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Programming Languages
2nd edition
Tucker and Noonan
Chapter 14
Functional Programming
It is better to have 100 functions operate one one data structure,
than 10 functions on 10 data structures.
A. Perlis
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Contents
14.1 Functions and the Lambda Calculus
14.2 Lisp
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14-2
1
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Overview of Functional Languages
• They emerged in the 1960’s with Lisp
• Functional programming mirrors mathematical
functions: domain = input, range = output
• Computation is viewed as a mathematical function
mapping inputs to outputs.
• Pure functional programming is state-free: no
assignment
• Referential transparency: a function’s result
depends only upon the values of its parameters.
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14-3
14.1 Functions and the Lambda Calculus
The function Square has R (the reals) as domain and range.
Square : R  R
Square(n) = n2
A function is total if it is defined for all values of its domain.
Otherwise, it is partial. E.g., Square is total.
A lambda expression is a particular way to define a function:
LambdaExpression  variable | ( M N) | (  variable . M )
M  LambdaExpression
N  LambdaExpression
E.g., (  x . x2 ) represents the Square function.
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Properties of Lambda Expressions
In ( x . M), x is bound. Other variables in M are free.
A substitution of N for all occurrences of a variable x in M is
written M[x  N]. Examples:
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Lambda Expressions
Lambda expressions describe nameless functions
Lambda expressions are applied to parameter(s) by
placing the parameter(s) after the expression
e.g. (x . x * x * x) 2 → 2 * 2 * 2 → 8
The process of evaluating a lambda expression is
called beta (β)-reduction.
A lambda expression which may be reduced is called
a redex (reducible expression).
A lambda expression which may not be further
reduced (i.e., has no redexes) is said to be in
normal form.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4
rede
x
15-7b
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Wesley, 2010.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4
15-7c
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
15-7d
Concepts of Programming Languages,
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Robert W.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
Result is a
function!
15-7e
Concepts of Programming Languages,
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
rede
x
15-7f
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
15-7g
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
15-7h
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
15-7i
Concepts of Programming Languages,
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
15-7j
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4
15-7k
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4
rede
15-7l
x
Concepts of Programming Languages, 9th ed., by Robert W. Sebesta. Addison Wesley, 2010.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4
15-7m
Concepts of Programming Languages,
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Robert W.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4 → (x . (y . 3 + y) x) 4
15-7n
Concepts of Programming Languages,
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Robert W.
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4 → (x . (y . 3 + y) x) 4
15-7o
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Functional Forms
A higher-order function, or functional form,
is one that either takes functions as
parameters or yields a function as its
result, or both
(x . y . x + y) 3 4 → (y . 3 + y) 4
→3+4
→7
(x . y . x + y) 3 → y . 3 + y
(f . x . f x) (y . 3 + y) 4 → (x . (y . 3 + y) x) 4
→ (y . 3 + y) 4
15-7p
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Non-Terminating Lambda Expression
(x . x x) (x . x x) → (x . x x) (x . x x)
→ (x . x x) (x . x x)
→…
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Function Evaluation
Lazy evaluation = delaying argument evaluation in a
function call until the argument is needed.
– Advantage: flexibility
Eager evaluation = evaluating arguments at the beginning
of the call.
– Advantage: efficiency
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Status of Functions
In imperative and OO programming, functions have
different (lower) status than variables.
In functional programming, functions have same status
as variables; they are first-class entities.
– They can be passed as arguments in a call.
– They can transform other functions.
A function that operates on other functions is called a
functional form. E.g., we can define
g(f, [x1, x2, … ]) = [f(x1), f(x2), …], so that
g(Square, [2, 3, 5]) = [4, 9, 25]
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14.2 Lisp
LISP is an acronym for LISt Processor
LISP was invented by John McCarthy of MIT
(later Stanford) in 1958
Data object types: originally only atoms and
lists
List form: parenthesized collections of
sublists and/or atoms
e.g., (A B (C D) E)
Originally, LISP was a typeless language
LISP lists are stored internally as singlelinked lists
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LISP Interpretation
Lambda notation is used to specify functions and
function definitions. Function applications and data
have the same form.
e.g., If the list (A B C) is interpreted as data it is
a simple list of three atoms, A, B, and C
If it is interpreted as a function application,
it means that the function named A is
applied to the two parameters, B and C
The first LISP interpreter appeared only as a
demonstration of the universality of the
computational capabilities of the notation
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Primitive Functions
Arithmetic: +, -, *, /, ABS, SQRT,
REMAINDER, MIN, MAX
e.g., (+ 5 2) yields 7
QUOTE - takes one parameter; returns the
parameter without evaluation
–
–
QUOTE is required because the Lisp interpreter, named
EVAL, always evaluates parameters to function
applications before applying the function. QUOTE is
used to avoid parameter evaluation when it is not
appropriate
QUOTE can be abbreviated with the apostrophe prefix
operator
'(A B) is equivalent to (QUOTE (A B))
15-16
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Running Lisp
GCL – Gnu Common Lisp
Path - /usr/bin/gcl
Example
brb0164@faculty% gcl
GCL (GNU Common Lisp) 2.6.7 CLtL1 Feb 1 2012 09:07:26
>(+ 5 2)
7
>'(A B)
(A B)
>(A B)
Error in EVAL [or a callee]: The function A is undefined.
