CSE 460: Computabilty and Formal Languages Turing Machine (TM) S. Pramanik
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1 CSE 460: Computabilty and Formal Languages Turing Machine (TM) S. Pramanik 1
2 Definition of Turing Machine A 5-tuple: T = (Q, Σ, Γ, q 0, δ), where Q: a finite set of states h: the halt state, not included in Q Σ= input alphabet Γ= Tape symbols Both Σ and Γ are finite and Σ Γ is the blank symbol and not in Γ q 0 is the start state and q 0 ɛq δ is the transition function (it is a partial function): δ : Q (Γ { }) (Q {h}) (Γ { }) {R, L, S} δ(p, X) = (q, Y, D) Current state pɛq {h}, Next state qɛq {h} X, Y ɛγ { } Dɛ{R, L, S} is the direction p X/Y, D q Tape has addresses 0,1,2,3,... Initially, input string is on tape starting at location 1 and all other locations on tape has initially Tape read/write head points to location 0. 2
3 Sequence of TM Moves We will call the symbol a blank. and each location of the tape a square TM read/write head points to a current square and head moves one position to right or one position to the left or stay at the same square for each move. Configuration of TM: xqy where q is the current state, x is the string to the left of the current square, y either is null or starts in the current square, everything after xy on the tape is blank. Trace a sequence of TM moves by specifying the configuration at each step: xqy T Zrw or xqy T Zrw Q is the current state, r is the next state Starting configuration: q 0 x, xɛσ is the input string Acceptance configuration: zhw, where z, wɛ(γ { }) Rejection: q 0 x T zqaw and (q, a) is not a valid input to the transition function TM loops for ever 3
4 Turing Machines as Language Acceptors If T = (Q, Σ, Γ, q 0, δ) is a TM and xɛσ, then x is accepted by T if q 0 x T zhw for some strings z, wɛ(γ { }) A language L Σ is accepted by T if L = L(T ), where L(T ) = {xɛσ x is accepted by T } 4
5 An Example Language L = ab An FA: q a 0 q 1 b a b q 2 a, b A TM accepting the same language (partial function): /, R q q 0 1 a/a, R q 2 b/b, R /, S h Trace of the computation for the input string x = abb q 0 abb q 1 abb aq 2 bb abq 2 b abbq 2 abbh (ACCEPT) x=aba q 0 aba q 1 aba aq 2 ba abq 2 a (RE- JECT) 5
6 Variants of Turing Machine Language model and computational model of Turing machines Any variant will not add any more power: (Same computations) Some minor variations: Only moves: R and L, no S (Reduction does not reduce power) Doubly infinite tape (Addition does not increase power) Major variations: Increase the number of tapes Nondeterminism Random access memory 6
7 Multitape Turing Machines A 5-tuple: T = (Q, Σ, Γ, q 0, δ), where δ : Q (Γ { }) 2 (Q {h}) (Γ { }) 2 {R, L, S} 2 Two input/output tapes with independent heads (i.e., the two heads are not moving in synchrony. Input is in the first tape, second tape is initially blank. Similarly, the output will appear on the first tape, and the contents of the second tape at the end of the computation will be irrelevant. A single move can change the state, the symbols in the current squares on both tapes, and the i positions of the two tape heads. Configuration of a 2-tape TM : (q, x 1 a1y 1, x 2 a 2 y 2 ) TM is in state q, current character is a 1 on the first and a 2 on the second tape. X 1, Y 1, X 2, Y 2 ɛ(γ { }) Starting configaration: (q 0, x, ) δ(q, X, Y ) = (r, Z, W, L, R) Current state is q, input symbols in tape 1 and 2 are X and Y respectively. Next state is r and X is replaced by Z and Y by W. Head on the first tape move left and on the second tape move right. (q, x 1 a 1 y 1, x 2 a 2 y 2 (r, x 1 b 1 y 1, x 2 b 2 y 2 ) δ(q, a 1, a 2 ) = (r, b 1, b 2, S, S) q (a1,a2) / (b1,b2), (S,S) r 7
8 Example of a Two-Tape Turing Machine Copy input in the second tape starting in square 1. Bring head back to square 0 after copying. δ(q,, ) = (q 1,,, R, R) δ(q 1, a, ) = (q 1, a, a, R, R) δ(q 1, b, ) = (q 1, b, b, R, R) δ(q 1,, ) = (q 2,,, L, L) δ(q 2, a, a) = (q 2, a, a, L, L) δ(q 2, b, b) = (q 2, b, b, L, L) δ(q 2,, ) = (h,,, S, S) (b, ) / (b,b), (R,R) (, )/ (, ), (R,R) ( a, )/ (a,a), (R,R) q 0 q 1 h (, )/(, ), (S,S) q 2 (, )/(, ), (L,L) (a,a)/ (a,a), ( L,L) (b,b)/ (b,b ), (L,L) Theorem: For every 2-tape (also true for m-tape) TM T, there is an ordinary 1-tape TM T1, such that 1. For every xɛσ, T accepts x if and only if T1 accepts x, and T rejects x if and only if T1 rejects x That is, L(T)=L(T1). 2. For every xɛσ, if (q 0, x, ) T (h, yaz, ubv) for some strings y, z, u, vɛ(γ { }) and symbols a, bɛγ { }, then q 1 x T 1 yhaz 8
9 That is, the tape of the simple TM and the first tape of the 2-tape TM have the same string left when they halt. Theorem: For every nondeterministic TM there is an ordinary deterministic TM. 9
