## 7.4 Combining Turing Machines 7.6 The Church-Turing Thesis 7.7 Nondeterministic Turing Machines 7.8 Universal Turing Machines

CSC4510/6510 AUTOMATA 7.4 Combining Turing Machines 7.6 The Church-Turing Thesis 7.7 Nondeterministic Turing Machines 7.8 Universal Turing Machines ...
CSC4510/6510 AUTOMATA

7.4 Combining Turing Machines 7.6 The Church-Turing Thesis 7.7 Nondeterministic Turing Machines 7.8 Universal Turing Machines

Combining Turing Machines If T1 and T2 are TMs, the composition T1T2: first execute T1, then execute T2 on the result  States of T1T2: the union of the sets of states of T1 and T2 (relabeled if necessary)  Initial state: the initial state of T1  The transitions of T1T2: all of those of T2 and all of those of T1 that don’t go to ha  A transition in T1 that goes to ha is replaced by a similar transition that goes to the start state of T2  It is important that the output of T1 be a valid input configuration for T2 2

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Combining Turing Machines (cont’d.)

Combining Turing Machines (cont’d.)

 In state p, if the current symbol is a, then execute the TM T

 We have the following TMs:

 Execute T1, and if T1 halts in ha with current symbol a, then execute T2

 Accepting the Language of Palindromes

   

Copy: copy a string NB/PB: find the next/previous blank R: reverse a string Equal: compare two strings

Copy

Reverse

 T: Copy  NB  R  PB  Equal

a/a,R b/b,R

Copy 3

/,R

a/a,L b/b,L

Reverse

/,L

Equal 4

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The Church‐Turing Thesis

The Church‐Turing Thesis (cont’d.)

 “To say that the TM is a general model of computation implies that any algorithmic procedure that can be carried out at all, by a human computer or a team of humans or an electronic computer, can be carried out by a TM”  The statement was formulated by Alonzo Church in the 30’s  It is referred to as Church’s thesis or the Church‐ Turing thesis  It isn’t a statement that can be proved, but there is a lot of evidence for it

1. The nature of the model makes it seem that a TM can execute any algorithm a human can 2. Enhancements to the TM have been shown not to increase its power 3. Other theoretical models of computation proposed have been shown to be equivalent to a TM 4. No one has ever suggested any kind of computation that cannot be implemented on a TM

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Nondeterministic Turing Machines

Nondeterministic Turing Machines  Let L be the subset of {1}* of all strings whose length is a composite number, i.e., an element of L is a string of the form 1n, where n=p*q for some integers p and q with p,q≥2.

 There are multiple choices for each transition.  For each input x, the NTM may have more than one computation paths.

G2

 An input x is accepted if at least one computation path leads to the final state.  L(M) is the set of all accepted inputs.

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 NBG2NBG2PBMPBEqual Original: 111111111111111 After NB: 111111111111111 After G2: 111111111111111 11111 After NB: 11111111111111111111  After G2: 11111111111111111111 111 After PB: 11111111111111111111111 11111111 After M: 111111111111111 111111111111111 After PB: 111111111111111111111111111111 After Equal: Accept 8

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Nondeterministic Turing Machines

Universal Turing Machines

 Theorem 7.31: For every nondeterministic TM T there is an ordinary (deterministic) TM T1 with L(T1)=L(T)  Proof:  The idea is to use an algorithm that can test, if necessary, every possible sequence of moves of T on an input string x.  See book for details

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 Definition 7.32: A universal Turing machine is a Turing machine Tu that works as follows  Receive an input string of the form e(T)e(z), where T is an arbitrary TM, z is a string over the input alphabet of T, and e is an encoding function whose values are strings in {0, 1}*  The computation performed by Tu on this input string satisfies 1. Tu accepts e(T)e(z) if and only if T accepts z 2. If T accepts z and produces output y, then Tu produces output e(y) 10

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Universal Turing Machines (cont’d.)

Universal Turing Machines (cont’d.)

 The encoding function has several crucial features:  It is possible to decide algorithmically, for an arbitrary string w{0, 1}*, whether w is a legitimate value of e

 State labels will be replaced by numbers, and we will base the encoding on these numbers  In order to use numbers to encode symbols as well, we adopt this convention: there is an infinite set S={a1, a2, …} of symbols, including a1= , such that the tape alphabet of every TM T is a subset of S  The idea of the encoding is to represent a TM as a set of moves and associate each move with a 5‐tuple of numbers  Each number is in unary followed by a 0

 A string w should represent at most one TM, or at most one string  There should be an algorithm for decoding strings of the form e(T) or e(z) and reconstructing the TM or string it represents 11

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Universal Turing Machines (cont’d.)

Universal Turing Machines (cont’d.)

 Definition 7.33: If T=(Q, , , q0, ) is a TM and z is a string, define the strings e(T) and e(z) as follows  First assign numbers to each state, tape symbol, and tape head direction of T:  n(ha)=1, n(hr)=2, n(q0)=3  The other elements qQ get distinct numbers  4  n(R)=1, n(L)=2, n(S)=3  For each move m of the form (p, )=(q, , D), e(m)=1n(p)01n()01n(q)01n()01n(D)0  List the moves of T as m1, …, mk (the order is arbitrary), and e(T)=e(m1)0e(m2)0…0e(mk)0  If z=z1z2…zj is a string, where each ziS, e(z)=01n(z1)01n(z2) …01n(zj)0 13

 Encoding  Symbol: n()=1, n(a)=2, n(b)=3  State: n(ha)=1, n(hr)=2, n(q0)=3, n(p)=4, n(r)=5,  Direction: n(R)=1, n(L)=2, n(S)=3

 Move m: (q0, )=(p, , R)  e(m)=130101401010=111010111101010 e(m): 5‐tuple (11*0)5  e(T) = 1110101111010100 111101110111101110100 1111011011111011101100 1111010111110101100 111110111011111011101100 11111010101011100 e(T): a concatenation of one or more (11*0)50 14

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Universal Turing Machines (cont’d.)

Universal Turing Machines (cont’d.)

 Theorem 7.36: Let E={e(T) | T is a TM}. Then for every x{0,1}*, xE if and only if: 1. x matches the regular expression (11*0)50((11*0)50)*, so that it is a sequence of 5‐tuples 2. No two substrings of x representing 5‐tuples have the same first two parts 3. None of the 5‐tuples have first part 1 or 11 4. The last part of each 5‐tuple must be 1, 11, or 111

Construct a universal TM Tu  Tape 1: input and out, originally with e(T)e(z).  Tape 2: working tape of T containing current symbol.  Tape 3: T’s current state. 1110101110101001111011101111011101001111011011111011101100111101011… 10111011101101110110 1111

(p, a)= (r, b, L)

1110101110101001111011101111011101001111011011111011101100111101011… 101110111011101110110 11111

1. T’s computation on the input string z might never terminate 2. T might reject 3. T might accept 15

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