Unformatted text preview: Introduction to the Theory of Computation
AZADEH FARZAN SPRING 2010 Tuesday, February 2, 2010 STRUCTURAL INDUCTION Tuesday, February 2, 2010 DEFINING SETS RECURSIVELY
Deﬁne a set of objects: (i) deﬁne the smallest or smiplest object (or objects). (ii) deﬁne the ways in which larger or more complex objects can be built out of the smaller or simpler ones. Example: smallest set of simple expressions over x, y, z , call it E . (i) smiple objects: x, y, z ∈ E . (ii) more complex objects: if e1 , e2 ∈ E then so are (e1 + e2 ), (e1 − e2 ), (e1 × e2 ), and (e1 ÷ e2 ). Tuesday, February 2, 2010 CORRECTNESS
Theorem. Let S be a set, B ⊆ S , m a positive integer, and f1 , . . . , fm be operators on S of arity k1 , . . . , km , respectively. There is a unique U ⊆ S such that: (a) B ⊆ U , (b) U is closed under f1 , . . . , fm , and (c) any subset of S that satisﬁes (a) and (b) contains U . Tuesday, February 2, 2010 U0 U1 U2 U3 ... =B = U0 ∪ set obtained by applying f1 , . . . , fm to elements in U0 = U1 ∪ set obtained by applying f1 , . . . , fm to elements in U1 = U2 ∪ set obtained by applying f1 , . . . , fm to elements in U1 Ui = Ui−1 ∪ set obtained by applying f1 , . . . , fm to elements in Ui−1 U0 ∪ U1 ∪ U2 ∪ · · · ∪ Ui ∪ . . . Tuesday, February 2, 2010 PRINCIPLE OF SET DEFINITION BY INDUCTION
Let S be a set, B ⊆ S , m be a positive interger, and f1 , . . . , fm be operators on S of arity k1 , . . . , km respectively. Let if i = 0 if i > 0 Ui = B Ui−1 ∪ m j =1 {fj (a1 , . . . , akj ) : a1 , . . . akj ∈ Si−1 } Then i∈N Ui is the smallest subset of S that contains B and is closed under f1 , . . . , fm . Tuesday, February 2, 2010 STRUCTURAL INDUCTION
To prove that every element of X (deﬁned inductively) has a certain property P : Basis: Prove that every smallest or smiplest element of X satisﬁes P . Induction Step: Prove that each of the (ﬁnitely many) ways of constructing larger or more complex elements out of t smaller or simpler ones preserves the property. Tuesday, February 2, 2010 EXAMPLE Example 1: wellbalanced parentheses. Example 2: Binary Trees. Tuesday, February 2, 2010 ALTERNATIVE: COMPLETE INDUCTION
How can we use complete induction instead of structural induction? • Deﬁne a measure for each element of the set. • Show by complete induction that every element with measure n has the property P . Tuesday, February 2, 2010 EXAMPLE example: wellbalanced parentheses revisited: lp(e) = rp(e) example: binary threes revisited: nodes(T) ≤ 2h(T)+1 1 Tuesday, February 2, 2010 READING Chapter 4. Tuesday, February 2, 2010 ...
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This note was uploaded on 04/11/2010 for the course CSC CSC236 taught by Professor Farzanazadeh during the Spring '10 term at University of Toronto.
 Spring '10
 FarzanAzadeh

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