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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 ⊆ Σ * . 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: Ν → 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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