notes14 Variants of Turing Machines and the Church-Turing Thesis

Notes14 Variants of Turing Machines and the Church-Turing Thesis

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Unformatted text preview: CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1 1 Variants of Turing Machines 1.1 Multi-Tape TM Multi-Tape Turing Machine 1 1 t t 1 t t 1 t t finite-state control Input on Tape 1 Initially all heads scanning cell 1, and tapes 2 to k blank In one step: Read symbols under each of the k-heads, and depending on the current con- trol state, write new symbols on the tapes, move the each tape head (possibly in different directions), and change state. Multi-Tape Turing Machine Formal Definition A k-tape Turing Machine is M = ( Q, , ,,q ,q acc ,q rej ) where Q is a finite set of control states is a finite set of input symbols is a finite set of tape symbols. Also, a blank symbol t \ q Q is the initial state q acc Q is the accept state q rej Q is the reject state, where q rej 6 = q acc : ( Q \ { q acc ,q rej } ) k Q ( { L , R } ) k is the transition function. Computation, Acceptance and Language A configuration of a multi-tape TM must describe the state, contents of all k-tapes, and positions of all k-heads. Thus, c Q ( * {*} * ) k , where * denotes the head position. 2 Accepting configuration is one where the state is q acc , and starting configuration on input w is ( q , * w, *t ,..., *t ) Formal definition of a single step is skipped. w is accepted by M , if from the starting configuration with w as input, M reaches an accepting configuration. L ( M ) = { w | w accepted by M } Expressive Power of multi-tape TM Theorem 1. For any k-tape Turing Machine M , there is a single tape TM single ( M ) such that L ( single ( M )) = L ( M ) . Challenges How do we store k-tapes in one? How do we simulate the movement of k independent heads? Storing Multiple Tapes 1 1 t 1 t t finite-state control Figure 1: Multi-tape TM M (1 , , , * ) (0 , * , 1 , ) ( t , , 1 , ) t finite-state control Figure 2: 1-tape equivalent single( M ) Store in cell i contents of cell i of all tapes. Mark head position of tape with * . Simulating One Step Challenge 1: Head of 1-Tape TM is pointing to one cell. How do we find out all the k symbols that are being read by the k heads, which maybe in different cells? 3 Read the tape from left to right, storing the contents of the cells being scanned in the state , as we encounter them. Challenge 2: After this scan, 1-tape TM knows the next step of k-tape TM. How do we change the contents and move the heads? Once again, scan the tape, change all relevant contents, move heads (i.e., move * s), and change state. Overall Construction First we outline the high-level algorithm for the 1-tape TM. On input w 1. First the machine will rewrite input w to be in new format....
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This note was uploaded on 10/04/2011 for the course CS 373 taught by Professor Viswanathan,m during the Fall '08 term at University of Illinois, Urbana Champaign.

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Notes14 Variants of Turing Machines and the Church-Turing Thesis

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