p, q%1, etc. until a true term is
found; the value is then that of the term following the next %3then%1.
If none of the propositional terms is true, then the value is that of
the term following the %3else%1.
The function graphed in figure 4 is described by the equation
%2tri[%2x] = %3if %2x<-1 %3then 0 %3else if %2x<0 %3then 1+%2x
%3else if %2x<1 %3then 1-%2x %3else 0.
.BEGIN verbatim; group;
(0,1)
(-1,0) (1,0)
Figure 4.
.END
The internal form of conditional forms is made in a more
regular way than the publication form; the publication form was
altered to conform to ALGOL. We write
(COND (p1 e1) (p2 e2) ... (pn en))%1
with any number of terms. Its value is determined by evaluating the
p%1's successively until one is found which is true. Then the
corresponding e%1 is taken as the value of the conditional expression.
If none of the p%1's is true, then the value of the conditional
expression is undefined. Thus all the e%1's are treated the same
which makes programs for interpreting or compiling conditional
expressions easier to write. Putting parentheses around each p-e%1
pair was probably a mistake in the design of LISP, because it
unnecessarily increases the number of right parentheses. It should
have been (COND p1 e1 p2 e2 ... pn en)%1, but such a change should
be made now only as part of a massive improvement.
Conditional expressions may be compounded with functions to
get forms like
%3if n %2x %3then NIL %3else %3a %2x . append[%3d %2x,y]%1
involving a yet to be defined function %2append%1. The internal
form of this is
(COND ((NULL X) NIL) (T (CONS (CAR X) (APPEND (CDR X) Y))))%1.
One would normally have expected to write (QUOTE NIL)%1 and
(QUOTE T)%1, but there is a special convention that NIL%1
and T%1 may be written without QUOTE%1. This means that
these symbols cannot be used as variables. The value of T%1
is the propositional constant %3true%1, i.e. it is always true
so that it is impossible to "fall off the end" of a conditional
expression with T%1 as its last p%1. It is the translation
into internal form of the final %3else%1 of a conditional
expression.
.ss Boolean expressions.
In making up the propositional parts of conditional
expressions, it is often necessary to combine elementary
propositional expressions using the Boolean operators and, or, and
not. We use the symbols ∧%1, ∨%1, and ¬%1 respectively, for these
operators.
The Boolean operators may be described simply by listing the
values of the elementary Boolean expressions for each case of the
arguments. Thus we have:
.NOFILL
T∧T = T
T∧F = F
F∧T = F
F∧F = F
T∨T = T
T∨F = T
F∨T = T
F∨F = F
¬T = F
¬F = T.
.FILL
Since both ∧ and ∨ are associative we can write Boolean forms
like %2p1∧p2∧p3%1 without worrying about grouping. The internal forms
are (AND p1 p2 ... pn)%1, (OR p1 p2 ... pn)%1 and (NOT p)%1.
It is also customary to use NIL%1 instead of F%1, because in most
LISP systems both Boolean %3false%1 and the null list are represented
by the number 0. This makes certain tests run faster, but introduces
so many complications in implementation that it was certainly a bad
idea.
%1The Boolean operators can be described by conditional
expressions in the following way:
.NOFILL
%2p∧%2q = %3if %2p %3then %2q %3else F
%2p∨%2q = %3if %2p %3then T %3else %2q
¬%2p = %3if %2p %3then F %3else T.
.FILL
%1Note that if %2p%1 is false %2p∧%2q%1 is false independent of the value of
%2q%1, and likewise if %2p%1 is true, then %2p∨%2q%1 is true independent of
%2q%1. If %2p%1 has been evaluated and found to be false, then %2q%1 does
not have to be evaluated at all to find the value of %2p∧%2q%1, and, in
fact, LISP does not evaluate %2q%1 in this case. Similarly, %2q%1 is not
evaluated in evaluating %2p∨%2q%1 if %2p%1 is true. This procedure is in
accordance with the above conditional expression descriptions of the
Boolean operators. The importance of this convention which contrasts
with that of ALGOL 60, will be apparent later when we discuss
recursive definitions of functions and predicates.
Boolean expressions can be combined with functions and conditional
exrpessions to get forms like
%3 if n %2x ∨ %3n d %2 x %3then %2x %3else %3a %2x . alt %3dd %2x%1
the internal form of which is
(COND ((OR (NULL X) (NULL (CDR X))) X) (T (CONS (CAR X) (ALT (CDDR
X)))))%1.
.ss Recursive function definitions.
In such languages as FORTRAN and Algol, computer progrrams
are expressed in the main as sequences of assignment statements and
conditional %3go to%1's. In LISP, programs are mainly expressed in the
form of recursively defined functions. We begin with an example.
The function %2alt[%2x]%1 gives a list whose elements are
alternate elements of the list %2x%1 beginning with the first. Thus
.NOFILL
%2alt[(A B C D E)] = (A C E),
%2alt[(((A B) (C D))] = ((A B)),
%2alt[(A)] = (A),
%1and
%2alt[NIL] = NIL.
.fill
In LISP %2alt%1 is defined by the equation
%2alt[x] ← %3if n %2x ∨ %3n d %2x %3then %2x %3else a %2x . alt[%3dd
%2x]%1.
In case you need a review of our precedence conventions, fully bracketed
it looks like
%2alt[x] ← %3if %2[%3n%2[x] ∨ %3n%2[%3d%2[x]]] %3then%2 x %3else%2
[%3a%2[x] . alt[%3dd%2[x]]]%1.
This definition uses no ways of forming expressions that we
haven't introduced previously, but it uses the function %2alt%1 in
the right hand side of its own defining equation. To get the value of
%2alt x%1 for some particular list, we evaluate the right hand side of
the definition substituting that list for %2x%1 whenever %2x%1 is
encountered and re-evaluating the right hand side with a new %2x%1
whenever a form %2alt e%1 is encountered.
This process is best explained by using it to
evaluate some expressions.
Consider evaluating %2alt[NIL]%1.
We do it by evaluating the expression to the right of the ←%1 sign
remembering that the variable %2x%1 has the value NIL%1. A conditional
expression is evaluated by first evaluating the first propositional
term ¬ in this case %3n %2x ∨ %3n d %2x%1. This expression is a disjunction and
in its evaluation we first evaluate the first disjunct, namely %3n_%2x%1.
Since %2x = NIL, %3n %2x%1 is true, the disjunction is true, and the value
of the conditional expression is the value of the term after the
first true propositiional term. The term is %2x%1, and its value is
NIL%1, so the value of %2alt[NIL]%1 is NIL%1 as stated above. Obeying
the rules about the order of evaluation of terms in conditional and
Boolean expressions is important in this case since if we didn't obey
these rules, we might find ourselves trying to evaluate %3n d %2x%1 or
%2alt[%3dd %2x]%1, and since %2x%1 is NIL%1, neither of these has a value. An
attempt to evaluate them in LISP would give rise to an error return.
As a second example, consider %2alt[(A B)]%1. Since neither
%3n %2x%1 nor %3n d %2x%1 is true in this case, the disjunction %3n %2x ∨ %3n d %2x
%1is false and the value of the expression is the value of
%3a %2x . alt[%3dd %2x]. %3a %2x%1 has the value A%1, and %3dd %2x%1 has the value
NIL%1, so we must now evaluate %2alt[NIL]%1, and we already know that the
value of this expression is NIL%1. Therefore, the value of
%2alt[(A B)]%1 is that of A.NIL%1 which is (A)%1.
We can describe this evaluation in a less wordy way by
writing
.NOFILL
%2alt[(A B)] = %3if n(A B) ∨ %3n d(A B) %3then (A B) %3else
%3a(A B).%2alt[%3dd(A B)]
= %3if NIL ∨ %3n d(A B) %3then (A B) %3else
%3a(A B).%2alt[%3dd(A B)]
= %3if NIL %3then (A B) %3else
%3a(A B).%2alt[%3dd(A B)]
= %3a(A B).%2alt[%3dd(A B)]
= A.%2alt[NIL]
= A.[%3if nNIL ∨ %3ndNIL %3then NIL %3else
aNIL.%2alt[%3ddNIL]]
= A.[%3if T ∨ %3ndNIL %3then NIL %3else
aNIL.%2alt[%3ddNIL]]
= A.[%3if T %3then NIL %3else
aNIL.%2alt[%3ddNIL]]
= A.[NIL]
= (A).
.FILL
%1This is still very long-winded, and now that the reader
understands the order of evaluation of conditional and Boolean
expressions, we can proceed more briefly to evaluate
%2alt[(A B C D E)] = A.%2alt[(C D E)]
= A.[C.%2alt[(E)]]
= A.[C.(E)]
= (A C E).
%1Here are three more examples of recursive functions and their
application: We define %2last%1 by
%2last[%2x] ← %3if n d %2x %3then a %2x %3else %2last[%3d %2x],
%1and we compute
%2last[(A B C)] = %3if nd(A B C) %3then a(A B C) %3else %2last[%3d(A B C)]
= %2last[(B C)]
= %2last[(C)]
= C.
%1Clearly %2last%1 computes the last element of a list. %2last[NIL]%1 is
undefined in the LISP language; the result of trying to compute it
may be an error message or may be some random result depending on the
implementation.
The function %2subst%1 is defined by
%2subst[%2x, y, z] ← %3if at %2z %3then [%3if %2z %3eq %2y %3then %2x %3else %2z]
%3else %2subst[x,y,%3a %2z].subst[x,y,%3d %2z]%1.
We have
%2subst[(A.B), X, ((X.A).X)] =
= %2subst[(A.B), X, (X.A)]%2subst[(A.B), X, X]
= [%2subst[(A.B), X, X].%2subst[(A.B), X, A]].(A.B)
= [[(A.B)].A].(A.B)]
= (((A.B).A).(A.B)).
%2subst%1 computes the result of substituting the S-expression %2x%1 for
the atom %2y%1 in the S-expression %2z%1. This operation is important
in all kinds of symbolic computation.
The function %2append[x,y]%1 which gives the concatenation of the lists
%2x%1 and %2x%1 is also important. It is also denoted by the
infixed expression %2x*%2y%1 since it
is associative. We have the examples
(A B C)*(D E F) = (A B C D E F),
%1and
NIL*(A B) = (A B) = (A B)*NIL,
%1and the formal definition
%2x*%2y ← %3if n %2x %3then %2y %3else [%3a %2x].[[%3d %2x]*%2y].
%1The Boolean operations can also be used in making recursive
definitions provided we observe the order of evaluation of
constituents. First, we define a predicate %2equal%1 by
%2equal[x,y] ← x %3eq %2y ∨ [¬%3at %2x ∧ ¬%3at %2y ∧
equal[%3a %2x,%3a %2y] ∧ equal[%3d %2x, %3d %2y]]%1.
%2equal[x,y]%1 is true if and only if %2x%1 and %2y%1 are the same
S-expression, and the use of this predicate makes up for the fact
that the basic predicate %3eq%1 is guaranteed to test equality only
when one of the operands is known to be an atom. We shall also use
the infixes =%1 and ≠%1.
Membership of an S-expression %2x%1 in a list %2y%1 is tested by
%2member[%2x, y] ← ¬%3n %2y ∧ [[%2x = %3a %2y] ∨ %2member[%2x, %3d %2y]].
%1This relation is also denoted by %2x_ε_y%1. Here are some computations:
.NOFILL
%2member[B, (A B)] = ¬%3n (A B) ∧ [[B = %3a (A B)]
∨ %2member[B, %3d (A B)]],
= %2member[B, (B)]
= T.
.FILL
%1Sometimes a function is defined with the help of auxiliary
functions. Thus, the reverse of a list %2x%1 , (e.g. %2reverse[(A B C
D)] = (D C B A)%1), is given by
.NOFILL
%2reverse[%2x] ← %2rev[%2x, NIL]
%1where
%2rev[%2x, y] ← %3if n %2x %3then %2y %3else %2rev[%3d %2x, [%3a %2x].%2y].
%1A computation is
%2reverse[(A B C)] = %2rev[(A B C), NIL]
= %2rev[(B C), (A)]
= %2rev[(C), (B A)]
= %2rev[NIL, (C B A)]
= (C B A).
%1A more elaborate example of recursive definition is given by
%2flatten[%2x] ← %2flat[%2x, NIL]
%2flat[%2x, y] ← %3if at %2x %3then %2x.y
%3else %2flat[%3a %2x, flat[%3d %2x, y]].
%1We have
%2flatten[((A.B).C)] = %2flat[((A.B).C), NIL]
= %2flat[(A.B), %2flat[C, NIL]]
= %2flat[(A.B), (C)]
= %2flat[A, %2flat[B, (C)]]
= %2flat[A, (B C)]
= (A B C).
.FILL
%1The reader will see that the value of %2flatten[%2x]%1 is a list of the
atoms of the S-expression %2x%1 from left to write. Thus
%2flatten[((A B) A)] = (A B NIL A NIL)%1.
Many functions can be conveniently written in more than one
way. For example, the function %2reverse%1 mentioned above can be
written without an auxiliary function as follows:
%2reverse[%2x] ← %3if n %2x %3then NIL %3else %2reverse[%3d %2x]*<a x>
%1but it will be explained later that the earlier definition involves
less computation.
The use of conditional expressions for recursive function
definition is not limited to functions of S-expressions. For
example, the factorial function and the Euclidean algorithm for the
greatest common divisor are expressed as follows:
.NOFILL
%2n! ← %3if %2n=0 %3then 1 %3else %2n(%2n-1)!