10 Universal Turing Machine 1. A computer runs many programs, not just one particular program. 2. Data for the program is stored in memory with the program 3. If we consider the moves of a TM as a program, a given TM runs a particular program. Where is this program stored? Where is the data for this program stored? 4. A TM is not general purpose like a modern computer which runs NOT just one program. 5. Universal Turing Machine, T u, is a general purpose TM in the sense that it runs any Turing machine on it s input (a sort of recursive concept, a Turing Machine running a Turing machine). 6. Because a TM can take any input string on it s tape, we will use different input strings for T u to simulate different Turing Machines. But T u is a particular TM (i.e., a fixed program). 7. A TM T and and an input, w, to T is encoded for T u, so that T and w can be placed on the tape of T u as T u s input. 8. After all, a T u is also a TM and it takes ainy string of characters on it s tape. 10
11 9. If e(t) and e(w) are the encoding of T and w respectively, then input to T u is e(t)e(w). 10. Initially, e(t)e(w) is placed on the tape of T u, and T u runs on this input string e(t)e(w) to simulate the functions of T running on w. 11
12 Encoding e(t) and e(w) for an Universal Turing Machine 1. Symbolic notation first: e(t)={m(t)} where, m(t)= a move of TM T move m: δ(p, σ) = (q, τ, D) e(m)=(p, σ, q, τ, D), a five tuple Example: T accepting the language a q /, R q 0 1 a /a, R /, S h {(, q 0,, R), (q 1, a, q 1, a, R), (q 1,, h,, S)} Moves for T stored on tape in any order. 12
13 Encoding Using Tape Symbols 1. we will use Σ = {0, 1} for T u 2. Assign numbers to each state, tape symbol, and tape head direction of T 3. Each tape symbol a i = 1 n(a i), where n(a i ) = i 4. we use a 1, thus, = 1 n(a 1) = 1 1 = 1 5. Similarly, for states, we can assign n(h) = 1, n(q 0 ) = 2, etc. Therefore, h = 1, q 0 = 11, etc. 6. Three directions: n(r) = 1, n(l) = 2, n(s) = 3 Therefore, R= 1, L=11, S= Encoding of a move m: δ(p, σ) = (q, τ, D) is represented by a five tuple, each element of the tuple separated by a 0 1 n(p) 01 n(σ) 01 n(q) 01 n(τ) 01 n(d) 0 Example: δ(, q 0 ) = (, R) = (, q 0,, R) = Encoding of T: Moves of T in some order as m 1, m 2,..., m k e(t ) = e(m 1 )0e(m 2 )0...0e(m k )0 9. Encoding of input z: if z = z 1 z 2...z j is a string, where each z i ɛs is the infinite number of symbols. e(z) = o1 n(z 1) 01n(z 2 ) n(z j) o 13
14 Encoding of the TM accepting a q /, R q 0 1 a /a, R /, S h 1. Symbolic: e(t): (q 0,, q 1,, R), (q 1, a, q 1, a, R), (q 1,, h,, S) Using Tape symbols: 2. Two 1 s (11) to delimit consecutive moves. 3. Four 1 s (i.e., 1111) separating e(t) and e(w)? 4. e(t): e(t)e(w) where w = aba When it is placed on the input tape of the UTM, it will look like: Tape 1: Tape 2: Tape 3: 14
15 Halting Problem 1. What is the language that is accepted by the Universal Turing Machine? H={e(T )e(w) T accepts w} If T halts on w, T u halts on (e(t ), e(w)). If T does not accept w, T u does not accept (e(t ), e(w)), i.e., T u either crashes or goes into an infinite loop. 2. What is the Halting Problem? Given a Turing Machine T and an input w, does T accept w? 3. This is a decision problem with yes/no answer 4. Input to this decision problem is xɛσ 5. Σ can be partitioned int three subsets as follows: Subset that the T accepts, subset that the T does not accept by crashing and the subset that T does not accept by infinite looping. 6. Y(T)= the set of input T accepts N(T): the set of input T rejects, i.e., it says no I(T): the set of input T infinite loops on; T never halts 15
16 Halting Problem is unsolvable 1. Recursively enumerable language L: If there exists a TM that accepts L, i.e., if xɛl then TM for L halts on x, else it will either crash or loop. 2. Recursive language L: If there exists a TM for L that recognizes L, i.e., if xɛl then TM for L halts on x, else the TM crashes (i.e. it does not accept the input by only crashing and NOT by going into a LOOP). 3. If a language is recursive then it s compliment is recursive. 4. If a language is recirsively enumerateble and it s complement is also recursively enumerable then the language is recursive. Take a string and run it on T and T. Either T or T will halt on any input. 5. If a language is not recursive then it s compliment is not recursive. 6. Proof of the halting problem: 16
17 CHURCH-TURING THESIS 1. Any effective computation whatsoever, any algorithmic procedure that can be carried out by a human being or a team of human beings or a computer can be carried out by some Turing machine. 2. This statement was first formulated by a logician Alonzo Church in the 1930s. 3. One cannot prove it because there is no precise definition of an effective computation or algorithmic procedure. 4. A good bit of evidence has accumulated over the last 80 years that has caused this thesis to be generally accepted. 17
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