>^D
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Function Definition: DEFUN
A Function for Constructing Functions
DEFUN - To bind names to lambda
expressions
e.g.,
(DEFUN cube (x) (* x x x))
Example use:
(cube 4)
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Lisp Example: cube
brb0164@faculty% gcl
GCL (GNU Common Lisp) 2.6.7 CLtL1 Feb 1 2012 09:07:26
>(DEFUN cube (x) (* x x x))
CUBE
>(cube 4)
64
>^D
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Function Definition: LAMBDA
Lambda Expressions
– Form is based on  notation
e.g., (LAMBDA (x) (* x x x))
x is called a bound variable
Lambda expressions can be applied
e.g., ((LAMBDA (x) (* x x x)) 4)
((LAMBDA (x y) (+ x y)) 3 4)
((LAMBDA (x) (LAMBDA (y) (+ x y))) 3)
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Lisp Example: LAMBDA
brb0164@faculty% gcl
GCL (GNU Common Lisp) 2.6.7 CLtL1 Feb 1 2012 09:07:26
>((LAMBDA (x) (* x x x)) 4)
64
>((LAMBDA (x y) (+ x y)) 3 4)
7
>^D
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Evaluation
Parameters are evaluated, in no particular
order
The values of the parameters are substituted
into the function body
The function body is evaluated
The value of the last expression in the body
is the value of the function
15-15
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Control Flow: COND
Multiple Selection - the special form, COND
General form:
(COND
(predicate_1 expr {expr})
(predicate_2 expr {expr})
...
(predicate_n expr {expr})
(T expr {expr}))
Returns the value of the last expression in
the first pair whose predicate evaluates to
true
15-22
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Lisp Example: min
brb0164@faculty% cat min.lsp
(defun min (x y)
(cond
((<= x y) x)
((>= x y) y)
)
)
brb0164@faculty% gcl
GCL (GNU Common Lisp) 2.6.7 CLtL1
Feb 1 2012 09:07:26
>(load "min")
Loading min.lsp
Warning: MIN is being redefined.
Finished loading min.lsp
T
>(min 17 45)
17
>(min 0 1)
0
>(min 65 3)
3
>^D
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Lisp Example: factorial
brb0164@faculty% cat factorial.lsp
(defun factorial (n)
(cond
((= n 0) 1)
(t (* n (factorial (- n 1))))
)
)
brb0164@faculty% gcl
>(load "factorial")
Loading factorial.lsp
Finished loading factorial.lsp
T
>(factorial 5)
120
>(factorial 10)
3628800
>^D
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Representation of Lists
'(A B C)
•
•
•
•
A
B
'((A B) C D)
•
•
•
•
A
∘
•
C
•
•
∘
C
•
•
∘
D
B
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List Functions: CAR and CDR
CAR takes a list parameter; returns
element of that list
e.g., (CAR '(A B C)) yields A
(CAR '((A B) C D)) yields (A
CDR takes a list parameter; returns
after removing its first element
e.g., (CDR '(A B C)) yields (B
(CDR '((A B) C D)) yields (C
the first
B)
the list
C)
D)
15-25
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List Functions: CAR and CDR Extensions
(CADR L) ≡ (CAR (CDR L))
e.g., (CADR '(A B C)) yields B
(CADR '((A B) C D)) yields C
(CDDR L) ≡ (CDR (CDR L))
e.g., (CDDR '(A B C)) yields (C)
(CDDR '((A B) C D)) yields (D)
(CAAR L) ≡ (CAR (CAR L))
e.g., (CAAR '(A B C)) yields an error
(CAAR '((A B) C D)) yields A
Any number of A’s and D’s may be combined
e.g., (CADADR '((A B) (C D))) yields D
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List Predicates: NULL, ATOM and LISTP
NULL returns true (T) if the argument is null
(NIL) and false (NIL) otherwise.
ATOM returns true (T) if the argument is an
atom and false (NIL) otherwise.