%1and
%2gcd(%2m, n) ← %3if %2m>%2n %3then %2gcd(%2n, m) %3else if %2m=0 %3then %2n
%3else %2gcd(%2n mod m, m)
.FILL
%1where %2n mod m%1 denotes the remainder when %2n%1 is divided by %2m%1 and
may itself be expressed recursively by
%2n mod m ← %3if %2n<%2m %3then %2n %3else (%2n-%2m) %2mod m.%1
The internal form of function definitions depends on the implementation.
Stanford LISP and UCI LISP for the PDP-10 computer use the form
(DE <function name> <list of variables> <right hand side>)%1.
In these LISPs, the above definition of %2subst%1 is
(DE SUBST (X Y Z) (COND ((ATOM Z) (COND ((EQ Z X) Y) (T Z)))
(T (CONS (SUBST X Y (CAR Z)) (SUBST X Y (CDR Z))))))%1,
and the definition of %2alt%1 is
(DE ALT (X) (COND ((OR (NULL X) (NULL (CDR X))) X)
(T (CONS (CAR X) (ALT (CDDR X))))))%1.
Another notation for function definition called the DEFPROP%1 notation
will be explained after λ-expressions have been introduced.
%1 Exercises
1. Consider the function %2drop%1 defined by
%2drop[%2x] ← %3if n %2x %3then NIL %3else <%3a %2x>.%2drop[%3d %2x].
%1Compute (by hand) %2drop[(A B C)]%1. What does %2drop%1 do to lists in
general? Write %2drop%1 in internal notation using DE%1.
2. What does the function
%2r2[%2x] ← %3if n %2x %3then NIL %3else %2reverse[%3a %2x].%2r2[%3d %2x]
%1do to lists of lists? How about
%2r3[%2x] ← %3if at %2x %3then %2x %3else %2reverse[%2r4[%2x]]
%2r4[%2x] ← %3if n %2x %3then NIL %3else %2r3[%3a %2x].%2r4[%3d %2x]?
%13. Compare
%2r3'[%2x] ← %3if at %2x %3then %2x %3else %2r3'[%3d %2x]*<%2r3'[%3a %2x]>
%1with the function %2r3%1 of the preceding example.
4. Consider %2r5%1 defined by
.NOFILL
%2r5[%2x] ← %3if n %2x ∨ %3n d %2x %3then %2x
%3else [%3a %2r5[%3d %2x]]
. %2r5[%3a %2x . r5[%3d %2r5[%3d %2x]]].
.FILL
%1Compute %2r5[(A B C D)]%1. What does %2r5%1 do in general? Needless to
say, this is not a good way of computing this function even though it
involves no auxiliary functions.
.ss Lambda expressions and functions with functions as arguments.
.once turn on "↑";
It is common to use phrases like "the function 2%2x+%2y%1". This
is not a precise notation because we cannot say 2%2x+%2y(3, 4)%1 and know
whether the desired result is 2↑.3+4 or 2↑.4+3 regarding the
expression as a function of two variables. Worse yet, we might have
meant a one-variable function of %2x%1 wherein %2y%1 is regarded as a
parameter.
The problem of giving names to functions is solved by
Church's λ-notation. In the above example, we would write
λ%2x y: 2%2x+%2y%1 to denote the function of two variables with first
argument %2x%1 and second argument %2y%1 whose value is given by the
expression 2%2x+%2y%1. Thus,
[λ%2x y: 2%2x+%2y][3, 4] = 10%1.
%1Likewise, [λ%2y x: 2%2x+%2y][3, 4] = 11%1. Like variables of integration
and the bound variables of quantifiers in logic, variables following λ
are bound or dummy and the expression as a whole may be replaced by
any others provided the replacement is done consistently and does not
make any variable bound by λ the same as a free variable in the
expression. Thus λ%2x y: 2%2x+%2y%1 represents the same function as
λ%2y x: 2%2y+%2x%1 or λ%2u v: 2%2u+%2v%1 , but in the function of one argument
λ%2x: 2%2x+%2y%1 , we cannot replace the variable %2x%1 by %2y%1 , though we could
replace it by %2u%1.
λ-notation plays two important roles in LISP. First, it
allows us to rewrite an expression containing two or more occurrences
of the same sub-expression in such a way that the expression occurs
only once. Thus (2%2x+1)%54+3(2%2x+1)%53%1 can be written
[λ%2w: %2w%54+3%2w%53][2%2x+1]%1. This can save considerable
computation, and
corresponds to the practice in ordinary programming of assigning to a
variable the value of a sub-expression that occurs more than once in
an expression and then writing the expression in terms of the
variable.
The second use of λ-expressions is in using functions that
take functions as arguments. Suppose we want to form a new list from
an old one by applying a function %2f%1 to each element of the list.
This can be done using the function %2mapcar%1 defined by
%2mapcar[%2x, f] ← %3if n %2x %3then NIL %3else %2f[%3a %2x]
. %2mapcar[%3d %2x, f].
%1Suppose the operation we want to perform is squaring, and we want to
apply it to the list (1 2 3 4 5 6 7)%1. We have
%2mapcar[(1 2 3 4 5 6 7), λ%2x: %2x%52] = (1 4 9 16 25 36 49).
%1A more generally useful operation than %2mapcar%1 is %2maplist%1
in which the function is applied to the successive sublists of the
list rather than to the elements. %2maplist%1 is defined by
%2maplist[%2x, f] ← %3if n %2x %3then NIL %3else %2f[%2x]
. %2maplist[%3d %2x, f].
%1As an application of %2maplist%1 and functional arguments, we
shall define a function for differentiating algebraic expressions
involving sums and products. The expressions are built up from atoms
denoting variables and integer constants according to the syntax
.NOFILL
<expression> ::= <variable> | <integer> |
(PLUS <explist>) | (TIMES <explist>)
<explist> ::= <expression> | <expression><explist>
.FILL
Here, PLUS%1 followed by a list of arguments denotes the sum of these
arguments and TIMES%1 followed by a list of arguments denotes their
product. The function %2diff[%2e, v]%1 gives the partial derivative of
the expression %2e%1 with respect to the variable %2v%1. We have
.NOFILL
%2diff[%2e, v] ← %3if at %2e %3then [%3if %2e %3eq %2v %3then 1 %3else 0]
%3else if a %2e %3eq PLUS %3then
PLUS . %2mapcar[%3d %2e, λ%2x: %2diff[%2x, v]_]
%3else if a %2e %3eq TIMES %2then
PLUS.%2maplist[%3d %2e, λ%2x: TIMES
. %2maplist[%3d %2e, λ%2y: %3if %2x %3eq %2y %3then
%2diff[%3a %2y, v] %3else a %2y]].
.FILL
%1The term that describes the rule for differentiating products
corresponds to the rule
.begin nofill
∂/∂%2v %2e%5i%1 = [%3if %2i=%2j %3then ∂%2e%5j/∂%2v %3else %2e%5j].
.end
and %2maplist%1 has to be used rather than %2mapcar%1 since whether to
differentiate in forming the product is determined by equality of the
indices %2i%1 and %2j%1 rather than equality of the terms %2e%5i%1 and
%2e%5j%1.
Two more useful functions with functions as arguments are the
predicates %2andlis%1 and %2orlis%1 defined by the equations
%2andlis[%2u, p] ← %3n %2u ∨ [%2p[%3a %2u] ∧ %2andlis[%3d %2u, p]]
%1and
%2orlis[%2u, p] ← ¬%3n %2u ∧ [%2p[%3a %2u] ∨ %2orlis[%3d %2u, p]].
The internal form for a λ-expression is
(LAMBDA <list of variables> <expression to be evaluated>)%1.
Thus λ x: diff[x,v]%1 is written (LAMBDA (X V) (DIFF X V))%1. When
a function specified with λ is written as an argument of a function
taking a functional argument, then the λ-expression is marked with the
identifier FUNCTION%1. Thus the above definition of %2diff%1 translates
to
.NOFILL
(DE DIFF (E V) (COND
((ATOM E) (COND ((EQ E V) 1) (T 0)))
((EQ (CAR E) (QUOTE PLUS))
.FILL
(CONS (QUOTE PLUS) (MAPCAR (CDR E) (FUNCTION
(LAMBDA (X) (DIFF X V))))))
.NOFILL
.FILL
((EQ (CAR E) (QUOTE TIMES))
(CONS (QUOTE PLUS) (MAPLIST (CDR E) (FUNCTION
(LAMBDA (X) (CONS (QUOTE TIMES) (MAPLIS (CDR E) (FUNCTION (LAMBDA (Y)
(COND ((EQ X Y) (DIFF (CAR Y) V)) (T (CAR Y))))))))))))))%1.
Another way of writing function definitions in internal notation
uses LAMBDA%1 to make a function of the right side of a definition.
It is like writing %2subst%1 and %2alt%1 as
%2subst = λx y z.[%3if at %2z %3then %2[%3if %2z %3eq%2 y %3then%2 x
%3else %2z] %3else %2subst[x,y,%3a%2 z] . subst[x,y,%3d%2 z]]%1
and
%2alt = λx.[%3if n %2x ∨ %3n d %2x %3then%2 x %3else a %2x .alt %3dd%2 x]%1.
The definitions of %2subst%1 and %2alt%1 take the forms
(DEFPROP SUBST (LAMBDA (X Y Z) (COND ((ATOM Z) (COND ((EQ Z X) Y) (T Z)))
(T (CONS (SUBST X Y (CAR Z)) (SUBST X Y (CDR Z)))))))%1,
and
(DEFPROP ALT (LAMBDA (X) (COND ((OR (NULL X) (NULL (CDR X))) X)
(T (CONS (CAR X) (ALT (CDDR X)))))))%1.
The general form is (DEFPROP <function name> <defining λ-expression>)%1
and is often used by programs that output LISP.
.if lines < 8 then skip to column 1
%1 Exercises
1. Compute %2diff[(TIMES X (PLUS Y 1) 3), X]%1 using the above
definition of %2diff%1. Now do you see why algebraic simplification is
important?
2. Compute %2orlis[((A B) (C D) E), %3at]%1.
.ss Label.
The λ mechanism is not adequate for providing names for
recursive functions, because in this case there has to be a way of
referring to the function name within the function. Therefore, we
use the notation %3label[%2f, e]%1 to denote the expression %2e%1 but where
occurrences of %2f%1 within %2e%1 refer to the whole expression. For
example, suppose we wished to define a function that takes alternate
elements of each element of a list and makes a list of these. Thus,
we want
%2glub[((A B C) (A B C D) (X Y Z))] = ((A C) (A C) (X Z)).%1
We can make the definition
.begin nofill
%2glub[%2x] ← %2mapcar[%2x, %3label[%2alt, λ%2x: %3if n %2x ∨ %3n d %2x %3then %2x
%3else a %2x . alt[%3dd %2x]]].
.end
The internal form of %2label[<name>,<function expression>]%1
is (LABEL <name> <function expression>)%1, so that the internal form
of the definition of %2glub%1 is
(DE GLUB (X) (MAPCAR X (LABEL ALT (LAMBDA (X) (COND (OR
(NULL X) (NULL (CDR X))) X) (T (CONS (CAR X) (ALT (CDDR X))))))))%1.
The identifier %2alt%1 in the above example is bound by the label and
is local to that expression, and this is the general rule. The
label construction is not often used in LISP since it is more usual
to give functions global definitions. D. M. R. Park pointed out that
if we allow variables to represent functions and use a suitable λ
construction, the use of label could be avoided.
.ss Numerical computation.
Numerical calculation and symbolic calculation must often
be combined, so LISP provides for numerical computation also.
In the first place, we need to include numbers as parts
of symbolic expressions. LISP has both integer and floating point
numbers which are regarded as atoms.
These numbers may be included as atoms in writing S-expressions.
Thus we can have the lists
(1 3 5)%1, (3.5 6.1 -7.2E9)%1, and (PLUS X 1.3)%1; the first
is a list of integers, the second a list of floating point numbers,
and the third a symbolic list containing both numberical and
non-numerical atoms. Integers are written without decimal points
which are used to signal floating point numbers. As in Fortran,
the letter E%1 is used to signal the exponent of a floating point
number which is a signed integer. The sizes of numbers admitted
depends on the implementation. When a dotted pair, say (1 . 2)%1
is wanted, the spaces around the dot distinguish it from the
list (1.2)%1 whose sole element is the floating point number 1.2.
In publication language we will use ordinary mathematical
notation for numerical functions. For exponentiation we will use
the usual superscript notation %2x%5y%1 when typographically convenient and
the linear Algol notation %2x↑y%1 when it isn't. Of course, numerical
and symbolic calculation must often be combined, so that the function
giving the length of a list can be written
%2length u ← %3if n %2u %3then 0 %3else 1 %2+ length %3d %2u%1.
The internal notation for numerical functions is that used
in the examples given: (PLUS X Y ... Z)%1 for %2x+y+...+z%1,
(TIMES X ... Z)%1 for %2xy...z%1, (MINUS X)%1 for %2-x%1,
(DIFFERENCE X Y)%1 for %2x-y%1, (QUOTIENT X Y)%1 for %2x/y%1,
and (POWER X Y)%1 for %2x%5y%1.
Since numbers that form part of list structures must be
represented by pointers anyway, there is room for a flag distinguishing
floating point numbers and integers. Therefore, the arithmetic
operations are programmed to treat types dynamically, i.e. a variable
may take an integer value at one step of computation and a real value
at another. The subroutines realizing the arithmetic functions make
the appropriate tests and create results of appropriate types.
This is slow compared to direct use of the machine's arithmetic
instructions, so that LISP can be efficiently used only when
the numerical calculations are small or at least small compared
to the symbolic calculations in a problem.
