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Inductive Sets of Data

Programming Language Essentials

2nd edition

Chapter 1.1 Recursively Specified Data

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Inductive Specification

specific value is in the set.

if some value is in the set, some other also is.

S: smallest set of natural numbers with

0 in S

if x in S then x+3 in S

M: multiples of 3

smallest guarantees uniqueness

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list-of-numbers?L: smallest set of values withempty list in Lif x in L and n a number, (n . x) in L

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

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list-of-numbers?L: smallest set of values withempty list in Lif x in L and n a number, (n . x) in L

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

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Backus-Naur Forml-of-nums: '()'

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

grammar

nonterminals l-of-nums

terminals '(' 'Number’ ‘.’ ‘)’

rules, productions l-of-nums: '()'

context-free

notations differ

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Extended Backus-Naur Forml-of-ns: '()' | '(' 'Number' '.' l-of-ns ')'

l-of-ns: '()'

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

l-of-ns: '(' 'Number'* ')'

parentheses for grouping

optional term?

zero or more term*

one or more term+

separated {<term>}*(,)

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Syntactic Derivationlist-of-numbers

=> ( Number . list-of-numbers )

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Syntactic Derivationlist-of-numbers

=> ( Number . list-of-numbers )

=> ( 14 . list-of-numbers )

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Syntactic Derivationlist-of-numbers

=> ( Number . list-of-numbers )

=> ( 14 . list-of-numbers )

=> ( 14 . () )

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Syntactic Derivationlist-of-numbers

=> ( Number . list-of-numbers )

=> ( 14 . list-of-numbers )

=> ( 14 . () )

order of substitution does not matter

done once only terminals remain

need to cheat about quoting terminals

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s-lists-list: '(' symbol-expression* ')'

symbol-expression: 'Symbol' | s-list

(a b c)

(an (((s-list)) (wth () lots) ((of) nests)))

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bintreebintree: 'Number' | '(' 'Symbol' bintree bintree ')'

1(foo 1 2)(bar 1 (foo 1 2))(baz (bar 1 (foo 1 2)) (biz 4 5))

search-tree: '()' | '(' 'Key' search-tree search-tree ')'

needs restriction for key ordering

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Scheme Datalist: '(' datum* ')'

dotted: '(' datum+ '.' datum ')'

vector: '#(' datum* ')'

datum: 'Number' | 'Symbol' | 'Boolean'

| 'String' | list | dotted | vector

what's missing?

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Sample Derivationlist

=> ( datum datum datum )

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Sample Derivationlist=> ( datum datum datum )=> ( Boolean datum datum )

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Sample Derivationlist=> ( datum datum datum )=> ( Boolean datum datum )=> ( #t datum datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

=> ( #t ( Symbol . datum ) datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

=> ( #t ( Symbol . datum ) datum )

=> ( #t ( foo . datum ) datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

=> ( #t ( Symbol . datum ) datum )

=> ( #t ( foo . datum ) datum )

=> ( #t ( foo . list ) datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

=> ( #t ( Symbol . datum ) datum )

=> ( #t ( foo . datum ) datum )

=> ( #t ( foo . list ) datum )

=> ( #t ( foo . () ) datum )

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Sample Derivationlist

=> ( datum datum datum )

=> ( Boolean datum datum )

=> ( #t datum datum )

=> ( #t dotted datum )

=> ( #t ( datum+ . datum ) datum )

=> ( #t ( Symbol . datum ) datum )

=> ( #t ( foo . datum ) datum )

=> ( #t ( foo . list ) datum )

=> ( #t ( foo . () ) datum )

=> ( #t ( foo . () ) Number )

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Sample Derivationlist=> ( datum datum datum )=> ( Boolean datum datum )=> ( #t datum datum )=> ( #t dotted datum )=> ( #t ( datum+ . datum ) datum )=> ( #t ( Symbol . datum ) datum )=> ( #t ( foo . datum ) datum )=> ( #t ( foo . list ) datum )=> ( #t ( foo . () ) datum )=> ( #t ( foo . () ) Number )=> ( #t ( foo . () ) 3 )

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Lambda Calculusexpr: 'Symbol'

| '(' 'lambda' '(' 'Symbol' ')' expr ')'

| '(' expr expr ')'

small language

variable references

function definition with single parameter

function call with one argument

http://www.cs.rit.edu/~ats/projects/oops/edu/doc/edu/rit/cs/oops/examples/Lambda.html

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Proof by Inductionbintree: 'Number' | '(' 'Symbol' bintree bintree ')'

has an odd number of nodes:(0) Hypothesis: trees of size <= k have odd number

of nodes. Show that this holds for any k.(1) k=0: there are no such bintrees. (0) true.(2) assume (0) true up to some k. Look at tree of

size k+1: bintree: 'Number' has one node. (0) true.| '(' 'Symbol' bintree bintree ')' has one plus size of two smaller tress, i.e., 1+odd+odd, nodes. (0) true.

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Proof by Structural Induction

Strategy:

(0) some hypothesis.

(1) show (0) on a simple structure, i.e., on one without substructures.

(2) show (0) on a structure with substructures: if it is true on the substructures it is true on the composite.