LISTP returns true (T) if the argument is a list
and false (NIL) otherwise.
15-27
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Lisp Example: length
brb0164@faculty% cat length.lsp
(defun length (list)
(cond
((null list) 0)
(t (+ 1 (length (cdr list))))
)
)
brb0164@faculty% gcl
>(load "length")
Loading length.lsp
Warning: LENGTH is being redefined.
Finished loading length.lsp
T
>(length '(a b c d))
>(length nil)
4
0
>(length '(a (b c)))
>(length '(a (b c) d (e f (g h))))
2
4
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List Functions: CONS
CONS takes two parameters, the first of which
can be either an atom or a list and the
second of which should be a list; returns a
new list that includes the first parameter as
its first element and the second parameter
as the remainder of its result
e.g., (CONS 'A '(B C)) returns (A B C)
(CONS 'A 'B) returns (A . B)
•
•
A
B
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List Functions: LIST
LIST takes any number of parameters;
returns a list with the parameters as
elements
e.g., (LIST 'A 'B 'C) returns (A B C)
(LIST '(A B C)) returns ((A B C))
15-24b
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List Functions: APPEND
APPEND takes two parameters, the first of
which is a list and the second of which
should be a list; returns a new list with the
CDR of the last node in the first parameter
being the second parameter
e.g., (APPEND '(A) '(B C)) returns
(A B C)
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Defining Local Scope: PROG, RETURN
and SETQ
(PROG (VAR1 VAR2 … VARn)
EXP1 EXP2 … EXPm)
VAR1, VAR2, … VARn have local scope in
EXP1, EXP2, … EXPm
The variables may be assigned values using
the (SETQ VAR EXP) function.
The last expression in the body, EXPm, is
returned using a RETURN function which
may also appear anywhere an expression
may be used inside the block (e.g., to
return a different value).
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Lisp Examples: List Manipulation
brb0164@faculty% gcl
> (setq a (cons 'x nil))
(x)
> (setq b (cons 'y a))
(y x)
> (setq c (append b 'z))
(y x . z)
> (car c)
y
> (cadr c)
x
> (cddr c)
z
> (setq d (append b '(z)))
(y x z)
> (setq e (cons d d))
((y x z) y x z)
> (setq f (append d d))
(y x z y x z)
> (setq g (cons d (list d)))
((y x z) (y x z))
> (car g)
(y x z)
> (cdr g)
((y x z))
> (cadr g)
(y x z)
> (caadr g)
y
> (cdadr g)
(x z)
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Functions That Build Code
It is possible in Lisp to define a function that
builds Lisp code and requests its
interpretation
This is possible because the interpreter is a
user-available function, EVAL
A similar function is APPLY which takes two
parameters, the first a function F and the
second a list of parameters L, and applies F
to L.
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Lisp Examples: Lambda Expression
Manipulation
brb0164@faculty% gcl
>(setq add '(lambda (x y) (+ x y)))
(LAMBDA (X Y) (+ X Y))
>(car add)
>(cdadr add)
LAMBDA
(Y)
>(cdr add)
>(caddr add)
((X Y) (+ X Y))
(+ X Y)
>(cadr add)
>(caaddr add)
(X Y)
+
>(cddr add)
>(apply add '(3 4))
((+ X Y))
7
>(caadr add)
>(eval '(apply add '(3 4)))
X
7
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Lisp Example: postfix
brb0164@faculty% cat postfix.lsp
; postfix converts a postfix expression,
; represented as a list of operands and
; operators, into a syntax tree.
