2001Lec10 - CSE 2001: Introduction to Theory of Computation...

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3/28/2006 CSE 2001, Winter 2006 1 CSE 2001: Introduction to Theory of Computation Winter 2006 Suprakash Datta [email protected] Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/2001 Some of these slides are adapted from Wim van Dam’s slides ( www.cs.berkeley.edu/~vandam/CS172/ ) 3/28/2006 CSE 2001, Winter 2006 2 Next Towards undecidability: The Halting Problem Countable and uncountable infinities Diagonalization arguments 3/28/2006 CSE 2001, Winter 2006 3 The Halting Problem The existence of the universal TM U shows that A TM = { < M,w > | M is a TM that accepts w } is TM-recognizable, but can we also decide it? The problem lies with the cases when M does not halt on w. In short: the halting problem . We will see that this is an insurmountable problem: in general one cannot decide if a TM will halt on w or not, hence A TM is undecidable. 3/28/2006 CSE 2001, Winter 2006 4 Counting arguments • We need tools to reason about undecidability. • The basic argument is that there are more languages than Turing machines and so there are languages than Turing machines. Thus some languages cannot be decidable 3/28/2006 CSE 2001, Winter 2006 5 Baby steps • What is counting? – Labeling with integers – Correspondence with integers • Let us review basic properties of functions 3/28/2006 CSE 2001, Winter 2006 6 Mappings and Functions The function F:A B maps one set A to another set B: A B F F is one-to-one (injective) if every x A has a unique image F(x): If F(x)=F(y) then x=y. F is onto (surjective) if every z B is ‘hit’ by F: If z B then there is an x A such that F(x)=z. F is a correspondence (bijection) between A and B if it is both one-to-one and onto.
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3/28/2006 CSE 2001, Winter 2006 7 Cardinality A set S has k elements if and only if there exists a bijection between S and {1,2,…,k}. S and {1,…,k} have the same cardinality . If there is a surjection possible from {1,…,n} to S, then n |S|. We can generalize this way of comparing the sizes of sets to infinite ones. 3/28/2006 CSE 2001, Winter 2006 8 How Many Languages? For Σ ={0,1}, there are 2 k words of length k. Hence, there are languages L ⊆Σ k . Proof : L has two options for every word ∈Σ k ; L can be represented by a string . That’s a lot, but finite. There are infinitely many languages * . But we can say more than that… Georg Cantor defined a way of comparing infinities. ) 2 ( k 2 ) 2 ( k } 1 , 0 { 3/28/2006 CSE 2001, Winter 2006 9 Countably Infinite Sets A set S is countable if there exists a surjective function F: Ν→ S “The set S has not more elements than 1 .” A set S is infinite if there exists a surjective function F:S →Ν . “The set Ν has no more elements than S.” A set S is countably infinite if there exists a bijective function F: S. “The sets Ν and S are of equal size.” 3/28/2006 CSE 2001, Winter 2006 10 Counterintuitive facts • Cardinality of even integers – Bijection i 2i – A proper subset of N has the same cardinality as N !
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This note was uploaded on 12/11/2010 for the course CSE CSE 2001 taught by Professor N during the Winter '10 term at York University.

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2001Lec10 - CSE 2001: Introduction to Theory of Computation...

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