17 - 17 Extensions of TM and Church-Turing Thesis...

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17. Extensions of TM and Church-Turing Thesis Extensions of TM 1. allow both write and move in one step 2. multiple tapes 3. two-way infinite tape 4. nondeterminism Theorem 1 The operation of a machine allowing some or all of the above extensions can be simulated by a standard Turing machine. These extensions do not produce more powerful machines than the standard Turing machine. That is, any language that can be decided (or semidecided) by these extensions can also be decided (or semidecided) by a standard Turing machine. Note : These extensions are nevertheless helpful when design- ing Turing machines to solve specific problems. 1
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Proving equivalence in power A simple example: allow both write and move in one step Suppose we allow the TM the ability to both write and move (left or right) in one transition step. The transition function would have the form δ : ( K H ) × Σ −→ K × Σ × {→ , ←} . Might this feature allow TMs to recognize additional lan- guages? Obviously no, because we can convert any TM with the “write AND move in one step” feature to one that only allows write OR move in each step. We do so by replacing each such transition with 2 transi- tions – one that writes and the second one that moves. This example illustrates the idea of such proofs: to show that an extension of TMs is no more powerful than a stan- dard TM, we only need to show that we can simulate the operations of the extended machine on a standard TM. 2
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Multitape Turing Machine Each tape has its own read/write head. Initially, the input appears on tape 1, and the other tapes start out blank. In each step, the TM reads the symbols pointed to by all its heads simultaneously. Depending on its current state and all the symbols read, the machine decides for each head whether to write onto the current square, or move , h q q 1 2 q 0 a b a a b a b a (q0, (b, a, a)) = (q2, (a, b, )) δ a b b b a a a a h q q 1 2 q 0 3-tape TM 3
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Multitape Turing Machines Theorem 2 Every multitape Turing machine has an equiv- alent single-tape Turing machine. Proof: We shall prove that any k -tape TM M 1 can be simulated by a standard Turing machine M 2 . M 2 uses the new symbol # as a delimiter to separate the contents of the different tapes.
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