Turing Machines CS 130 Theory of ComputationHMU Textbook: 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 and recursively enumerable languages • Model 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 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)
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 accepts 0n1n 0,0, Y,Y, 0,0, q1 Y,Y, 0,B, 1,Y, q0 q2 B,B, Y,Y, Y,Y, q4 q3 B,B,
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
TM that increments a bit-string go to the rightmost non-blank move to the left,replacing 1s with 0s 0,0, 1,1, 1,0, 0,1, q0 B,B, q1 q2 B,1, once a B or 0 isencountered, replace it with a 1
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
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 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
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, qiQ)& (qi,x) is undefined (no transition applies) • The TM can be viewed as a function; f(w) = uxv
TM as decider • It could be arranged so that the TM either leaves a YES or a NO on the input tape after processing an input string • YES means accept, NO means reject • The TM decides whether a string is acceptable • The TM in this case defines a corresponding language, in a stronger sense than a TM recognizer, because it also halts for rejected strings • The language in this case is called a recursive language (different from recursively enumerable)
Hierarchy of languages TM recognizer recursively enumerable type 0 recursive TM decider context-sensitive type 1 context-free type 2 type 3 regular
About TMs • Church-Turing Thesis: TMs represent what can be solved by a computer program(a mathematically unprovable statement) • Some problems cannot be solved/decided 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)
