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Understanding Nondeterministic Turing Machines and their Equivalence with Deterministic TMs

This text discusses the concept of Nondeterministic Turing Machines (NTMs), highlighting their transition functions and how they accept languages. It explains how an input string is accepted if an accepting configuration can be reached within the computation tree. Additionally, it defines key terms such as acceptance, rejection, and looping within the context of NTMs. The equivalence of NTMs to Deterministic Turing Machines (DTMs) is also explored, emphasizing how both types of machines can decide the same languages. The process of simulating an NTM with a DTM is also outlined.

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Understanding Nondeterministic Turing Machines and their Equivalence with Deterministic TMs

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Presentation Transcript

  1. Nondeterministic TMs

  2. Definition of NTMs • Must change transition function: •  : Q x   2(Q x  x {L, R})

  3. Language of an NTM • String accepted if it’s possible to reach accepting configuration • Set of computations – Computation Tree

  4. Deciding a Language • Accept – The computation tree contains an accepting configuration. • Reject – All leaves in the computation tree are rejecting configurations. • Loop – The computation tree contains no accepting configuration and there exists an infinitely long path in the tree.

  5. Equivalence of NTMs and TMs For a language L • NTMM s.t. L = L(M) TMM’ s.t. L = L(M’) • NTMM that decides L  TMM’ that decides L

  6. Simulating NTM with TM • Let M = (Q, , , , q0, , F) be any NTM • Will create TM M = (Q, , , , q0, , F) • On input w  *, M creates computation tree for M • By simulating M‘s moves.

  7. TM Simulating NTM 1 2...n Input NTM transitions ((q,),{(q1,1,D1), (q2,2,D2), ... , (qk,k,Dk)})... Finite Control (States) Queue of configs q01 2...n$1 q12 ... n$ ... Scratch

  8. Pseudocode Step 1: Put M‘s state transitions on tape 2 Step 2: Put start configuration on tape 3 (q0 followed by copy of tape 1) Step 3: Create computation tree

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