(defun postfix (exp)
(car (postfixstack exp nil)))
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Lisp Example: postfix
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
((eq (car exp) '–) (postfixstack (cdr exp)
(cons (list '– (cadr stack) (car stack))
(cddr stack))))
((eq (car exp) '*) (postfixstack (cdr exp)
(cons (list '* (cadr stack) (car stack))
(cddr stack))))
((eq (car exp) '/) (postfixstack (cdr exp)
(cons (list '/ (cadr stack) (car stack))
(cddr stack))))
(t (postfixstack (cdr exp) (cons (car exp) stack)))
))))
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Lisp Example: postfix
brb0164@faculty% gcl
>(load "postfix")
Loading postfix.lsp
Finished loading postfix.lsp
T
>(postfix '(a b c * +))
(+ A (* B C))
>(postfix '(a b + c * d / e –))
(– (/ (* (+ A B) C) D) E)
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Lisp Example: postfix (cont)
(defun postfix (exp)
(car (postfixstack exp nil)))
(postfix '(a b c * +))
(car (postfixstack '(a b c * +) nil))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
…
(t (postfixstack (cdr exp)
(cons (car exp) stack)))
))))
(postfixstack '(a b c * +) nil)
(postfixstack (cdr '(a b c * +))
(cons (car '(a b c * +)) nil))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
…
(t (postfixstack (cdr exp)
(cons (car exp) stack)))
))))
(postfixstack '(b c * +) '(a))
(postfixstack (cdr '(b c * +))
(cons (car '(b c * +)) '(a)))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
…
(t (postfixstack (cdr exp)
(cons (car exp) stack)))
))))
(postfixstack '(c * +) '(b a))
(postfixstack (cdr '(c * +))
(cons (car '(c * +)) '(b a)))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
…
((eq (car exp) '*) (postfixstack (cdr exp)
(cons (list '* (cadr stack) (car stack))
(cddr stack))))
…
))))
(postfixstack '(* +) '(c b a))
(postfixstack (cdr '(* +))
(cons (list '* (cadr '(c b a)) (car '(c b a)))
(cddr '(c b a)))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
…
(t (postfixstack (cdr exp)
(cons (car exp) stack)))
))))
(postfixstack '(+) '((* b c) a))
(postfixstack (cdr '(+)) (cons
(list '+ (cadr '((* b c) a) (car '((* b c) a)))
(cddr '((* b c) a))))
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Lisp Example: postfix (cont)
(defun postfixstack (exp stack)
(cond
((null exp) stack)
(t
(cond
((eq (car exp) '+) (postfixstack (cdr exp)
(cons (list '+ (cadr stack) (car stack))
(cddr stack))))
…
(t (postfixstack (cdr exp)
(cons (car exp) stack)))
))))
(postfixstack nil '((+ a (* b c))))
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Lisp Example: postfix (cont)
(defun postfix (exp)
(car (postfixstack exp nil)))
(postfix '(a b c * +))
(car (postfixstack '(a b c * +) nil))
…
'(+ a (* b c))
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Tail Recursion in Lisp
Definition: A function is tail recursive if its
recursive call is the last operation in the
function
A tail recursive function can be automatically
converted by a compiler to use iteration,
making it faster
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14.2.8 Example: Semantics of Clite
Semantics - the meaning of the expressions,
statements, and program units
Static semantics – compiler time
“correctness” of semantics, i.e., type
checking, shows correctness of
position := initial + rate * 60
with type conversion of 60 to floating
point
Concepts of Programming Languages,
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Robert W.
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14.2.8 Example: Semantics of Clite
Dynamic semantics – run-time behavior
- fetch the value of rate from memory
- multiply by 60.0
- fetch the value of initial and add to
previous result
- store the result in position
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Robert W.
Sebesta. Addison
Wesley, 2010.
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14.2.8 Example: Semantics of Clite
Operational Semantics
– Describe the meaning of a program by
executing its statements on a machine, either
simulated or actual. The change in the state of
the machine (memory, registers, etc.) defines
the meaning of the statement.
Concepts of Programming Languages,
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Robert W.
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Wesley, 2010.
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14.2.8 Example: Semantics of Clite
Program memory store can be modeled as a function
which maps locations to values.
For simplicity, we map identifiers to integer values only.
Initial store indicates all variables are undefined.
(lambda (id) (cond (t 'undefined)))
Successive stores bind ids to their values.
(lambda (id)
(cond
((eq id 'y) 5)
((eq id 'x) 32)
(t 'undefined)
)
)
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Accessing a Variable’s Value in the Store
Function to retrieve the value of a variable from the state:
(defun access (store id)
(apply store (list id)))
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Updating a Variable’s Value in the Store
Function to store a new value for a variable in the store:
(defun update (store id value)
(prog (currentenv newbinding newstore)
(setq currentenv (cdar (cddr store)))
(setq newbinding (cons
(list 'eq 'id (list 'quote id))
(list value)))
(setq newstore (list 'lambda (list 'V)
(cons 'cond
(cons newbinding currentenv))))
(return newstore)
)
)
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Updating a Variable’s Value in the Store
(update '(lambda (id)
(cond ((eq id 'x) 32) (t 'undefined)))) 'y 5)
(setq currentenv (cdar (cddr store)))
currentenv = ((eq id 'x) 32) (t 'undefined))
(setq newbinding (cons
(list 'eq 'id (list 'quote id)) (list value)))
newbinding = (((eq id 'y) 5) ((eq id 'x) 32)
(t 'undefined))
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Updating a Variable’s Value in the Store
(setq newstore (list 'lambda (list 'id)
(cons 'cond (cons newbinding currentenv))))
newstore = (lambda (id)
(cond
((eq id 'y) 5)
((eq id 'x) 32)
(t 'undefined))
)
)
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