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Inductive Sets of Data

Programming Language Essentials

2nd edition

Chapter 1.2 Recursively Specified Programs

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Recursion

recursion can lead to divide-and-conquer:

solution for a small problem 'by hand'.

solution for a bigger problem by recursively invoking the solution for smaller problems.

Examples:

Factorial.

Greatest common divisor (Euclid).

Tower of Hanoi.

Quicksort.

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Recursively Specified Programs

patterned after proof by induction.

to compute xn:

e(0, x) = 1, because x0 is 1.

e(n, x) = x * e(n-1, x), because xn is x * xn-1.

induction proves that e(n,x) is xn:

(0) Hypothesis: e(k, x) is xk.

(1) true for k=0, see above.

(2) assume true up to k. Consider e(k+1, x).

This is x * e(k, x) by definition, x * xk by assumption, and therefore xk+1.

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Recursively Specified Programs (2)

to compute xn:

e(0, x) = 1, because x0 is 1.

e(n, x) = x * e(n-1, x), because xn is x * xn-1.

(define e

(lambda (n x)

(if (zero? n)

1

(* x

(e (- n 1) x)

) ) ) )

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count-nodesbintree: 'Number'

| '(' 'Symbol' bintree bintree ')'

(define count-nodes

(lambda (bintree)

(if (number? bintree)

1

(+ (count-nodes (cadr bintree))

(count-nodes (caddr bintree))

1

) ) ) )

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Deriving Programs from BNFl-of-nums: '()'l-of-nums: '(' 'Number' '.' l-of-nums ')'

pattern program after data (structural induction)one procedure for one nonterminal

(define list-of-numbers? (lambda (x) (if (null? x) #t (and (pair? x) (number? (car x)) (list-of-numbers? (cdr x))) ) ) )

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Proof by Induction

(0) Hypothesis: l-of-n? works on lists of length k.

(1) k=0: empty list, l-of-n? returns true. ok.

(2) assume true up to k. Consider k+1.

By grammar, car must be a number and cdr a list of numbers.

cdr of list of k+1 elements has k elements, i.e.,

l-of-n? can be applied. ok.

proof does not discover that pair? is necessary

because proof assumes list arguments

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nth-elt

like list-ref, returns 0..th element of list.

(define nth-elt

(lambda (list n)

(if (null? list)

(error 'nth-elt "List too short."

(+ n 1) "elements")

(if (zero? n)

(car list)

(nth-elt (cdr list) (- n 1))

) ) ) )

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errorhttp://srfi.schemers.org/

lists Scheme Request for Implementation

$ scheme48> ,open srfi-23Newly accessible in user: (error)> (define nth-elt …> (nth-elt '(1 2 3) -4)

Error: nth-elt "List too short." -6 "elements"

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list-length

like length, returns number of elements in list

(define list-length

(lambda (list)

(if (null? list)

0

(+ 1 (list-length (cdr list)))

) ) )

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fragile vs. robustfragile procedures do not check the types of their

arguments.

(define list-length ; robust version (lambda (list) (if (list? list) (if (null? list) 0 (+ 1 (list-length (cdr list))) ) (error 'list-length "Not a list:" list)) ) )

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remove-first

returns list without first occurrence of symbol

list: '()' | '(' symbol '.' list ')'

(define remove-first ; fragile

(lambda (symbol list)

(if (null? list)

list

…

) ) )

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remove-first

returns list without first occurrence of symbol

list: '()' | '(' symbol '.' list ')'

(define remove-first ; fragile

(lambda (symbol list)

(if (null? list)

list

(if (eqv? (car list) symbol)

(cdr list)

…

) ) ) )

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remove-first

returns list without first occurrence of symbol

list: '()' | '(' symbol '.' list ')'

(define remove-first ; fragile

(lambda (symbol list)

(if (null? list)

list

(if (eqv? (car list) symbol)

(cdr list)

(cons (car list)

(remove-first symbol (cdr list))

) ) ) ) )

