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# 2001Lec9 - CSE 2001 Introduction to Theory of Computation...

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3/14/2006 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/ ) 3/14/2006 CSE 2001, Winter 2006 2 Outline Last class: Introduction to Turing machines TM-computable/recognizable languages Today: Recall the definition of TMs Variants of TMs Church-Turing thesis Decidability 3/14/2006 CSE 2001, Winter 2006 3 Turing Machine (Def. 3.3) A Turing machine M is defined by a 7-tuple (Q, Σ , Γ , δ ,q 0 ,q accept ,q reject ), with • Q finite set of states Σ finite input alphabet (without “_”) Γ finite tape alphabet with { _ } ∪ Σ ⊆ Γ • q 0 start state Q • q accept accept state Q • q reject reject state Q δ the transition function δ : Q × Γ → Q × Γ × {L,R} 3/14/2006 CSE 2001, Winter 2006 4 Configuration of a TM The configuration of a Turing machine consists of • the current state q Q • the current tape contents ∈ Γ * • the current head location {0,1,2,…} This can be expressed as an element of Γ * × Q ×Γ *: L _ _ 1 # 0 _ 1 1 0 1 q 9 becomes “101 q 9 1_0#1” 3/14/2006 CSE 2001, Winter 2006 5 Turing recognizable vs decidable A language L is Turing-recognizable if and only if there is a TM M such that L=L(M). Also called: a recursively enumerable language. A language L=L(M) is decided by the TM M if on every w, the TM finishes in a halting configuration. (That is: q accept for w L and q reject for all w L.) A language L is Turing-decidable if and only if there is a TM M that decides L. Also called: a recursive language. 3/14/2006 CSE 2001, Winter 2006 6 Describing TM Programs Three Levels of Describing algorithms: • formal (state diagrams, CFGs, et cetera) • implementation (pseudo-Pascal) • high-level (coherent and clear English) Describing input/output format: TMs allow only strings ∈Σ * as input/output. If our X and Y are of another form (graph, Turing machine, polynomial), then we use < X,Y > to denote ‘some kind of encoding ∈Σ *’.

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3/14/2006 CSE 2001, Winter 2006 7 Turing machine variants Multi-tape TMs Non-deterministic TMs Uses: Proves robustness of the model Extra tools for problem solving 3/14/2006 CSE 2001, Winter 2006 8 Multi-tape Turing Machines A k-tape Turing machine M has k different tapes and read/write heads. It is thus defined by the 7-tuple (Q, Σ , Γ , δ ,q 0 ,q accept ,q reject ), with Q finite set of states Σ finite input alphabet (without “_”) Γ finite tape alphabet with { _ } ∪ Σ ⊆ Γ • q 0 start state Q • q accept accept state Q • q reject reject state Q δ the transition function δ : Q × Γ k Q × Γ k × {L,R} k 3/14/2006 CSE 2001, Winter 2006 9 k-tape TMs versus 1-tape TMs Theorem 3.13 : For every multi-tape TM M, there is a single-tape TM M’ such that L(M)=L(M’).
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2001Lec9 - CSE 2001 Introduction to Theory of Computation...

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