7.2 Turing Machines as Language Acceptors 7.3 Turing Machines that Compute Partial Functions

Transcription

1 CSC4510/6510 AUTOMATA 7.1 A General Model of Computation 7.2 Turing Machines as Language Acceptors 7.3 Turing Machines that Compute Partial Functions A General Model of Computation Both FA and PDA are models of computation An FA cannot accept SimplePal={xcx r x {a, b}*} or L={xcx x {a,b}*} A PDA cannot accept AnBnCn={a n b n c n n 0} An FA with a queue instead of a stack can accept L A PDA-like machine with two stacks can accept AnBnCn Either one is a reasonable candidate for a model of general-purpose computation 2 1

2 A General Model of Computation (cont d.) Turing machine Not obtained by adding data structures onto an FA Predates the FA and PDA models (Alan Turing s contributions date from the 1930 s) A Turing machine is not just the next step beyond a PDA According to the Church-Turing thesis, it is a general model of computation, potentially able to execute any algorithm A General Model of Computation (cont d.) For simplicity, Turing specified a linear tape which has a left end and is potentially infinite to the right head Unbounded tape 3 4 2

3 A General Model of Computation (cont d.) A single move is determined by the current state and the current tape symbol and has three parts Changing from the current state to another state Replacing the symbol in the square by another Leaving the tape head where it is (S), moving it one square to the left (L), or moving it one square to the right (R) The input string is assumed to be on the tape initially A Turing machine has two halt states: acceptance, rejection Turing machines may never stop 5 A General Model of Computation (cont d.) FINITE STATE CONTROL q 10 A I N P U T 3

4 A General Model of Computation (cont d.) Definition 7.1: A Turing Machine (TM) is a 5-tuple T=(Q,,, q 0, ), where: Q is a finite set of states. Halt states h a and h r are not elements of Q The input alphabet and the tape alphabet are both finite sets, with. The blank symbol is not an element of. q 0, the initial state, is an element of Q The transition function is : Q ( { }) (Q {h a, h r }) ( { }) {R, L, S} A General Model of Computation (cont d.) (p, X)=(q, Y, D): when T is in state p and the symbol in the current square is X, T replaces X by Y in that square, changes to state q, and moves the tape head one square to the right, or moves one square to the left, or doesn t move If q=h a /h r, this move causes T to halt Once it halts, it cannot move L Transition diagram If the TM attempts to move the tape head to the left when it is on square 0, the TM halts in state h r, leaving the tape head in square 0 and leaving the tape unchanged X p q Y 7 8 4

5 A General Model of Computation (cont d.) 1/1,R 0/0,R; 1/1,R s 0/0,R p 0/0,R q /,R h a 1/1,R M=(Q, Σ, Γ, q 0, δ) Q = {s, p, q} Σ = {0, 1} Г = {0, 1} q 0 =s δ 0 1 s (p, 0, R) (s, 1, R) - p (q, 0, R) (s, 1, R) - q (q, 0, R) (q, 1, R) (h,, R) A General Model of Computation (cont d.) Normally a TM begins with an input string starting in square 1 and all other squares blank In any case, the set of nonblank squares on the tape must always be finite Current configuration of a TM: a single string xqy, where q is the current state, x is the string of symbols to the left of the current square, y is either null or a string starts in the current square, and everything after xy on the tape is blank xqy T zrw / xqy T *zrw: T moves from the 1 st configuration to the 2 nd in one move, or in 0 or more moves, respectively (q, a)=(r,, L): aabqa a T aarb a Initial configuration corresponding to input x: q 0 x 10 5

6 A General Model of Computation (cont d.) CONFIGURATION 11010q Turing Machines as Language Acceptors Definition 7.2: If T=(Q,,, q 0, ) is a TM and x *, x is accepted by T if q 0 x T * wh a y for some w, y ( { })* A language L * is accepted by T if L=L(T), where L(T)={x * x is accepted by T} An FA and a TM that accept the same language: q

7 Turing Machines as Language Acceptors (cont d.) If the language were not regular, the TM could not move its tape head to the right on every move. The 2 nd diagram does not show any of the moves to the reject state They all have the same form, and there is one from each of the states p, q, and s (the nonhalting states other than q 0 that correspond to nonaccepting states in the FA), as shown below A string could be accepted as soon as an occurrence of ab is found, without reading the rest of the input. Turing Machines as Language Acceptors (cont d.) TMs vs. FAs TM can both write to and read from the tape The head can move left and right The string does not have to be read entirely Accept and Reject take immediate effect

8 Turing Machines as Language Acceptors (cont d.) A TM accepting XX={xx x {a, b}*} Input: aba Turing Machines as Language Acceptors (cont d.) A TM accepting XX={xx x {a, b}*} Input: aba q 1 q 2 q 2 q 4 q 3 q 2 A a b a A a b A a

9 Turing Machines as Language Acceptors (cont d.) A TM accepting XX={xx x {a, b}*} Input: aba Turing Machines as Language Acceptors (cont d.) A TM accepting XX={xx x {a, b}*} Input: aba q 4 q 1 q 2 h q r : 3 Reject q 2 A a B b A a A a B b A a

10 Turing Machines as Language Acceptors (cont d.) q 0 ab q 1 ab Aq 2 b Abq 2 Aq 3 b q 4 AB Aq 1 B q 5 AB q 5 ab q 6 ab Aq 8 B Ah r B Reject Turing Machines that Compute Partial Functions A TM that produces an output string for every legal input string is said to compute a partial function on * We ll also consider TMs that compute partial functions on ( *) k, i.e., functions of k variables q 0 aa q 1 aa Aq 2 a Aaq 2 Aq 3 a q 4 AA Aq 1 A q 5 AA q 5 aa q 6 aa Aq 8 A q 9 A Aq 6 Ah a Accept The most important issue is what output strings are produced for input strings in the domain of f

11 Turing Machines that Compute Partial Functions (cont d.) Definition 7.9: Let T=(Q,,, q 0, ) be a TM, k a natural number, and f a partial function from ( *) k to *. We say that T computes f if for every (x 1, x 2,, x k ) in the domain of f, q 0 x 1 x 2 x k T * h a f(x 1, x 2,, x k ) and no other input that is a k-tuple of strings is accepted by T. A partial function f is Turing-computable if there is a TM that computes f Turing Machines that Compute Partial Functions (cont d.) A TM may compute a partial function whose domain and codomain are sets of numbers Consider partial functions on N k with values in N Use unary notation for numbers The official definition is similar to Definition 7.9, except that the input alphabet is {1}, the initial configuration looks like q 0 1 n 1 1 n 2 1 n k and the final configuration is h a 1 f(n 1, n 2,, n k )

12 Turing Machines that Compute Partial Functions (cont d.) n mod 2 Turing Machines that Compute Partial Functions (cont d.) Reversing a string q * h a 1 q * h a 23 q 0 abb * h a bba q 0 baba * h a abab 24 12