Besides functions of numbers we need predicates on numbers
and the usual =, <, >, ≤, and ≥ are used with the internal
names EQUAL, LESSP, GREATERP, LESSEQP%1, and GREATEREQP%1, respectively.
Not all are implemented in all LISP systems, but of course the
remaining ones can be defined. Besides that, the predicate %2numberp%1
is used to distinguish numbers from other atoms.
It is worth remarking that including type flags in numbers
would benefit many programming languages besides LISP and would
not cost much in either storage or hardware.
As a first example of a combined numeric and symbolic
computation, here is an interpreter for expressions with sums and
products. Assume that the values of variables are given in
an %2association list%1, having the form
(<variable1>.<value1>) ... (<variablen>.<valuen>)),
e.g. ((X . 5) (Y . 9.3) (Z . 2.1))%1.
The function is
%2numval[e,a] ← %3if %2numberp e %3then %2e
%3else if at %2e %3then d %2assoc[e,a]
%3else if a %2e %3eq PLUS %3then %2evplus[%3d %2e,a]
%3else if a %2e %3eq TIMES %3then %2evtimes[%3d %2e,a]%1
where
%2evplus[u,a] ← %3if n %2u %3 then 0 %3else %2numval[%3a %2u,a] +
evplus[%3d %2u,a]%1,
%2evtimes[u,a] ← %3if n %2u %3 then 1 %3else %2numval[%3a %2u,a] +
evtimes[%3d %2u,a]%1,
and
%2assoc[x,a] ← %3if n %2a %3 then NIL
%3else if aa %2a %3eq %2x %3then da %2a %3else %2assoc[x,%3d %2a]%1.
.s HOW TO WRITE RECURSIVE FUNCTION DEFINITIONS
.ss Static and dynamic ways of programming.
In order to write recursive function definitions, one must
think about programming differently than is customary when writing
programs in languages like Fortran or Algol or in machine language.
In these languages, one has in mind the state of the computation as
represented by the values of certain variables or locations in the
memory of the machine, and then one writes statements or machine
instructions in order to make the state change in an appropriate way.
When writing LISP recursive functions one thinks differently.
Namely, one thinks about the value of the function, asks for what
values of the arguments the value of the function is immediate, and,
given an arbitrary values of the arguments, for what simpler
arguments must the function be known in order to give the value of
the function for the given arguments. Let us take a numerical
example; namely, suppose we want to compute the function %2n!%1. For
what argument is the value of the function immediate. Clearly, for
%2n = 0 or %2n = 1%1, the value is immediately seen to be 1. Moreover,
we can get the value for an arbitrary %2n%* if we know the value for
%2n-1. Also, we see that knowing the value for %2n = 1 is redundant,
since it can be obtained from the %2n = 0 case by the same rule as
gets it for a general %2n%* from the value for %2n-1%1. All this talk
leads to the simple recursive formula:
%2n! ← %3if %2n = 0 %3then 1 %3else %2n(%2n-1)!.
%1We may regard this as a static way of looking at programming.
We ask what simpler cases the general case of our function depends on
rather than how we build up the desired state of the computation.
One often is led to believe that static = bad and dynamic = good,
but in this case, the static way is often better than the dynamic
way. Perhaps this distinction is equivalent to what some people call
the distinction between %2top-down%1 and %2bottom-up%1 programming
with %2static%1 corresponding to %2top-down%1.
LISP offers both, but the static style is better developed in LISP,
and we will emphasize it.
.if lines < 10 then skip to column 1
Compare the above recursive definition with the following
obvious Algol program for computing %2n!:
.NOFILL
%3integer procedure %2factorial(%2n); %3integer %2s;
%3begin
%2s := 1;
%2loop: %3if %2n = 0 %3then go to %2done;
%2s := %2n*%2s;
%2n := %2n-1;
%3go to %2loop;
%2done: %2factorial := %2s;
%3end;
.FILL
%1The LISP program is shorter and clearer in this particularly
favorable case. Actually, when we discuss the mechanism of
recursion, it will turn out that the LISP program will be inefficient
in using the pushdown mechanism unnecessarily and should be replaced
by the following somewhat longer program that corresponds to the
above Algol program rather precisely:
%2n! ← %2fact(%2n, %2s),
%1where
%2fact(%2n, %2s) ← %3if %2n = 0 %3then %2s %3else %2fact(%2n-1, %2n*%2s).
%1In fact, compilers should produce the same object code from the two
programs.
.ss Simple list recursion.
About the simplest form of recursion in LISP occurs when one
of the arguments is a list, the result is immediate when the argument
is null, and otherwise we need only know the result for the %3d%*-part of
that argument. Consider, for example, %2u*%2v%1, the result of %2append%1ing
the list %2v%1 to the list %2u%1.
The result is immediate for the case %3n %2u%1 and
otherwise depends on the result for %3d %2u%1. Thus, we have
%2u*%2v ← %3if n %2u %3then %2v %3else a %2u . [%3d %2u * %2v].
%1On the other hand, if we had tried to recur on %2v%1 rather than on %2u%1
we would not have been successful. The result would be immediate for
%3n %2v%1, but %2u*%2v%1 cannot be constructed in any direct way from %2u*%3d %2v%1
without a function that puts an element onto the end of a list.
(From a strictly list point of view, such a function would be as
elementary as %2cons%1 which puts an element onto the front of a list,
but, when we consider the implementation of lists by list structures,
we see that the function is not so elementary. This has led some
people to construct systems in which lists are bi-directional, but,
in the main, this has turned out to be a bad idea). Anyway, it is
usually easier to recur on one argument of a function than to recur
on the other. The %2append%1 function machine coded in most Lisp systems
in a way that allows an arbitrary number of arguments, e.g.
(APPEND (QUOTE (A B)) (QUOTE (C D)) (QUOTE (E F))) is (A B C D E F).
It is often necessary to represent a correspondence between
the elements of a small set of atoms and certain S-expressions by a
list structure. This is conveniently done by means of an association
list which is a list of pairs, each pair consisting of an atom and
the corresponding S-expression. Thus the association list
((X . (PLUS A B)) (Y . C) (Z . (TIMES U V)),
%1which would print as
((X PLUS A B)) (Y . C) (Z TIMES U V)),
%1pairs X%1 with (PLUS A B), Y%1 with C%1, etc. We need a function to
tell whether anything is associated with the atom %2x%1 in the
association list %2a%1, and, if so, to tell us what. We have
.NOFILL
%2assoc[%2x, a] ← %3if n %2a %3then NIL %3else if %2x %3eq aa %2a %3then a %2a
%3else %2assoc[%2x, %3d %2a].
.FILL
%1Its value is NIL%1 if nothing is associated with %2x%1 and the
association pair otherwise. E.g. %2assoc[X, ((X.W) (Y.V))] = (X.W)%1.
%2assoc%1 is also built in to most Lisp systems.
It commonly happens that a function has no simple recursion,
but there is a simple recursion for a function with one more variable
that reduces to the desired function when the extra variable is set
to NIL%1. Thus
%2reverse[%2u] ← %2rev1[%2x, NIL],
%1where
%2rev1[%2u, v] ← %3if n %2u %3then %2v %3else %2rev1[%3d %2u, %3a %2u . v].
%2reverse%1 has a direct recursive definition as discovered by S. Ness,
but no-one would want to use the following in actual computation nor
does it generate much understanding, only appreciation of Mr. Ness's
ingenuity:
.NOFILL
%2reverse[%2u] ← %3if n %2u ∨ %3n d %2u %3then %2u %3else
a %2reverse[%3d %2u] . %2reverse[%3a %2u. %2reverse[%3d %2reverse[%3d %2u]]].
%1
.FILL
Exercises
1. Using the function %2member[%2x, u]%1 defined in Chapter I
which may also be written %2x ε %2u%1, write function definitions for the
union %2u ∪ %2v%1 of lists %2u%1 and %2v%1, the intersection %2u ∩ %2v%1, and the
set difference %2u-%2v%1. What is wanted should be clear from the
examples:
(A B C) ∪ (B C D) = (A B C D),
(A B C) ∩ (B C D) = (B C),
%1and
(A B C) - (B C D) = (A).
%1Pay attention to getting correct the trivial cases in which some of
the arguments are NIL%1. In general, it is important to understand
clearly the trivial cases of functions.
2. Suppose %2x%1 takes numbers as values and %2u%1 takes as
values lists of numbers in ascending order, e.g. (2 4 7)%1. Write a
function %2merge[%2x, u]%1 whose value is obtained from that of %2u%1 by
putting %2x%1 in %2u%1 in its proper place. Thus
%2merge[3, (2 4)] = (2 3 4)%1, and %2merge[3, (2 3)] = (2 3 3)%1.
3. Write functions giving the union, intersection, and set
difference of ordered lists; the result is wanted as an ordered list.
Note that computing these functions of unordered lists takes
a number of comparisons proportional to the square of the number of
elements of a typical list, while for ordered lists, the number of
comparisons is proportional to the number of elements.
4. Using %2merge%1, write a function %2sort%1 that transforms an
unordered list into an ordered list.
5. Write a function %2goodsort%1 that sorts a list using a
number of comparisons proportional to %2n log n%1, where %2n%1 is the
length of the list to be sorted.
.ss Simple S-expression recursion.
In another class of problems, the value of the function is
immediate for atomic symbols, and for non atoms depends only on the
values for the a-part and the d-part of the argument. Thus %2subst%1
was defined by
.NOFILL
%2subst[%2x, y, z] ← %3if at %2z %3then [%3if %2z %3eq %2y %3then %2x %3else %2z]
%3else %2subst[%2x, y, %3a %2z] . %2subst[%2x , y , %3d %2z].
.FILL
%1Two other examples are %2equal%1 which gives the equality of
S-expressions and %2flat%1 which spreads an S-expression into a list
of atoms: They are defined by
%2x=%2y ← %2x %3eq %2y ∨ [¬%3at %2x ∧ ¬%3at %2y ∧ %3a %2x = %3a %2y ∧ %3d %2x = %3d %2y],
%1and
%2flat[%2x] ← %2flata[%2x, NIL]
%1where
%2flata[x,u] ← %3if at %2x %3then %2x.y %3else %2flata[%3a %2x,flata[%3d %2x,y]]%1.
The kind of double recursion used in %2flat%1 is often useful. %2flat%1 could
also be written
%2flat x ← %3if at %2 x %3then%2 <x> %3then%2 flat %3a%2 x * flat %3d%2 x,
which is a bit easier to follow although less efficient, because the %2append%1
function copies unnecessarily.
.if lines < 7 then skip to column 1
EXERCISES
1. Write a predicate to tell whether a given atom occurs in a
given S-expression, e.g. %2occur[B, ((A.B).C)] = T%1.
2. Write a predicate to tell how many times a given atom
occurs in an S-expression.
3. Write a function to make a list without duplications of
the atoms occurring in an S-expression.
4. Write a function to make a list of all atoms that occur
more than once in a given S-expression paired with their
multiplicities.
5. Write a predicate to tell whether an S-expression has more
than one occurrence of a given S-expression as a sub-expression.
.ss Other structural recursions.
When lists are used to represent algebraic expressions,
functions of these algebraic expressions often have a
recursive form closely related to the inductive definition of the
expressions. Suppose, for example, that sums and products are represented
respectively by the forms (PLUS X Y)%1 and (TIMES X Y)%1 and that the
values of variables are given on an a-list. We can write a recursive
formula for the value of an expression as follows:
.NOFILL
%2value[%2e, a] ← %3if at %2e %3then d %2assoc[%2e, a]
%3else if a %2e %3eq PLUS %3then %2value[%3ad %2e, a] + %2value[%3add %2e, a]
%3else if a %2e %3eq TIMES %3then %2value[%3ad %2e, a] * %2value[%3add %2e, a].
.FILL
%1On the other hand, suppose that sums and products are not restricted
to have just two arguments; then we must use auxiliary functions
to define the value of an expression, as follows:
.NOFILL
%2value[%2e, a] ← %3if at %2e %3then d %2assoc[%2e, a]
%3else if a %2e %3eq PLUS %3then %2vplus[%3d %2e, a]
%3else if a %2e %3eq TIMES %3then %2vtimes[%3d %2e, a].
%1where
%2vplus[%2u, a] ← %3if n %2u %3then 0 %3else %2value[%3a %2u, a] + %2vplus[%3d %2u, a],
%1and
%2vtimes[%2u, a] ← %3if n %2u %3then 1 %3else %2value[%3a %2u, a] * %2vtimes[%3d %2u, a].
.FILL
%1In both cases, the recursion form is related to the structure of the
algebraic expressions more closely than to the structure of
S-expressions or lists.
.ss Tree search recursion.
.FILL
We begin with a general depth first tree search function.
It can be used to search specific trees of possibilities by defining
three auxiliary functions in a way that depends on the application.
We have
%2search p ← qif lose p qthen %1LOSE%2
qelse qif ter p qthen p qelse searchlis[successors p]%1
where
%2searchlis u ← qif qn u qthen %1LOSE%2 qelse {search qa
u}[λx: qif x = %1LOSE%2 qthen searchlis qd u qelse x]%1.
In the applications, we start with a position %2p%40%1, and we
are looking for a win in the successor tree of %2p%40%1. Certain
positions lose and there is no point looking at their successors.
This is decided by the predicate %2lose%1. A position is a win if
it doesn't lose and it satisfies the predicate %2ter%1. The
successors of a position is given by the function %2successors%1,
and the value of %2search p%1 is the winning position. No
non-losing position should have the name LOSE or the function won't
work properly.
