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Logical Foundations of CS

by: Dorothy Bahringer DVM

Logical Foundations of CS CS 5303

Dorothy Bahringer DVM
GPA 3.85

Martine Ceberio

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Martine Ceberio
Class Notes
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This 2 page Class Notes was uploaded by Dorothy Bahringer DVM on Thursday October 29, 2015. The Class Notes belongs to CS 5303 at University of Texas at El Paso taught by Martine Ceberio in Fall. Since its upload, it has received 40 views. For similar materials see /class/231290/cs-5303-university-of-texas-at-el-paso in ComputerScienence at University of Texas at El Paso.

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Date Created: 10/29/15
Logical Foundations of CS 085303 Sinduction These notes were written using Mathematiqnes ponr I Inforniatiqne by Andre Arnold and Irene Gnessarian They complete the course given during the rst week of class on proofs inductive de nitions and proofs by induction 1 Inductive de nitions Let s review a couple of examples Example 1 The part X ofN inductively de ned by B 0 e X I nEX n1 X is simply N4 Therefore B and I constitute an inductive de nition of N Example 2 Let A the alphabet made of both parentheses The set D Q Aquot of wellformed expressions with parenthesis called the Dyck language is de ned by B eED I IiiED xzy D Example 3 The setE of wellparenthesed expressions made of identi ers from A and operators and X is the subset of AU X inductively de ned as follows BAgE 15 fEE 5f exf EE Example 4 The set BT of binary trees whose elements are elements of alphabet A is the subset of AU 2 inductively de ned as follows E lie AB it is the empty tree I l r e AB Va e A alr e AB the tree of root 1 of left subtree l and right subtree r De nition 1 An inductive de nition of a set X is non ambiguous is for all x E X there exist an unique way to get at by applying the inductive rulest Example 5 The following inductive de nition of N2 is ambiguous B 0 0 E N2 11 mm EN2 71 1711 E N2 12 mm EN2 71711 1 E N2 2 Proof by induction Example 6 Let s prove by induction that all strings in the Dyck language D contain as many as 3 For all x E D 195 denotes the number of and rx denotes the number of 3 Let Px the property we want to prove Poc 95 roc B the only element of the basis is e the empty string So it satis es P we we 0 I Let x y E D such that Px and Py Let 2 my 12 196 My We My NZ Therefore Pay Similarly we show that Px4 We deduce from the above that Vac E D 95 roc 3 Inductively de ned functions Example 7 The factorial function N A N is inductively de ned by B fact0 1 I factn 1 n 1 X factn Here we use the inductive de nition of N as a support for the inductive de nition of fact We can also note it as follows if n 0 1 fac nl n X factn E 1 otherwise Example 8 Let us consider the set E inductively de ned in Example 34 The expressions of E are in in x notationt Post x notation can also be used to de ne parenthesed expressions For instance the post x expression of a X b c d is abc gtltd The transformation from in x to post x can be inductively de ned as follows


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