Hopcroft, Motawi, Ullman, Chap 8 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 Represent recursive languages Models contemporary programs
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
TM definition A Turing Machine M is a tuple M = (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 B is the blank symbol (default symbol on input tape) F Q is the set of accepting states or final states of M (if applicable)
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})
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
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…Xn which means the TM is in state q, the tape contains X1X2…Xn and the tape head is at Xi
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)
TM as recognizer A TM accepts a string w if there exists a sequence of moves from ID 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
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 from ID q0w to ID uqixv (u,v*, x, qiQ) & (qi,x) is undefined (no transition applies) The TM can be viewed as a function; f(w) = uxv
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)