Simple S-expression recursion can be regarded as a special case
of depth first recursion. It is special in that there exactly two
branches, but even more important, the tree is the tree of parts of
the S-expression and is present at the beginning of the calculation.
In case of tree search recursion, the tree is generated by the
%2successors%1 function.
Our first application is finding a path from an initial node to
a final node in a graph represented by a list structure as described
in chapter I. A position is a path starting from the initial node
and continuing to some intermediate node and is represented by a
list of its nodes in reverse order. The three functions for this
application are
%2lose p ← %3a %2p ε %3d %2p,
ter p ← [%3a %2p = final],%1
and
%2successors p ← mapcar[qd assoc[qa p, graph], λx: x . p]%1.
Another example is the so-called %2Instant Insanity%1 puzzle.
There are four cubical blocks, and each face of each block is colored with
one of four colors. The object of the puzzle is to build a tower of all four
blocks such that each vertical face of the tower involves all four colors.
In order to use the above defined function %2search%1 for this purpose,
we must define the representation of positions and give the functions
%2lose, ter, %1and %2successors%1. A position is represented by a list
of lists ¬ one for each face of the tower. Each sublist is the list of colors
of the faces of the blocks showing in that face. We shall assume that the
blocks are described in the following longwinded but convenient way. (We'll
take up precomputing this description later.) For each block there is a list
of the 24 orientations of the block where each orientation is described as
a list of the colors around the vertical faces of the block when it is in that
orientation. Thus the puzzle is described by a list of lists of lists which
we shall call %2puzz%1.
We now have
%2p%40%1 = (NIL NIL NIL NIL),
%2lose p ← orlis[p, λu: qa u ε qd u]%1,
%2ter p ← [length qa p = 4]%1,
and
%2successors p ← mapcar[qa nth[puzz, 1 + length qa p], λx: mapcar2[p, x, λyz: z.y]]%1,
where
%2mapcar2[u, v, f] ← qif qn u qthen qNIL qelse
f[qa u, qa v] . mapcar2[qd u, qd v, f]%1.
Getting the initial position in the desired form is as complicated
a computation as the actual tree search. It can be
conveniently done by a sequence of assignment
statements starting with a description of the blocks:
%2puzz1 ← %1((G B B W R G) (G G B G W R) (G W W R B R) (G G R B W W)).
Here each block is represented by a list of the colors of the faces starting
with the top face, going around the sides in a clockwise direction and
finishing with the bottom face.
We need to go from this description of the blocks to a list of the
possible cycles of colors on the vertical faces for the 24 orientations of
the block. This not easy, because the order in which we have given the colors
is not invariant under rotations of the block. An easy way out is to start with
a block whose faces are assigned the numbers 1 thru 6 starting with the top,
going clockwise around the sides and finishing with the bottom. We write down
one cycle of side colors for each choice of the face put on top and get the
list of all 24 cycles by appending the results of generating the cyclic
permutations of the cycles. All this is accomplished by the assignment.
%2puzz2 ← cycles[(2 3 4 5)] * cycles[(2 5 4 3)] * cycles[(1 2 6 4)]
* cycles[(1 4 6 2)] * cycles[(1 3 6 5)] * cycles[(1 5 6 3)]%1,
where the function %2cycles%1 is defined by
%2cycles u ← maplist[u, λx: x * upto[u, x]]%1
with the auxiliary function
%2upto[u, x] ← qif x qeq u qthen qNIL qelse qa u . upto[qd u, x]%1.
Next we create for each block a list of substitutions expressing the
colors of the six numbered faces. We have
%2puzz3 ← mapcar[puzz1, λx: prup[(1 2 3 4 5 6), x]]%1,
and we use these substitutions to get for each block the list of 24 orientations
of the block. Thus
%2puzz4 ← mapcar[puzz3, λs: sublis[s, puzz3]]%1.
%2puzz4%1 has all 24 orientations of the first block while for symmetry
reasons we need only consider three as distinct, say the first, ninth, and
seventeenth. So we finally get
%2puzz ← <qa nth[qa puzz4, 1], qa nth[qa puzz4, 9],
qa nth[qa puzz4, 17]> . qd puzz4.%1
The program when compiled runs about 11 seconds on the KA-10.
A more sophisticated representation of face cycles and partial towers
makes a factor of ten speedup without changing the basic search algorithm. A
face cycle is represented by 16 bits in a word four for each face a particular
one of which being turned on tells us the color of the face. If we %3or%1
these words together for the blocks in a partial tower we get a word which
tells us for each face of the tower what colors have been used. A
particular face cycle from the next block can be added to the tower if
the %3and%1 of its word with the word representing the tower is zero.
We represent a position by a list of words representing its partial
towers with 0 as the last element and the highest partial tower as the
first element. The virtue of this representation is that it makes
the description of the algorithm short.
The initial position is (0). The new %2puzz%1 can be formed from the
old one by the assignment
%2puzza ← mapcar[puzz, λx: mapcar[x, zap]], %1
where
%2zap v ← qif qn v qthen 0 qelse poo qa v + 16 * zap qd v%1.
and
%2poo x ← qif x = %1R%2 qthen 1 qelse if x = %1W%2 qthen
2 qelse qif x = %1G%2 qthen 4 qelse 8%1.
Now we need the new functions %2lose, ter, %1and
%2successors%1. These are
%2lose p ← %3false%2,
%2ter p ← [length p = 5], %1
and
%2successors p ← mapchoose[qa nth[puzza, length p],
λx: qa p %3and%2 x = 0, λx: [qa p %3or%2 x] . p].%1
where
%2mapchoose[u, pred, fn] ← qif qn u qthen qNIL qelse qif
pred qa u qthen fn[qa u] . mapchoose[qd u, pred, fn]
qelse mapchoose[qd u, pred, fn].%1
%2lose%1 is trivial, because the %2mapchoose%1 is used to make sure
that only non-losing new positions are generated by %2successors%1.
This version runs in a little less than one second on the KA-10. A greater
speedup can be made by the application of some mathematics. In fact, with
enough mathematics, extensive tree search is unnecessary in this problem.
%2search%1 is used when we want to search a tree of possibilities
for a solution to a problem.
Naturally we can do other things with tree search recursion than just
search. For example we may want
to find all solutions
to a problem. This can be done with a function %2allsol%1 that uses
the same %2lose, ter, %1and %2successors%1 functions as does %2search%1.
The simplest way to write %2allsol%1 is
%2allsol p ← qif lose p qthen qNIL qelse qif ter p
qthen <p> qelse mapapp[successors p, allsol]%1,
where
%2mapapp[u, fn] ← qif qn u qthen qNIL qelse fn[qa u] . mappap[qd u, fn]%1.
This form of the function is somewhat inefficient because of all the
%2append%1ing. A more efficient form uses an auxiliary function as follows:
%2allsol p ← allsola[p, qNIL]
allsola[p, found] ← qif lose p qthen found
qelse qif ter p qthen p . found
qelse allsolb[successors p, found]%1,
%2allsolb[u, found] ← qif qn u qthen found
qelse allsolb[qd u, allsola[qa u, found]].%1
The recursive program structure that arises here is common when a list
is to be formed by recurring through a list structure.
.ss Game trees.
.FILL
The positions that can be reached from an initial position in
a game form a tree, and deciding what move to make often involves
searching this tree. However, game trees are searched in a different
way than the trees we have looked at, because the opposing interests
of the players makes it not a search for a joint line of play that
will lead to the first player winning, but rather a search for a
strategy that will lead to a win regardless of what the other player
does.
The simplest situation is characterized by a function
%2successors%1 that gives the positions that can be reached in one
move, a predicate %2ter%1 that tells when a position is to be
regarded as terminal for the given analysis, and a function
%2imval%1 that gives a number approximating the value of the
position to one of the players. We shall call this player the
maximizing player and his opponent the minimizing player. Usually,
the numerical values arise, because the search cannot be carried out
to the end of the game, and the analysis stops with reasonably static
positions that can be evaluated by some rule. Naturally, the
function %2imval%1 is chosen to be easy to calculate and to
correlate well with the probability that the maximizing player can
win the position.
The simplest rule for finding the correct move in a position
uses auxiliary functions
%2valmax%1 and %2valmin%1 that give a value to a position
by using %2imval%1 if the position is terminal and taking the max
or min of the successor positions otherwise.
For this we want functions for getting the maximum or the
minimum of a function on a list, and they are defined as follows:
%2maxlis[u, f] ← %3if n %2u %3then %2-∞ %3else %2
max[f[%3a %2u], maxlis[%3d %2u, f]], %1
and
%2minlis[u, f] ← %3if n %2u %3then %2∞ %3else %2
min[f[%3a %2u], minlis[%3d %2u, f]].%1
In these functions, -∞ and ∞ represent numbers that are smaller and
larger than any actual values that will occur in evaluating %2f%1.
If these numbers are not available, then it is necessary to prime
the function with the values of %2f%1 applied to the first element
of the list, and the functions are meaningless for null lists.
Iterative versions of the functions are generally better; we give only
the iterative version of %2maxlis%1, namely
%2maxlis[u, f] ← maxlisa[u, -∞, f], %1
where
%2maxlisa[u, x, f] ← %2if n %2u %3then %2x %3else
%2maxlisa[%3d %2u, max[x, f[%3a %2u]], f].%1
We now have
%2valmax p ← %3if %2ter p %3then %2imval p %3else
%2maxlis[successors p, valmin]%1,
and
%2valmin p ← %3if %2ter p %3then %2imval p %3else
%2minlis[successors p, valmax]%1.
.NOFILL
The best move is now chosen by
%2movemax p ← bestmax[succesors p, valmin], %1
or
%2movemin p ← bestmin[succesors p, valmax], %1
where
%2bestmax[u, f] ← bestmaxa[%3d %2u, %3a %2u, f[%3a %2u], f],
%2bestmaxa[u, sofar, valsofar, f] ← %3if n %2u %3then %2sofar
.FILL
%3else if %2f[%3a %2u] ≤ %2bestsofar %3then
%2bestmaxa[%3d %2u, sofar, bestsofar, f]
.NOFILL
%3else %2bestmaxa[%3d %2u, %3a %2u, f[%3a %2u], f],
%2bestmin[u, f] ← bestmina[%3d %2u, %3a %2u, f[%3a %2u], f], %1
and
%2bestmina[u, sofar, valsofar, f] ← %3if n %2u %3then %2sofar
.FILL
%3else if %2f[%3a %2u] ≥ %2bestsofar %3then
%2bestmina[%3d %2u, sofar, bestsofar, f]
.NOFILL
%3else %2bestmina[%3d %2u, %3a %2u, f[%3a %2u], f].%1
.FILL
However, this straight minimax computation of the best move does
much more computation in general than is usually necessary. The simplest
way to see this is from the move tree of Figure 2.6.
.begin verbatim
8
max
1
min
6
9
X
X
Figure 2.6
.end
%1We see from this figure that it is unnecessary to examine the moves marked
x, because their values have no effect on the value of the game or on the
correct move since the presence of the 9 is sufficient information at this
point to show that the move leading to the vertex preceding it should not
be chosen by the minimizing player.
The general situation is that it is unnecessary to examine further
moves in a position once a move is found that refutes moving to the
position in the first place. This idea is called the αβ-heuristic and
expressed in the following recursive definitions:
.begin nofill
%2maxval p ← maxval1[p, -∞, ∞],
maxval1[p, αα, β] ← %3if %2ter p %3then %2max[αα, min[β, imval p]]
%3else %2maxval2[successors p, αα, β],
maxval2[u, αα, β] ← {minval1[%3a %2u, αα, β]}[λw: %3if %2w = β %3then
%2β %3else %2maxval2[%3d %2u, w, β]],
%2minval p ← minval1[p, -∞, ∞],
minval1[p, αα, β] ← %3if %2ter p %3then %2max[αα, min[β, imval p]]
%3else %2minval2[successors p, αα, β],
minval2[u, αα, β] ← {maxval1[%3a %2u, αα, β]}[λw: %3if %2w = αα %3then
%2αα %3else %2minval2[%3d %2u, αα, w].%1
.end
The reduction in number of positions examined given by the ααβ algorithm
over the simple minimax algorithm depends on the order in which the moves are
examined. In the worst case, the moves happen to be examined in inverse order
of merit in every position on the tree, i.e. the worst move first. In that case,
there is no improvement over minimax. The best case is the one in which the
best move in every position is examined first. If we look %2n%1 moves
deep on a tree that has %2k%1 successors to each position, then minimax looks
at %2k%5n%1 positions while ααβ looks at about only %2k%5n/2%1. Thus a
program that looks at 10%54%1 moves with ααβ might have to look at 10%58%1
with minimax. For this reason, game playing programs using ααβ make a big
effort to include as good as possible an ordering of moves into the %2successors%1
function. When there is a deep tree search to be done, the way to make the
ordering is with a short look-ahead to a much smaller depth than the main
search. Still shorter look-aheads are used deeper in the tree, and beyond
a certain depth, non-look-ahead ordering methods are of decreasing complexity.
A version of ααβ incorporating optimistic and pessimistic evaluations
of positions was first proposed by McCarthy about 1958. Edwards and Hart at
M.I.T. about 1959 proved that the present version of ααβ works and calculated the
improvement it gives over minimax. The first publication, however, was
Brudno (1963). It is worth noting that ααβ was not used in the early chess
playing programs in spite of the fact that it is clearly used in any human
play. Its non-use, therefore, represents a failure of self-observation.
Very likely, there are a number of other algorithms used in human thought
that we have not noticed an incorporated in our programs. To the extent
that this is so, artificial intelligence will be %2a posteriori%1 obvious
even though it is %2a priori%1 very difficult.
