Equivalence of TMs and Multitape TMs. Theorem 3.13 and Corollary 3.15 By: Joseph Lauman

Transcription

1 Equivalence of TMs and Multitape TMs Theorem 3.13 and Corollary 3.15 By: Joseph Lauman

2 Turing Machines First proposed by Alan Turing in 1936 Similar to finite automaton, but with an unlimited and unrestricted memory Uses an infinite tape as its unlimited memory Has a tape head that can read and write symbols and move around on the tape Initially contains only the input string and is blank everywhere else If the machine needs to store information, it writes it on the tape To read, it can move its head back over it

3 Turing Machines It continues computing until it decides to produce an output The output s accept and reject are obtained by entering designated accepting and rejecting states If it doesn t enter a state, it will go on forever

4 Definition of a Turing Machine (Q, Σ, Γ, δ, q 0, q accept, q reject ) Q, Σ, Γ are all finite sets Q is the set of states Σ is the input alphabet not containing the blank symbol Γ is the tape alphabet δ: Q x Γ Q x Γ x {L, R} is the transition function q 0 is a subset of Q and is the start state q accept is a subset of Q and is the accept state q reject is a subset of Q and is the reject state, where q reject!= q accept

5 Multitape Turing Machines Like a Turing machine, but with several tapes Each tape has its own head for reading and writing Each tape head moves independently of the other heads Initially, the input appears on tape 1 and the rest start out blank The transition function is changed to allow for reading, writing, and moving the heads on some or all of the tapes simultaneously

6 Definition of a Multitape Turing Machine (Q, Σ, Γ, δ, q 0, q accept, q reject ) Q, Σ, Γ are all finite sets Q is the set of states Σ is the input alphabet not containing the blank symbol Γ is the tape alphabet δ: Q x Γ k Q x Γ k x {L, R, S} k is the transition function where k is the number of tapes q 0 is a subset of Q and is the start state q accept is a subset of Q and is the accept state q reject is a subset of Q and is the reject state, where q reject!= q accept

7 Theorem 3.13 Every multitape Turing machine has an equivalent single-tape Turing machine Convert a multitape Turing machine M to an equivalent single-tape Turing machine S. The key idea is to show how to simulate M with S

8 Theorem 3.13 Say that M has K tapes. Then S simulates the effect of k tapes by storing their information on its single tape Uses # as a delimiter to separate the contents of the different tapes S must also keep track of the locations of the heads. Does this by writing a tape symbol with a dot above it to mark the place where the head on that tape would be The dotted tape symbols are new symbols that are added to the tape alphabet Treat them as virtual tapes and heads

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10 Theorem 3.13 Representing three tapes with one S = On input w = w 1 w n : First S puts its tape into the format that represents all k tapes of M The formatted tape contains #w 1* w 2 w n #_ * #_ * #...# To simulate a single move, S scans its tape from the first #, which marks the left-hand end, to the (k+1)st #, which marks the right-hand end, in order to determine the symbols under the virtual heads. Then S makes a second pass to update the tapes according to the way that M s transition function dictates.

11 Theorem 3.13 If at any point S moves one of the virtual heads to the right onto a #, this action signifies that M has moved the corresponding head onto the previously unread blank portion of that tape. So S writes a blank symbol on this tape cell and shifts the tape contents, from this cell until the rightmost #, one unit to the right. Then it continues the simulation as before

12 Corollary 3.15 A language is Turing-recognizable if and only if some multitape Turing machine recognizes it A Turning-recognizable language is recognized by an ordinary (singletape) Turing machine, which is a special case of multitape Turing machine. For k=1 The rest follows from Theorem 3.13 For k=n, construct a single-tape Turing machine out of a k-tape Turing machine in the same manner

13 Questions?