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Turing Machines

Turing Machines. Hopcroft, Motawi, Ullman, Chap 8. Models of computation. Finite automata and regular expressions Represent regular languages Can’t “count” Grammars and pushdown automata Represent context free languages Can count and remember symbols once Turing machines

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Turing Machines

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  1. Turing Machines Hopcroft, Motawi, Ullman,Chap 8

  2. Models of computation • Finite automata and regular expressions • Represent regular languages • Can’t “count” • Grammars and pushdown automata • Represent context free languages • Can count and remember symbols once • Turing machines • Represent recursive languages • Models contemporary programs

  3. Turing Machine Model • Input tape surrounded by infinitely many blanks • Tape head can move back and forth the tape and replace current symbol … … BBBBBBB000111BBBBB TM

  4. TM definition • A Turing Machine M is a tupleM = (Q, , , , q0, B, F), where: • Q is a set of states •  is the input alphabet •  is the tape alphabet =   {B}  other tape symbols • : Q    Q    D is the state transition function mapping (state, symbol) to (state, symbol, direction);D = {,};  may be undefined for some pairs • q0 is the start state of M • Bis the blank symbol (default symbol on input tape) • F  Q is the set of accepting states or final states of M(if applicable)

  5. TM that accepts 0n1n • Q={q0,q1 ,q2,q3,q4}, ={0,1}, ={0,1,B,Y} •  defined as follows: • (q0,0) = (q1,B,) erase leftmost 0 • (q1,0) = (q1,0,) move to right until a 1(q1,Y) = (q1,Y,) is encountered, replace(q1,1) = (q2,Y,) that 1 with Y • (q2,Y) = (q2,Y,) move to left until a blank(q2,0) = (q2,0,) is encountered, then go(q2,B) = (q0,B,) back to initial state • (q0,Y) = (q3,Y,) if Y on tape go to state 3 • (q3,Y) = (q3,Y,) ensure only Y’s remain on tape(q3,B) = (q4,B,) accept once B is seen (F={q4})

  6. TM that increments a bit-string • Q={q0,q1 ,q2,q3}, ={0,1}, ={0,1,B,X} •  defined as follows: • (q0,0) = (q0,0,) go to rightmost(q0,1) = (q0,1,) non-blank(q0,B) = (q1,B,) • (q1,1) = (q1,0,) replace 1’s with 0’s(q1,0) = (q2,1,) until 0/B is encountered, (q1,B) = (q2,1,) replace that 0/B with a 1 • No applicable transitions from q2 means the turing machine halts

  7. Instantaneous descriptions • Instantaneous description (ID): depicts the characteristics of the machine as transitions are carried out • For finite automata, the state of the machine and the remaining input is sufficient • For TM’s, the following are needed for an ID: • State • Symbols on the tape • Position of the tape head • Can be expressed as X1X2…Xi-1qXiXi+1…Xnwhich means the TM is in state q, the tape contains X1X2…Xn and the tape head is at Xi

  8. ID example • Suppose for the first TM example,the input is 0011 • The initial ID is q00011 • After applying the transition(q0,0) = (q1,B,), ID: q1011 • Depict this as a move: q00011 |q1011 • Next 3 transitions:(q1,0) = (q1,0,), ID: 0q111(q1,1) = (q2,Y,), ID: q20Y1(q2,0) = (q2,0,), ID: q2B0Y1 • Eventually, ID will be YYBq4 (TM accepts)

  9. TM as recognizer • A TM accepts a string w if there exists a sequence of moves fromID q0w to ID uqfv (u,v  *, qf  F) • q0w |* uqfv • In the previous example, q00011 |* YYBq4 • Given a TM M, L(M) is the set of all strings that M accepts • A language recognized by a TM is called a recursively enumerable language

  10. TM halting on input • TMs are also useful for computation • In this case, what is important is the machine halts on input w (and leaves the appropriate output on the tape) • A TM halts on input w if there exists a sequence of moves fromID q0w to ID uqixv (u,v*, x, qiQ)& (qi,x) is undefined (no transition applies) • The TM can be viewed as a function; f(w) = uxv

  11. About TMs • Church-Turing Thesis: TMs represent what can be solved by a computer program(a mathematically unprovable statement) • Some problems cannot be solved by a TM (e.g., the Halting Problem) • TMs can be deterministic or nondeterministic; the variation helps in modeling problem complexity classes (P and NP)

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