.s COMPILING IN LISP
.ss Introduction
Compiling is an important example of symbolic computation and
has received much study. Much of the study has been devoted
to parsing which is essentially the transformation of an input string
in the source language into an internal form. The internal form used
depends on the compiler. Sometimes it is Polish prefix or postfix notation,
sometimes it is list structure, and sometimes it consists of entries in
a collection of tables.
When internal notation LISP is being compiled, the parsing is
trivial, because the input is S-expressions and the internal form wanted
is list structure, so what parsing there is is done by the ordinary LISP
%2read%1 routine. Therefore, compilers can be very compact and
transparent in structure, and we can concentrate our attention on the code
generation phase. This is as it should be, because, in my opinion,
parsing is basically a side issue in compiling, and code generation is the
matter of main scientific interest.
We shall describe two compilers in this chapter called LCOM0 and
LCOM4 which compile S-expression LISP into machine language for the PDP-10
computer according to the conventions of Stanford LISP 1.6. For now we
shall take these conventions for granted. Before describing the
compilers, we must describe these conventions.
The target language is called LAP for LISP assembly program. Each
function is compiled separately into a separate LAP program, and these
programs are written onto a disk file. These files can later be read into
a LISP core image and assembled in place by the LAP assembler which is
part of the LISP runtime routines. The compiler, on the other hand, is a
separate LISP core image that can be instructed to compile several input
files. For an input file called <name>, it produces and output file
called <name>.LAP. All this is specific to the PDP-10 time-sharing system
and fortunately works the same whether the time-sharing system is the
D.E.C. system, the Stanford system, or TENEX.
The LAP program produced is a list of words; the last word is NIL,
and the first word is a header of the form (LAP %2fname%1 SUBR) where
%2fname%1 is the name of the function compiled. This header tells the
DSKIN program that it should read S-expressions till it comes to NIL and
then submit what it has collected to the LAP assembler which will assemble
it into core. The rest of the words are either atoms representing labels
or lists representing single PDP-10 instructions.
.ss Some facts about the PDP-10.
The following facts about the PDP-10 may be of use:
The PDP-10 has a 36 bit word and an 18 bit address. In
instructions and in accumulators used as index registers the address is
found in the right part of the word where the least significant bits in
arithmetic reside.
There are 16 general registers which serve simultaneously as
accumulators (receiving the results of arithmetic operations), index
registers (modifying the nominal addresses of instructions to form
effective addresses), and as the first 16 registers of memory (if the
effective address of an instruction is less than 16, then the
instruction uses the corresponding general register as its operand.)
All instructions have the same format and are written for the
LAP assembly program in the form
(<op name> <accumulator> <address> <index register>).
Thus _(MOVE 1 3 P)_ causes accumulator _1_ to receive the contents of
a memory register whose address is _3+c(P), i.e. 3+<the contents of
general register P>.
If no index register is specified, then the address used is just <address>.
If the op name is followed by _@, then the memory
reference is indirect, i.e. the effective addrress is the contents of
the right half of the word directly addressed by the instruction
(modified by the index register and indirect bit of that word). In
the following description of some instructions useful in constructing
the compiler, <ef> denotes the effective address of an instruction.
.BEGIN INDENT 8,30,8
.PREFACE 0
.AT 0 ⊂BREAK⊃
.AT 24 ⊂""⊃
.TURN ON "∂"
.VARIABLE I
.I ← 31
MOVE ∂(I)(ac) ← c(<ef>)
MOVEI ∂(I)c(ac) ← <ef>
MOVEM ∂(I)c(<ef>) ← c(ac)
HLRZ ∂(I)right half of c(ac) ← left half of c(<ef>)
HRRZ ∂(I)right half of c(ac) ← right half of c(<ef>)
∂(16)(These two are used indirectly for _car_ and _cdr_ respectively.)
ADD ∂(I)c(ac) ← c(ac) + c(<ef>)
SUB ∂(I)c(ac) ← c(ac) - c(<ef>)
JRST ∂(I)go to <ef>
JUMPE ∂(I)if c(ac) = 0 then go to <ef>
JUMPN ∂(I)if c(ac) ≠ 0 then go to <ef>
CAME ∂(I)if c(ac) = c(<ef>) then <skip next instruction>
CAMN ∂(I)if c(ac) ≠ c(<ef>) then <skip next instruction>
PUSH ∂(I)c(c(right half of ac)) ← c(<ef>); the contents
of each half of ac is increased by one and if
the contents of the left half is then _0, a stack
overflow interrupt occurs. (PUSH P ac) is
is used in LISP to put c(ac) on the
stack.
POP ∂(I)c(<ef>) ←c(c(right half of ac)); the
contents of each half of ac are then decreased
by 1. This may be used for removing the
top element of the stack. Thus (POP P 1) puts
the top element of the stack in accumulator 1.
The i-th element of the stack is obtained by
(MOVE@ 1 i P).
POPJ ∂(I)(POPJ P) is used for returning from a subroutine
.END
These instructions are adequate for compiling basic LISP code
with the addition of the subroutine calling pseudo-instrucion. (CALL
%2n (E <subr) S) is used for calling the LISP subroutine <subr> with %2n%1
arguments. The convention is that the arguments will be stored in
successive accumulators beginning with accumulator _1, and the result
will be returned in accumulator _1. In particular the functions
_ATOM_ and _CONS_ are called with _(CALL 1 (E ATOM) S)_ and (CALL 2 (E
CONS) S) respectively, and programs produced by you compiler will be
called similarly when their names are referred to.
.ss Code produced by LISP compilers.
We will discuss two compilers, a simple one called LCOM0 and a
more optimising compiler called LCOM4. LCOM4 produces about half as many
instructions for a given function as does LCOM0. Besides these, there is
the standard PDP-10 LISP compiler written at M.I.T. by Richard Greenblatt
and Stuart Nelson and modified by Whitfield Diffie.
.SKIP TO LINE 1
.BEGIN VERBATIM SELECT 1
Examples of output of LCOM0, LCOM4, and the regular Lisp compiler
For the file containing:
(DEFPROP DROP
(LAMBDA(X)
(COND ((NULL X) NIL) (T (CONS (LIST (CAR X)) (DROP (CDR X))))))
EXPR)
******************************************************************
LCOM0 produces the following code:
(LAP DROP SUBR)
(PUSH P 1)
(MOVE 1 0 P)
(PUSH P 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 1 (E NULL) S)
(JUMPE 1 G0163)
(MOVEI 1 0)
(JRST G0162)
G0163
(MOVEI 1 (QUOTE T))
(JUMPE 1 G0164)
(MOVE 1 0 P)
(PUSH P 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 1 (E CAR) S)
(PUSH P 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 1 (E LIST) S)
(PUSH P 1)
(MOVE 1 -1 P)
(PUSH P 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 1 (E CDR) S)
(PUSH P 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 1 (E DROP) S)
(PUSH P 1)
(MOVE 1 -1 P)
(MOVE 2 0 P)
(SUB P (C 2 0 2 0))
(CALL 2 (E CONS) S)
(JRST G0162)
G0164
G0162
(SUB P (C 1 0 1 0))
(POPJ P)
NIL
LCOM4 produces the following code:
(LAP DROP SUBR)
(PUSH P 1)
(MOVE 1 0 P)
(JUMPE 1 G0162)
(HLRZ@ 1 0 P)
(CALL 1 (E LIST) S)
(PUSH P 1)
(HRRZ@ 1 -1 P)
(CALL 1 (E DROP) S)
(MOVE 2 1)
(MOVE 1 0 P)
(SUB P (C 1 0 1 0))
(CALL 2 (E CONS) S)
G0162
(SUB P (C 1 0 1 0))
(POPJ P)
NIL
******************************************************************
The ILISP compiler produces the following code:
(LAP DROP SUBR)
(PUSH P 1)
(JUMPE 1 TAG1)
(HLRZ@ 1 0 P)
(CALL 1 (E NCONS) S)
(PUSH P 1)
(HRRZ@ 1 -1 P)
(CALL 1 (E DROP) S)
(POP P 2)
(CALL 2 (E XCONS) S)
TAG1 (SUB P (C 1 0 1 0))
(POPJ P)
NIL
.END
Note that all three compilers produce the same first line of
heading. This is necessary for the LAP assembly program which also
requires the _NIL_ at the end as punctuation. The standard compiler
uses a function called _XCONS_ that has the same effect as CONS_
except that it receives its arguments in the reverse order which is
sometimes convenient. This requires, of course, that the compiler be
smart enough to decide where it is more convenient to put the
arguments of the %2cons%1 function.
My two compilers are similar in structure, but the better
one, which is twice as long, uses the following optimisations.
1. When the argument of CAR or CDR is a variable, it compiles
a (HLRZ@ 1 i P) or (HRRZ@ 1 i P) which gets the result through the
stack without first compiling the argument into an accumulator.
2. When it has to set up the arguments of a function in the
accumulators, in general, the program must compute the arguments one
at a time and save them on the stack, and then load the accumulators
from the stack. However, if one of the arguments is a variable, is a
quoted expression, or can be obtained from a variable by a chain of
_cars_ and _cdrs, then it need not be computed until the time of
loading accumulators since it can be computed using only the
accumulator in which it is wanted.
3. The Diffy compiler produces about 10 percent less code
than LCOM4; the main difference seems to be that it sometimes notices
when it has an expression that it will need later. Sometimes this
feature leads it astray. For example, it may save _%3a %2u%1_
for later use at greater cost than re-computing it.
It should further be noted that quoted S-expressions can be
compiled with the instruction
(MOVEI 1 (QUOTE α)).
.ss Listings of LCOM0 and LCOM4
First are listings of both compilers in blackboard notation.
Following these is an an annotated listing of LCOM0 in S-expression
notation, and a listing of LCOM4 in that notation.
.SKIP TO LINE 1
LCOM0:
.REQUIRE "LC0.BB" SOURCE_FILE
.SKIP TO LINE 1
LCOM4:
.REQUIRE "LC4.BB" SOURCE_FILE
.SKIP TO LINE 1
Annotated listing of LCOM0:
.BEGIN VERBATIM SELECT 1
(DEFPROP LC0FNS
(LC0FNS COMPL
COMP
PRUP
MKPUSH
COMPEXP
COMPLIS
LOADAC
COMCOND
COMBOOL
COMPANDOR)
VALUE)
~COMPL is the user-callable driver. It is a FEXPR. It takes as
~ an argument a single file name, which must have no extension.
~ EXPRs on a file called FILNAM will be compiled into LAP and
~ written on the file FILNAM.LAP. Other types of function
~ definitions and non-definitions are simply copied to output.
(DEFPROP COMPL
(LAMBDA(FILE)
(PROG (Z)
(EVAL
(CONS (QUOTE OUTPUT)
(CONS (QUOTE DSK:)
(LIST (CONS (CAR FILE) (QUOTE LAP))))))
(EVAL (CONS (QUOTE INPUT) (CONS (QUOTE DSK:) FILE)))
(INC T NIL)
LOOP (SETQ Z (ERRSET (READ)))
(COND ((ATOM Z) (GO DONE)))
(SETQ Z (CAR Z))
(COND ((OR (EQ (CAR Z) (QUOTE DE))
(AND (EQ (CAR Z) (QUOTE DEFPROP))
(EQ (CADDDR Z) (QUOTE EXPR))))
(PROG (PROG)
(SETQ PROG
(COND ((EQ (CAR Z) (QUOTE DE))
(COMP (CADR Z)
(CADDR Z)
(CADDDR Z)))
(T
(COMP (CADR Z)
(CADR (CADDR Z))
(CADDR (CADDR Z))))))
(OUTC T NIL)
(MAPC (FUNCTION PRINT) PROG)
(OUTC NIL NIL)
(PRINT (LIST (CADR Z) (LENGTH PROG)))))
(T (OUTC T NIL) (PRINT Z) (OUTC NIL NIL)))
(GO LOOP)
DONE (OUTC T NIL)
(OUTC NIL T)
(INC NIL T)
(RETURN (QUOTE ENDCOMP))))
FEXPR)
~COMP compiles a single function definition, returning a list of
~ the LAP code corresponding to the definition.
~ FN is the atomic name of the function being compiled.
~ VARS is the formal parameter list for the function.
~ EXP is the function body.
(DEFPROP COMP
(LAMBDA(FN VARS EXP)
((LAMBDA(N)
(APPEND (LIST (LIST (QUOTE LAP) FN (QUOTE SUBR)))
(MKPUSH N 1)
(COMPEXP EXP (MINUS N) (PRUP VARS 1))
(LIST
(LIST (QUOTE SUB) (QUOTE P) (LIST (QUOTE C) N 0 N 0)))
(QUOTE ((POPJ P) NIL))))
(LENGTH VARS)))
EXPR)
~PRUP returns an A-LIST formed by pairing successive elements of
~ VARS with consecutive integers beginning with N.
(DEFPROP PRUP
(LAMBDA(VARS N)
(COND ((NULL VARS) NIL)
(T (CONS (CONS (CAR VARS) N) (PRUP (CDR VARS) (PLUS N 1))))))
EXPR)
~MKPUSH returns a list of N (PUSH P i) instructions, where i runs
~ from M to M+N-1. Used to push arguments onto the stack.
(DEFPROP MKPUSH
(LAMBDA(N M)
(COND ((LESSP N M) NIL)
(T
(CONS (LIST (QUOTE PUSH) (QUOTE P) M)
(MKPUSH N (PLUS M 1))))))
EXPR)
~COMPEXP is the heart of LCOM0. It determines precisely
~ what an expression is, and compiles appropriate code
~ for it. It returns a list of that code.
