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Multitape Turing Machines

Multitape Turing Machines. TM - Tape Head can Stay (TMS). Modify definition of transition function:  : Q x   Q x  x {L, R, S } Claim 1: If L is the language of some TM M then  TMS M S such that L = L( M S ) If M decides L then  TMS M S that decides L

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Multitape Turing Machines

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  1. Multitape Turing Machines

  2. TM - Tape Head can Stay (TMS) • Modify definition of transition function: •  : Q x   Q x  x {L, R, S} • Claim 1: If L is the language of some TM M then •  TMS MS such that L = L(MS) • If M decides L then  TMS MS that decides L • Claim 2: If L is the language of some TMS MS then •  TM M such that L = L(M) • If MS decides L then  TM M that decides L

  3. Formal Defn of a k-tape TM A 7-Tuple (Q, , , , q0, B, F) where • Q – Set of states •  – Input alphabet (B  ) •  – Tape alphabet (  and B  ) •  – Transition Function •  : Q x k  Q x k x {L,R,S}k • q0 – Start state (q0  Q ) • B - Blank symbol • F  Q – accepting states

  4. Equivalence of MTMs and TMs • Claim 1: For all languages L, L = L(M) for some TM M if and only if L = L(M‘) for some MTM M‘ • Claim 2: For all languages L, L is decided by some TM M if and only if L is decided by some MTM M‘

  5. Construct TMS from MTM • Input: MTM M = (Q, , , , q0,B, F) • Output TMS M’ = (Q’, , ’, ’, q’0,B, F’)where ’ =   {  |    }  {#} and remainder is built using following steps • Format tape • Scan and remember • Update tape and tape head • Change to appropriate state

  6. Correctness of Constructions • Claim 1: L(M) = L(M‘) • Claim 2: For all w  * M halts on w if and only if M‘ halts on w.

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