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remove

returns list without all occurrences of symbol

list: '()' | '(' symbol '.' list ')'

(define remove ; fragile

(lambda (symbol list)

(if (null? list)

list

(if (eqv? (car list) symbol)

(remove symbol (cdr list))

(cons (car list)

(remove symbol (cdr list))

) ) ) ) )

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substreturns s-list with any old replaced by new> (subst 'a 'b '((b c) (b () d)))'((a c) (a () d))

s-list: '(' symbol-expression* ')'symbol-expression: 'Symbol' | s-list

s-list: '()' | '(' symbol-expression '.' s-list ')'symbol-expression: 'Symbol' | s-list

pair avoids need for another rule

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substs-list: '()'

| '(' symbol-expression '.' s-list ')'

(define subst ; fragile

(lambda (new old slist)

(if (null? slist)

slist

(cons

(subst-se new old (car slist))

(subst new old (cdr slist))

) ) ) )

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subst-sesymbol-expression: 'Symbol' | s-list

(define subst-se

(lambda (new old se)

(if (symbol? se)

(if (eqv? se old) new se)

(subst new old se)

) ) )

mutual recursion

divide-and-conquer because of grammar

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subst(define subst ; fragile

(lambda (new old slist)

(define symbol-expression ; local procedure

(lambda (se) ; shares parameters

(if (symbol? se)

(if (eqv? se old) new se)

(subst new old se)

) ) )

(if (null? slist)

slist

(cons

(symbol-expression (car slist))

(subst new old (cdr slist))

) ) ) )

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notate-depth

returns s-list with symbols annotated for depth> (notate-depth '((b c) (b () d)))

'(((b 1) (c 1)) ((b 1) () (d 1)))

notate-depth: s-list

s-list: '()'

| '(' symbol-expression '.' s-list ')'

symbol-expression: 'Symbol' | s-list

notate-depth needed to mark start at level zero

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notate-depthnotate-depth: s-list

(define notate-depth ; fragile (lambda (slist) (s-list slist 0)) )

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notate-depths-list: '()' | '(' symbol-expression '.' s-list ')'

(define notate-depth (lambda (slist) (define s-list (lambda (slist d) (if (null? slist) slist (cons (symbol-expression (car slist) d) (s-list (cdr slist) d) ) ) ) ) (s-list slist 0)) )

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notate-depthsymbol-expression: 'Symbol' | s-list

(define notate-depth

(lambda (slist)

(define s-list …

(define symbol-expression

(lambda (se d)

(if (symbol? se)

(list se d)

(s-list se (+ d 1))

) ) )

(s-list slist 0)

) )

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notate-depth(define notate-depth (lambda (slist) (define s-list (lambda (slist d) (if (null? slist) slist (cons (symbol-expression (car slist) d) (s-list (cdr slist) d) ) ) ) ) (define symbol-expression (lambda (se d) (if (symbol? se) (list se d) (s-list se (+ d 1)) ) ) ) (s-list slist 0)) )

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Other Patterns of Recursionl-of-nums: '()'

l-of-nums: '(' 'Number' '.' l-of-nums ')'

compute sum by structural induction:

(define list-sum

(lambda (x)

(if (null? x)

0

(+ (car x))

(list-sum (cdr x))

) ) ) )

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Other Patterns of Recursion (2)vector is not suitable for structural induction, so:(define vector-sum (lambda (v) (partial-vector-sum v (vector-length v))) )(define partial-vector-sum (lambda (v n) (if (zero? n) 0 (+ (vector-ref v (- n 1)) ; last (partial-vector-sum v (- n 1))) ) ) )

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Other Patterns of Recursion (3)(define vector-sum (lambda (v) (letrec ((partial-vector-sum (lambda (n) (if (zero? n) 0 (+ (vector-ref v (- n 1)) ; last (partial-vector-sum (- n 1)) ))) ) ) (partial-vector-sum (vector-length v))) ) )

proof by induction on vector length