~ EXP is the expression to be compiled.
~ M is minus the number of entries on the stack. When
~ added to a value retrieved from the A-LIST VPR, it
~ can be used to locate a variable on the stack.
~ VPR is an A-LIST, associating variable names with
~ numbers which, when added to M, give stack offsets.
~ Both M and VPR maintain these definitions throughout.
(DEFPROP COMPEXP
(LAMBDA(EXP M VPR)
(COND ((NULL EXP) (QUOTE ((MOVEI 1 0)))) ~NIL
((EQ EXP T) (QUOTE ((MOVEI 1 (QUOTE T))))) ~T
((ATOM EXP) ~variable
(LIST
(LIST (QUOTE MOVE)
1
(PLUS M (CDR (ASSOC EXP VPR)))
(QUOTE P))))
((OR (EQ (CAR EXP) (QUOTE AND)) ~boolean expression
(EQ (CAR EXP) (QUOTE OR))
(EQ (CAR EXP) (QUOTE NOT)))
((LAMBDA(L1 L2)
(APPEND
(COMBOOL EXP M L1 NIL VPR)
(LIST (QUOTE (MOVEI 1 (QUOTE T)))
(LIST (QUOTE JRST) 0 L2)
L1
(QUOTE (MOVEI 1 0))
L2)))
(GENSYM)
(GENSYM)))
((EQ (CAR EXP) (QUOTE COND)) ~COND
(COMCOND (CDR EXP) M (GENSYM) VPR))
((EQ (CAR EXP) (QUOTE QUOTE)) ~QUOTE
(LIST (LIST (QUOTE MOVEI) 1 EXP)))
((ATOM (CAR EXP)) ~function call
((LAMBDA(N)
(APPEND
(COMPLIS (CDR EXP) M VPR)
(LOADAC (DIFFERENCE 1 N) 1)
(LIST
(LIST (QUOTE SUB) (QUOTE P) (LIST (QUOTE C) N 0 N 0)))
(LIST
(LIST (QUOTE CALL)
N
(LIST (QUOTE E) (CAR EXP))
(QUOTE S)))))
(LENGTH (CDR EXP))))
((EQ (CAAR EXP) (QUOTE LAMBDA)) ~LAMBDA expression
((LAMBDA(N)
(APPEND
(COMPLIS (CDR EXP) M VPR)
(COMPEXP
(CADDAR EXP)
(DIFFERENCE M N)
(APPEND (PRUP (CADAR EXP) (DIFFERENCE 1 M)) VPR))
(LIST
(LIST (QUOTE SUB) (QUOTE P) (LIST (QUOTE C) N 0 N 0)))))
(LENGTH (CDR EXP))))
(T NIL))) ~oops
EXPR)
~COMPLIS compiles code to evaluate each expression in a list of
~ expressions and to push those values onto the stack. It
~ returns a list of that code. It is used to compile code
~ to evaluate arguments to called functions or LAMBDA expressions.
~ U is a list of expressions.
(DEFPROP COMPLIS
(LAMBDA(U M VPR)
(COND ((NULL U) NIL)
(T
(APPEND (COMPEXP (CAR U) M VPR)
(QUOTE ((PUSH P 1)))
(COMPLIS (CDR U) (DIFFERENCE M 1) VPR)))))
EXPR)
~LOADAC returns a list of (MOVE i j P) instructions, loading
~ consecutive accumulators from the top of the stack.
~ K indexes the accumulator loaded.
~ N indexes the stack offset.
(DEFPROP LOADAC
(LAMBDA(N K)
(COND ((GREATERP N 0) NIL)
(T
(CONS (LIST (QUOTE MOVE) K N (QUOTE P))
(LOADAC (PLUS N 1) (PLUS K 1))))))
EXPR)
~COMCOND compiles a COND.
~ U is a list of clauses in the COND.
~ L is a label to be emitted at the end of all code for
~ the COND.
(DEFPROP COMCOND
(LAMBDA(U M L VPR)
(COND ((NULL U) (LIST L))
(T
((LAMBDA(L1)
(APPEND (COMBOOL (CAAR U) M L1 NIL VPR)
(COMPEXP (CADAR U) M VPR)
(LIST (LIST (QUOTE JRST) L) L1)
(COMCOND (CDR U) M L VPR)))
(GENSYM)))))
EXPR)
~COMBOOL compiles code for a single predicate. That is, the
~ code generated evaluates the predicate and branches somewhere,
~ depending on the value.
~ P is the predicate.
~ L is a label which represents the branch point.
~ FLG is a flag. If FLG is NIL, code is to fall thru on non-NIL
~ result and branch to L on NIL result. If FLG is non-NIL,
~ code is to fall thru on NIL result and branch to L on
~ non-NIL result.
(DEFPROP COMBOOL
(LAMBDA(P M L FLG VPR)
(COND ((ATOM P) ~simple variable
(APPEND
(COMPEXP P M VPR)
(LIST
(LIST (COND (FLG (QUOTE JUMPN)) (T (QUOTE JUMPE))) 1 L))))
((EQ (CAR P) (QUOTE AND)) ~conjunction
(COND ((NOT FLG) (COMPANDOR (CDR P) M L NIL VPR))
(T
((LAMBDA(L1)
(APPEND (COMPANDOR (CDR P) M L1 NIL VPR)
(LIST (LIST (QUOTE JRST) 0 L))
(LIST L1)))
(GENSYM)))))
((EQ (CAR P) (QUOTE OR)) ~disjunction
(COND (FLG (COMPANDOR (CDR P) M L T VPR))
(T
((LAMBDA(L1)
(APPEND (COMPANDOR (CDR P) M L1 T VPR)
(LIST (LIST (QUOTE JRST) 0 L))
(LIST L1)))
(GENSYM)))))
((EQ (CAR P) (QUOTE NOT)) ~negation
(COMBOOL (CADR P) M L (NOT FLG) VPR))
(T ~other expression
(APPEND
(COMPEXP P M VPR)
(LIST
(LIST (COND (FLG (QUOTE JUMPN)) (T (QUOTE JUMPE)))
1
L))))))
EXPR)
~COMPANDOR compiles code for lists of predicates connected
~ conjunctively or disjunctively.
~ U is a list of predicates.
~ L is a label.
~ FLG is a flag. If FLG is NIL, we are to fall thru on non-NIL
~ results and branch to L on NIL results (AND case). If FLG
~ is non-NIL, we are to fall thru on NIL results and branch
~ to L on non-NIL results (OR case).
(DEFPROP COMPANDOR
(LAMBDA(U M L FLG VPR)
(COND ((NULL U) NIL)
(T
(APPEND (COMBOOL (CAR U) M L FLG VPR)
(COMPANDOR (CDR U) M L FLG VPR)))))
EXPR)
.END
.SKIP TO LINE 1
LCOM4:
.BEGIN VERBATIM SELECT 1
.REQUIRE "LCOM4" SOURCE_FILE
.END
.s COMPUTABILITY
.ss The function %2eval%1.
Except for speed and memory size all present day stored program
computers are equivalent in what computations they can do. A program
written for one computer can be translated to run on another. Indeed,
one can write a simulator for one computer to run on another. To put it
in commercial terms, no computer manufacturer can advertise that his machine
can do calculations impossible on the machine made by his competitors.
This is well known intuitively, and the first mathematical theorem
of this kind was proved by A.M. Turing (1936), who defined a primitive kind
of computer now called a Turing machine, and showed how to make a universal
machine that could do any computation done by any Turing machine when given
a description of the machine to be simulated and the initial tape of the
computation to be imitated.
In LISP the function %2eval%1 is a universal LISP function in the
sense that any computation done by any LISP function can be done by
%2eval%1 when %2eval%1 is given suitable arguments.
%2eval%1 has two arguments the first of which is a LISP expression
in the notation given in the previous section, while the second is a list of
pairs that give the values of any free variables that may occur in the
expression. Since any computation can be described as evaluating an expression
without free variables, the second argument plays a role mainly in the
recursive definition of %2eval%1, and we can start our computations with
the second argument NIL.
To illustrate this, suppose we want to apply the function %2alt%1
to the list (A B C D E), i.e. we wish to evaluate %2alt[(A B C D E)]%1.
This can be obtained by computing
.NOFILL
%2eval[((LABEL ALT (LAMBDA (X) (COND ((OR (NULL X) (NULL (CDR X))) X)
(T (CONS (CAR X) (ALT (CDDR X))))))) (QUOTE (A B C D E)), NIL]%1,
.FILL
and gives the expected result (A C E).
The second argument of %2eval%1, taken as NIL in the above example is a list
of dotted pairs where the first element of each pair is an atom representing
a variable and the second element is the value assigned to that variable.
A variable may occur more than once in the list and the value chosen is that
paired with the first occurrence of the variable. We illustrate this by
the equation
%2eval[(CAR X), ((X.(B.C)) (Y.A) (X.B))] = B%1,
i.e. we have evaluated %3a %2x%1 with %2 x = (B.C)%1. The value associated
with a variable in such a list of pairs is computed by the auxiliary function
%2assoc%1 which has the recursive definition
%2assoc[%2v, a] ← %3if n %2a %3then NIL %3else if aa %2a
%3 eq %2v %3then a %2a %3else %2alt[%2v, %3d %2a]%1.
.BEGIN PREFACE 0 NOFILL
Thus we have %2assoc[X, ((X.(B.C)) (Y.A) (X.B))] = (X.(B.C))%1.
A simplified version of the usual LISP %2eval%1 is the following:
%2eval[%2e, a] ←
.FILL
%3if at %2e %3then [%3if %2numberp %2e ∨ %2e
%3eq NIL ∨ %2e %3eq T %3then %2e %3else%2 assoc[%2e, a]]
.NOFILL
%3else if at a%2 e %3then
[%3if a %2e%3 eq CAR %3then a %2eval[%3ad %2e, a]
%3else if a %2e%3 eq CDR %3then d %2eval[%3ad %2e, a]
.FILL
%3else if a %2e%3 eq CONS %3then %2eval[%3ad %2e, a]
. %2eval[%3add %2e, a]
.NOFILL
%3else if a %2e%3 eq ATOM %3then at %2eval[%3ad %2e, a]
.FILL
%3else if a %2e%3 eq EQ %3then at %2eval[%3ad %2e, a]
%3eq %2eval[%3add %2e, a]
.NOFILL
%3else if a %2e%3 eq QUOTE %3then ad %2e
%3else if a %2e%3 eq COND %3then %2evcon[%3d %2e, a]
.FILL
%3else if a %2e%3 eq LIST %3then %2mapcar[%3d %2e,
λ%2x: %2eval[%2x, a]]
.NOFILL
%3else %2eval[%3d %2assoc[%3a %2e, a] . %3d %2e, a]]
.FILL
%3else if aa %2e %3eq LAMBDA %3then
%2eval[%3adda %2e, prup[%3ada %2e, mapcar[%3d %2e, λ%2x: %2eval[%2x, a]]] * %2a]
.BREAK
%3else if aa %2e %3eq LABEL %3then
%2eval[%3adda %2e . %3d %2e, [%3ada %2e . %3a %2e] . %2a]%1,
.NOFILL
where the auxiliary function %2evcon%1 is defined by
.FILL
%2evcon[%2u, a] ← %3if %2eval[%3aa %2u, a] %3then %2eval[%3ada %2u, a]
%3else %2evcon[%3d %2u, a]%1,
.END
and the auxiliary function %2prup%1 used for pairing up the elements of two
lists of equal length is defined by
.BEGIN PREFACE 0 NOFILL
.FILL
%2prup[%2u, v] ← %3if n %2u %3then NIL %3else [%3a %2u
. %3a %2v] . %2prup[%3d %2u,%3d %2v].%1
.NOFILL
.END
The way %2eval%1 works should be clear; atoms are either immediately
evaluable or have to be looked up on the list %2a%1; expressions whose first
term is one of the elementary functions are evaluated by performing the indicated
operation on the result of evaluating the arguments; %2list%1 has to be
handled specially, because it has an indefinite number of arguments; conditionals
are handled by an auxiliary function that evaluates the terms in the right order;
quoted S-expressions are trivial; non-elementary functions have their definitions
looked up on %2a%1 and substituted for their names; when a function is specified
by a λ, the inner expression is evaluated with a new %2a%1 which is obtained
by evaluating the arguments and pairing them up with the variables and putting
them on the front of the old %2a%1; and finally, %3label%1 is handled by
pairing the name of the function with the expression on %2a%1 and replacing
the whole function by the λ-part.
%2eval%1 plays both a theoretical and a practical role in LISP.
Historically, the list notation for LISP functions and %2eval%1 were first
devised in order to show how easy it is to define a universal function in LISP -
the idea was to advocate LISP as an alternative to Turing machines for doing
the elementary theory of computability. The notation used was chosen without
much regard for human convenience, because the original idea was purely
theoretical; the notation for conditional expressions, for example, has
an unnecessary extra level of parentheses. However, S. R. Russell noted that
%2eval%1 could serve as an interpreter for LISP and promptly programmed it
in machine language with minor modifications for practical purposes. Since
a compiler was long delayed, the interpreter was more easily modified and
handled some difficult cases with functional arguments better, an interpreter
based on %2eval%1 has remained a feature of most LISP systems.
The way %2eval%1 handles arguments corresponds to the call-by-value
method of parameter passing in ALGOL and similar languages. There is also
a form of %2eval%1 that corresponds to call-by-name. Here it is:
.BEGIN PREFACE 0 NOFILL
%2neval[%2e, a] ← %3if at %2e%3 then
[%3if %2e %3eq T %3 then T
%3else if %2e %3eq NIL %3then NIL
%3else if %2numberp e %3then %2e
%3else %2neval[%3ad %2assoc[%2e, a], %3dd %2 assoc[%2e, a]]]
%3else if at a %2e %3 then
[%3if a %2e %3eq CAR %3then a %2neval[%3ad %2e, a]
%3else if a %2e %3eq CDR %3then d %2neval[%3ad %2e, a]
.FILL
%3else if a %2e %3eq CONS %3then %2neval[%3ad %2e, a]
. neval[%3add %2e, a]
.NOFILL
%3else if a %2e %3eq ATOM %3then at %2neval[%3ad %2e, a]
.FILL
%3else if a %2e %3eq EQ %3then %2neval[%3ad %2e, a]
%3eq %2neval[%3add %2e, a]
.NOFILL
%3else if a %2e %3eq QUOTE %3then ad %2e
%3else if a %2e %3eq COND %3then %2nevcon[%3d %2e, a]
.FILL
%3else if a %2e %3eq LIST %3then %2mapcar[%3d %2e
, λ%2x: %2neval[%2x, a]]
.NOFILL
%3else %2neval[%3d %2assoc[%3a %2e, a] . %3d %2e, a]]
.FILL
%3else if aa %2e %3eq LAMBDA %3then %2neval[%3adda %2e,
nprup[%3ada %2e, %3d %2e, %2a]]
.NOFILL
.FILL
%3else if aa %2e %3eq LABEL %3then %2neval[%3adda %2e
. %3d %2e, [%3ada %2e . %3a %2e] . %2a], %1
.NOFILL
where the auxiliary function %2nevcon%1 is given by
.FILL
%2nevcon[%2u, a] ← %3if %2neval[%3aa %2u, %2a] %3then
%2neval[%3ada %2u, %2a] %3else %2nevcon[%3d %2u, %2a].%1
and nprup is
.FILL
%2nprup[%2u, v,a] ← %3if n %2u %3then %2a %3else [%3a %2u
. [%3a %2v . a]] . %2nprup[%3d %2u, %3d %2v,a].%1
.END
The difference between %2eval%1 and %2neval%1 is only in two terms.
%2eval%1 evaluates a variable by looking it up on the association list whereas
%2neval%1 looks it up on the association list and evaluates the result
in the context in which it was put on the association list.
Correspondingly, when a λ-expression is encountered, %2eval%1 forms a new
association list by pairing the values of the arguments with the variables
bound by the λ and putting the new pairs in front of the old association list,
whereas %2neval%1 pairs the arguments themselves with the
variables and puts them on the front of the association list. The function
neval also saves the current association list with each variable on the
association list, so that the variables can be evaluated in the correct context.
In most cases both give the same result with the same work, but %2neval%1
gives a result in some cases in which %2eval%1 loops. An example is
obtained by evaluating %2F[2, 1]%1 where %2F%1 is defined by
.NOFILL
%2F[%2x, y] ← %3if%2 x=0 %3then 0 %3else %2F[%2x-1, %2F[%2y-2, %2x]].%1
Exercises
.FILL
1. Write %2neval%1 and the necessary auxiliary functions in list
form, and try them out on some examples.
.ss Computability.%1
Some LISP calculations run on indefinitely. The most trivial
case occurs if we make the recursive definition
%2loop x ← %2loop x%1
and attempt to compute %2loop[%2x]%1 for any %2x%1 whatsoever. Don't
dismiss this example just because no-one would
write such an obviously useless
function definition. There is a sense in which it is the
"zero" of a large class of non¬terminating function definitions, and,
as the Romans experienced but never learned, leaving zero out of the number
system is a mistake.
Nevertheless, in most programming, non-terminating calculations are
the results of errors in defining functions. Therefore, it would be
useful to be able to tell whether a function definition gives a
result for all arguments. In fact, it would be useful to be able to tell
whether a function will terminate for a single argument. Let us make
this goal more precise.
Suppose that %2f%1 is a LISP function and %2a%1 is an S-expression,
and we would like to know whether the computation of %2f[%2a]%1 terminates.
Suppose %2f%1 is represented by the S-expression %2f*%1 in the usual
S-expression notation for LISP functions. Then the S-expression (f* (QUOTE a))%1
represents %2f[%2a]%1. Define the function %2term%1 by giving
%2term[%2e]%1 the value %3true%2 if %2e%1 is an S-expression of the
form (f* (QUOTE a))%1 for which %2f[%2a]%1 terminates and %3false%1 otherwise.
We now ask whether %2term%1 is a LISP function, i.e. can it be constructed
from %2car, cdr, cons, atom, %1 and %2eq%1 using λ, %3label%1, and
conditional expressions? Well, it can't, as we shall shortly prove,
and this means that it is not %2computable%1 whether a LISP calculation
terminates, since if %2term%1 were computable by any computer or in any
recognized sense, it could be represented as a LISP function. Here is the
proof:
Consider the function %2terma%1 defined from %2term%1 by
%2terma u ← %3if%2 term list[%2u, list[QUOTE%2, u]] %3then%2 loop u
%3else true%1,
and suppose that %2f%1 is a LISP function and that %2f*%1 is its
S-expression representation. What is %2terma f*%1?
Well %2terma f*%1 tells us whether the computation of %2f[%2f*]%1
terminates, and it tells us this by going into a loop if %2f[%2f*]%1
terminates and giving %3true%1 otherwise. Now if %2term%1 were
a LISP function, then %2terma%1 would also be a LISP function. Indeed if
%2term%1 were represented by the S-expression %2term*%1, then
%2terma%1 would be represented by the S-expression
%2terma* = (LAMBDA (U) (COND ((%2term* (LIST U
(LIST (QUOTE QUOTE) U))) (LOOP U)) (T T))).
Now consider %2terma[%2terma*]%1. According to the definition of %2terma%1,
this will tell us whether %2terma[%2terma*]%1 is defined, i.e. it tells
about itself. However, it gives this answer in a contradictory way; namely
%2terma[%2terma*]%1 looping tells us that %2terma[%2terma*]%1 terminates,
and %2terma[%2terma*]%1 being %3true%1 tells us that %2terma[%2terma*]%1
doesn't terminate. This contradiction tells us that %2term%1 is not a
LISP function, and there is no general procedure for telling whether
a LISP calculation terminates.
The above result does not exclude LISP functions that tell whether
LISP calculations terminate. It just excludes perfect ones. Suppose
we have a function %2t%1 that sometimes says calculations terminate,
sometimes says they don't terminate, and sometimes runs on indefinitely.
We shall further assume that when %2t%1 gives an
answer it is always right. Given such a function we can improve it a bit
so that it will always give the right answer when the calculation it is
asked about terminates. This is done by mixing the computation of %2t[%2e]%1
with a computation of %2eval[%2e, NIL]%1 doing the computations
alternately. If the %2eval[%2e, NIL]%1 computation ever terminates,
then the new function asserts termination.
Given such a %2t%1, we can always find a calculation that
does not terminate but %2t%1 doesn't say so. The construction is just like that
used in the previous proof. Given %2t%1, we construct
%2ta u ← %3if%2 t list[%2u, list[QUOTE%2, u]] %3then%2 loop u
%3else true%1,
and then we consider %2ta[%2ta*]%1. If this had the value %3true%1,
then it wouldn't terminate so therefore it doesn't terminate but is
not one of those expressions which %2t%1 decides. Thus for any
partial decider we can find a LISP calculation which doesn't terminate
but which the decider doesn't decide.
This can in turn be used to get a slightly better decider, namely
%2t%51[%2e] ← %3if%2 e = %2ta* %3then DOESN'T-TERMINATE
%3else%2 t[%2e]%1.
Of course, %2t%51%1 isn't much better than %2t%1, since it can decide
only one more computation, but we can form %2t%52%1 by applying the same
process, and so forth. In fact, we can even form %2t%5∞%1 which
decides all the cases decided by any %2t%5n%1. This can be further
improved by the same process, etc. How far can we go? The answer is
technical; namely, the improvement process can be carried out any
recursive ordinal number of times.
Unfortunately, this kind of improvement seems to be superficial,
since none of the new computations proved not to terminate are likely
to be of practical interest.
Exercises.
.ITEM←0;
#. Write a function that gives %2t%5n+1%1 in terms of %2t%5n%1.
#. Write a function that gives %2t%5∞%1 in terms of %2t%1.
#. If you know about Turing machines, write a LISP function to
simulate an arbitrary Turing machine given a description of the
machine in some convenient notation.
#. Write a LISP function that will translate a Turing machine
description into a LISP function that will do the same computation.
#. If you really like Turing machines, write a description of
a Turing machine that will interpret LISP internal notation.
.s PROVING LISP PROGRAMS CORRECT
In this chapter we will introduce techniques for proving
LISP programs correct.
The techniques will mainly be limited to what we may call %2clean%1
LISP programs. In particular, there must be no side effects,
because our methods depend on the ability to replace subexpressions
by equal expressions.
The necessary basic facts can be divided into several categories:
first order logic including conditional forms and first order
lambda-expressions,
algebraic facts about lists and S-expressions,
facts about the inductive structure of lists and
S-expressions, and general facts about functions defined by
recursion.
.ss First order logic with conditional forms and lambda-expressions.
The logic we shall use is called first order logic with equality,
but we will extend it by allowing conditional forms to be terms and
lambda-expressions to be function expressions. From the mathematical
point of view, these extensions are inessential, because, as we shall see,
every sentence that includes conditional forms or first order lambdas can
readily be transformed into an equivalent sentence without them. However,
the extensions are practically important, because they permit us to use
recursive definitions directly as formulas of the logic.
Formulas of the logic are built from constants, variables
predicate symbols, and function symbols using function application,
conditional forms, boolean forms, lambda expressions, and quantifiers.
%3Constants%1: We will use S-expresssions as constants standing for
themselves and also lower case letters from the first part
of the alphabet to represent constants in other domains than
S-expressions.
%3Variables%1: We will use the letters %2u%1 thru %2z%1 with or
without subscripts as variables. The variables %2u%1 and %2v%1
will usually stand for lists while %2x%1 thru %2z%1 will stand
for S-expressions.
%3Function symbols%1: The letters %2f%1, %2g%1 and %2h%1 with or
without subscripts are used as function symbols.
The LISP function symbols %3a%1, %3d%1 and . (as an infix) are also
used as function symbols.
We suppose that each function symbol takes the same definite number of
arguments every time it is used.
We will often use one argument functions symbols as prefixes, i.e.
without brackets, just as %3a%1 and %3d%1 have been used up to now.
%3Predicate symbols%1: The letters %2p%1, %2q%1 and %2r%1 with or
without subscripts are used as predicate symbols. We will
also use the LISP predicate symbol %3at%1 as a constant predicate
symbol. The equality symbol = is also used as an infix.
We suppose that each predicate symbol takes the same definite
number of arguments each time it is used.
Infix and prefix notation will also be used where this is
customary.
Next we define terms, sentences, function expressions
and predicate expressions inductively. A term is
an expression whose value will be an object like an S-expression
or a number while a sentence has value T%1 or F%1.
Terms are used in making sentences, and sentences occur in
terms so that the definitions are %2mutually recursive%1 where
this use of the word %2recursive%1 should be distinguished from
its use in recursive definitions of functions.
Function expressions are also involved in the mutual recursion.
%3Terms%1: Constants are terms, and variables are terms. If
%2f%1 is a function expression taking %2n%1 arguments, and %2t%51%2,_..._,t%5n%2%1
are terms, then %2f[t%51%2,_..._,t%5n%2]%1 is a term. If %2p%1 is
a sentence and %2t%51%2%1 and %2t%52%1 are terms, then
%3if%2_p_%3then%2_t%51%2_%3else%2_t%52%1%1 is a term. We soften the
notation by allowing infix symbols where this is customary.
%3Sentences%1:
If %2p%1 is a predicate expression taking %2n%1 arguments and
%2t%51%2,_..._,t%5n%2%1 are terms, then %2p[t%51%2,_..._,t%5n%2]%1 is
a sentence. Equality is also used as an infixed predicate
symbol to form sentences, i.e. %2t%51%2_=_t%52%1%1 is a sentence.
If %2p%1 is a sentence, then %2¬p%1 is also a sentence. If
%2p%1 and %2q%1 are sentences, then %2p∧q%1, %2p∨q%1, %2p⊃q%1,
and %2p≡q%1 are sentences. If %2p%1, %2q%1 and %2r%1 are
sentences, then %3if%2_p_%3then%2_q_%3else%2_r%1 is a sentence.
If %2x%51%2,_...,_x%5n%2%1 are variables, and %2p%1 is a term, then
%2∀x%51%2_..._x%5n%2:t%1 and %2∀x%51%2_..._x%5n%2:t%1 are sentences.
%3Function expressions%1: A function symbol is a function expression.
If %2x%51%2,_..._,x%5n%2%1 are variables and %2t%1 is a term, then
%2[λx%51%2,_..._,x%5n%2:t]%1 is a function expression.
%3Predicate expressions%1: A predicate symbol is a predicate expression.
If %2x%51%2,_..._,x%5n%2%1 are variables and %2p%1 is a sentence, then
%2[λx%51%2,_..._,x%5n%2:p]%1 is a predicate expression.
An occurrence of a variable %2x%1 is called bound if
it is in an expression of one of the forms %2[λx%51%2_..._x%5n%2:t]%1,
%2[λx%51%2_..._x%5n%2:p]%1, %2[∀x%51%2_..._x%5n%2:p]%1 or %2[∃x%51%2_..._x%5n%2:p]%1
where %2x%1 is one
of the numbered %2x%1's. If not bound an occurrence is called free.
The %2semantics%1 of first order logic consists of the rules
that enable us to determine when a sentence is true and when it
is false. However, the truth or falsity of a sentence is relative
to the interpretation assigned to the constants, the function and predicate
symbols and the free variables of the formula. We proceed as follows:
We begin by choosing a domain. In most cases we shall consider
the domain will include the S-expressions and any S-expression constants
appearing in the formula stand for themselves. We will allow for the
possibility that other objects than S-expressions exist, and some constants
may designate them. Each function or predicate symbol is assigned a
function or predicate on the domain. We will normally assign to the
basic LISP function and predicate symbols the corresponding basic LISP
functions and predicates. Each variable appearing free in a sentence is
also assigned an elemet of the domain. All these assignments constitute
an interpretation, and the truth of a sentence is relative to the
interpretation.
The truth of a sentence is determined from the values of its
constituents by evaluating successively larger subexpressions. The
rules for handling functions and predicates, conditional expressions,
equality, and Boolean expressions are exactly the same as those we
have used in the previous chapters. We need only explain quantifiers:
%2∀x%51%2_..._x%5n%2:e%1 is assigned true if and only if %2e%1 is assigned
true for all assignments of elements of the domain to the %2x%1's. If
%2e%1 has free variables that are not among the %2x%1's, then the
truth of the sentence depends on the values assigned to these remaining
free variables. %2∃x%51%2_..._x%5n%2:e%1 is assigned true if and only if %2e%1
is assigned true for %2some%1 assignment of values in the domain to
the %2x%1's. Free variables are handled just as before.
%2λx%51%2_..._x%5n%2:u%1 is assigned a function or predicate according
to whether %2u%1 is a term or a sentence. The value of
%2[λx%51%2_..._x%5n%2:u][t%51%2,...,t%5n%2] is obtained by evaluating the
%2t%1's and using these values as values of the %2x%1's in the
evaluation of %2u%1. If %2u%1 has free variables in addition to the %2x%1's,
the function assigned will depend on them too.
Those who are familiar with the lambda calculus should note
that λ is being used here in a very limited way. Namely, the variables
in a lambda-expression take only elements of the domain as values,
whereas the essence of the lambda calculus is that they take arbitrary
functions as values. We may call these restricted lambda expressions
%2first order lambdas%1.
.ss Conditional forms.
All the properties we shall use of conditional forms follow
from the relation
%2[p ⊃ [%3if%2 p %3then %2a %3else%2 b] = a] ∧ [¬p ⊃ [%3if%2 p %3then%2 a
%3else%2b] = b]%1.
(If we weren't adhering to the requirement that all terms be defined for
all values of the variables, the situation would be more complicated).
It is, however, worthwhile to list separately some properties
of conditional forms.
First we have the obvious
%3if_T_%3then%2_a_%3else%2_b%2 = a%1
and
%3if_F_%3then%2_a_%3else%2_b%2 = b%1.
Next we have a %2distributive law%1 for functions applied
to conditional forms, namely
%2f[%3if%2 p %3then%2 a %3else%2 b] = %3if%2 p %3then%2 f[a] %3else%2 f[b]%1.
This applies to predicates as well as functions and can also be used
when one of the arguments of a function of several arguments is a
conditional form. It also applies when one of the terms of
a conditional form is itself a conditional form.
Thus
%3if%2 [%3if%2 p %3then%2 q %3else%2 r] %3then%2 a %3else%2 b =
%3if%2 p %3then%2 [%3if%2 q %3then%2 a %3else%2 b] %3else%2 [%3if%2 r
%3then%2 a %3else%2 b]%1
and
%3if%2 p %3then%2 [%3if%2 q %3then%2 a %3else%2 b] %3else%2 c =
%3if%2 q %3then%2 [%3if%2 p %3then%2 a %3else%2 c] %3else%2 [%3if%2 p
%3then%2 b %3else%2 c]%1.
When the expressions following %3then%1 and %3else%1 are
sentences, then the conditional form can be replaced by a sentence
according to
%2[%3if%2 p %3then%2 q %3else%2 r] ≡ [p ∧ q] ∨ [¬p ∧ r]%1.
These two rules permit eliminating conditional forms from sentences
by first using distributivity to move the conditionals
to the outside of any functions and then replacing the conditional
form by a Boolean expression.
Note that the elimination of conditional forms may increase
the size of a sentence, because %2p%1 occurs twice in the right
hand side of the above equivalence. In the most unfavorable case,
%2p%1 is dominates the size of the expression so that writing it
twice almost doubles the size of the expression.
Suppose that %2a%1 and %2b%1 in
%3if%2_p_%3then%2_a_%3else%2_b%1
are expressions that may contain the sentence %2p%1. Occurrences
of %2p%1 in %2a%1 can be replaced by T%1, and occurrences of %2p%1
in %2b%1 can be replaced by F%1. This follows from the fact that
%2a%1 is only evaluated if %2p%1 is true and %2b%1 is evaluated only
if %2p%1 is false.
This leads to a strengthened form of the law of replacement
of equals by equals. The ordinary form of the law says that if
we have %2e_=_e'%1, then we can replace any occurrence of %2e%1
in an expression by an occurrence of %2e'%1. However, if we want
to replace %2e%1 by %2e'%1 within %2a%1 within
%3if%2_p_%3then%2_a_%3else%2_b%1,
then we need only prove %2p_⊃_e_=e'%1, and to make the replacement
within %2b%1 we need only prove %2¬p_⊃_e_=_e'%1.
Additional facts about conditional forms are given in (McCarthy 1963a)
including a discussion of canonical forms that parallels the canonical
forms of Boolean forms. Any question of equivalence of conditional
forms is decidable by truth tables analogously to the decidability of
Boolean forms.
.ss Lambda-expressions.
The only additional rule required for handling lambda-expressions
in first order logic is called %2lambda-conversion%1, essentially
%2[λx:e][a] =%1 <the result of substituting %2e%1 for %2x%1 in %2a%1>.
As examples of this rule, we have
%2[λx:%3a%2 x . y][u . v] = [%3a%2[u . v]] . y%1.
However, a complication requires modifying the rule. Namely,
we can't substitute for a variable and expression that has a free variable
into a context in which that free variable is bound. Thus it would be
wrong to substitute %2x_+_y%1 for %2x%1 in %2∀y:[x_+_y_=_z]%1 or into
the term %2[λy:x_+_y][u_+_v]%1. Before doing the substitution, the
variable %2y%1 would have to be replaced in all its bound occurrences by
a fresh variable.
Lambda-expressions can always be eliminated from sentences and
terms by lambda-conversion, but the expression may increase greatly
in length if a lengthy term replaces a variable that occurs more than
once in %2e%1. It is easy to make an expression of length %2n%1 whose
length is increased to 2%5n%1 by converting its %2n%1 nested lambda-expressions.
.ss Algebraic axioms for S-expressions and lists.
The algebraic facts about S-expressions are expressed
by the following sentences of first order logic:
%2∀x.(issexp x ⊃ %3at%2 x ∨ (issexp %3a%2 x ∧ issexp %3d%2 x
∧ x = (%3a%2 x . %3d%2 x)))%1
and
%2∀x y.(issexp x ∧ issexp y ⊃ issexp(x.y) ∧ ¬%3at%2(x.y)
∧ x = %3a%2(x.y) ∧ y = %3d%2(x.y))%1.
Here %2issexp e%1 asserts that the object %2e%1 is an S-expression
so that the sentences used in proving a particular program correct
can involve other kinds of entities as well.
If we can assume that all objects are S-expressions or can declare certain
variables as ranging only over S-expressions, we can simplify the axioms
to
%2∀x.[%3at%2 x ∨ x = [%3a%2 x . %3d%2 x]]%1
and
%2∀x y.[¬%3at%2[x.y] ∧ x = %3a%2[x.y] ∧ y = %3d%2[x.y]]%1.
The algebraic facts about lists are expressed by the following
sentences of first order logic:
%2∀x. islist x ⊃ x = NIL%2 ∨ islist %3d %2x%1,
%2∀x y. islist y ⊃ islist[x . y]%1,
%2∀x y. islist y ⊃ %3a%2[x . y] = x ∧ %3d%2[x.y] = y%1.
We can rarely assume that everything is a list, because the
lists usually contain atoms which are not themselves lists.
These axioms are analogous to the algebraic part of Peano's
axioms for the non-negative integers. The analogy can be made clear
if we write Peano's axioms using %2n'%1 for the successor of %2n%1
and %2n%5-%1 for the predecessor of %2n%1. Peano's algebraic axioms
then become
%2∀n: n' ≠ 0%1,
%2∀n: (n')%5-%2 = n%1,
and
%2∀n: n ≠ 0 ⊃ (n%5-%2)' = n%1.
Integers specialize lists if we identify 0 with NIL%1 and assume that
there is only one object (say 1) that can serve as a list element.
Then %2n'_=_cons[1,n]%1, and %2n%5-%2_=_%3d%2 n%1.
Clearly S-expressions and lists satisfy the axioms given for them,
but unfortunately these algebraic axioms are insufficient to
characterize them. For example, consider a domain of one element
a%1 satisfying
%3a%2 a = %3d%2 a = a . a = a%1.
It satisfies the algebraic axioms for S-expressions. We can exclude
it by an axiom %2∀x.(%3a%2 x ≠ x)%1, but this won't exclude other
circular list structures that eventually return to the same element
by some %3a-d%1 chain. Actually we want to exclude all infinite
chains, because most LISP programs won't terminate unless every
%3a-d%1 chain eventually terminates in an atom. This cannot be
done by any finite set of axioms.
.ss Axiom schemas of induction.
In order to exclude infinite list structures we need axiom schemas
of induction analogous to Peano's for the integers. Peano's induction schema
is ordinarily written
%2P(0) ∧ ∀n:(P(n) ⊃ P(n')) ⊃ ∀n:P(n)%1.
Here %2P(n)%1 is an arbitrary predicate of integers, and we get particular
instances of the schema by substituting particular predicates.
It is called an axiom schema rather than an axiom, because we consider
the schema, which is not properly a sentence of first order logic, as
standing for the infinite collection of axioms that arise from it by
substituting all possible predicates for %2P%1.
Peano's induction schema can also be written
%2∀n:(n = 0 ∨ P(n%5-%2) ⊃ P(n)) ⊃ ∀n:P(n)%1,
and the equivalence of the two forms is easily proved.
The S-expression analog is
%2∀x:[issexp x ⊃ [%3at%2 x ∨ P[%3a%2 x] ∧ P[%3d%2 x] ⊃ P[x]]] ⊃
∀x:[issexp x ⊃ P[x]]%1,
or, assuming everything is an S-expression
%2∀x:[%3at%2 x ∨ P[%3a%2 x] ∧ P[%3d%2 x] ⊃ P[x]] ⊃ ∀x:P[x]%1.
The corresponding axiom schema for lists is
%2∀u:[islist u ⊃ [%3n%2 u ∨ P[%3d%2 u] ⊃ P[u]]] ⊃ ∀u:[islist u
⊃ P[u]]%1.
These schemas are called principles of %2structural induction%1,
since the induction is on the structure of the entities involved.
.ss Proofs by structural induction.
Recall that the operation of appending two lists is defined by
!!si2: %2u * v ← %3if n%2 u %3then%2 v %3else a%2 u . [%3d%2 u * v]%1.
Let us assume that
%2u * v%1 is defined for all %2u%1 and %2v%1, i.e. the computation
described above terminates for all %2u%1 and %2v%1; we will
show how to prove it later. Then we can replace ({[5] si2})
by the sentence
!!si1: %2∀u v:[islist u ∧ islist v ⊃
[%2u * v = %3if n%2 u %3then%2 v %3else a%2 u . [%3d%2 u * v]]]%1.
Now suppose we would like to prove %2∀v:[NIL%2 * v = v]%1. This is
quite trivial; we need only substitute NIL%1 for %2x%1 in ({[5] si1}),
getting
→NIL%2 * v ∂(15)_= %3if n NIL %3then%2 v %3else a NIL%2 . [%3d NIL%2 * v]
∂(15)_= v%1.
Next consider proving %2∀u:[u * NIL%2 = u]%1. This cannot be done by
simple substitution, but it can be done as follows: First substitute
%2λu:[u * NIL%2 = u]%1 for %2P%1 in the induction schema
%2∀u:[islist u ⊃ [%3n%2 u ∨ P[%3d%2 u] ⊃ P[u]]] ⊃ ∀u:[islist u
⊃ P[u]]%1,
getting
%2∀u:[islist u ⊃ [%3n%2 u ∨ %2λu:[u * NIL%2 = u][%3d%2 u]
⊃ %2λu:[u * NIL%2 = u][u]]] ⊃ ∀u:[islist u
⊃ %2λu:[u * NIL%2 = u][u]]%1.
Carrying out the indicated lambda conversions makes this
!!si3: %2∀u:[islist u ⊃ [qn u ∨ qd u * qnil = qd u]
⊃ u * qnil = u] ⊃ ∀u:[islist u ⊃ u * qnil = u]%1.
Next we must use the recursive definition of %2u*v%1. There
are two cases according to whether qn %2u%1 or not.
In the first case, we substitute qnil for %2v%1 and get qnil*qnil_=_qnil,
and in the second case we use the hypothesis qd_u_*_qnil_=_qd_u and the
third algebraic axiom for lists to make the simplification
!!si4: %2qa u . [qd u * qnil] = qa u . qd u = u.%1
Combining the cases gives the hypothesis of ({eq si3}) and hence
its conclusion, which is the statement to be proved.
.ss First Order Theory of Partial Functions
.BACK;