Introduction to Turing Machines 1 1 1 1 1 1 1 1 1 1 1 1 1

The Turing Machine A TM consists of an infinite length tape, on which 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A TM consists of an infinite length tape, on which input is provided as a finite sequence of symbols. A head reads the input tape. The TM starts at start state s0. On reading an input symbol it optionally replaces it with another symbol, changes its internal state and moves one cell to the right or left. In the above diagram, TM states are depicted by different colours. So, when the head moves, its colour also changes indicating that it has changed its internal state.

The Turing Machine A TM is defined as: TM = <S, T, s0, d, H> where, S is a set of TM states T is a set of tape symbols s0 is the start state H  S is a set of halting states d : S x T S x T x {L,R} is the transition function A TM begins computation at state s0. At each computational step, it reads the present tape symbol, changes its internal state, optionally writes another symbol onto tape, and moves one cell to the left or right in the tape. The TM halts computation (and accepts the input string) when it reaches one of the halt states in H.

Simple TM Examples Turing Machine U+1: Given a string of 1s on a tape (followed by an infinite number of 0s), add one more 1 at the end of the string. #111100000000…….  #1111100000000………. A tape is denoted by #11110000…. Where # is the start of the tape.

Simple TM Examples TM: U+1 d(s0, 1) |-- (s0, 1, R) d(s0, 0) |-- (h, 1, STOP) #s0111100000…..  #1s011100000…..  #11s01100000…..  #111s0100000…..  #1111s000000…..  #11111h0000….. STOP TM computations are illustrated in the form #s11110000… which shows the TM to be in state s, and at the beginning of the tape.

Exercice state symbol Δ(state, symbol) S0 b (halt, a, stop) a (S1 , a, right ) S1 (halt, b, stop) (S0 , a, right ) TM computations are illustrated in the form #s11110000… which shows the TM to be in state s, and at the beginning of the tape. Input = “aaaabb” What is the output for this input?

Solution s0 “ aaaabb” s1 “ aaaabb ” s0 “ aaaabb ” s1 “ aaaabb ” halt “ aaaaab ” TM computations are illustrated in the form #s11110000… which shows the TM to be in state s, and at the beginning of the tape. Input = a finite sequence of “a” symbol, followed by an infinite sequence of “b”. Describe what the output this machine generates.

Turing’s Thesis Any mathematical problem solving that can be described by a mechanical procedure (algorithm) can be modeled by a Turing machine. All computers today perform only mechanical problem solving. They are no more expressive than a Turing machine. A philosophical question: Do digital computers today perform only mechanical computations? What about our thinking? Do we think mechanically? Can we reduce our thought processes to an algorithm?

Turing’s Thesis Turing’s thesis is not a “theorem” there is no “proof” for the thesis. The theorem may be refuted by showing at least one task that is performed by a digital computer which cannot be performed by a Turing machine. Many contentions have been made to this end. However, till date there have not been any conclusive evidence to refute Turing’s thesis. Although Turing’s a-machine were the most popular, there are many more machines proposed by Turing. Basically they are meant to be more powerful than a-machines. These include the c-machines (choice machines), o-machines (oracle machines) and u-machines (unorganized machines). However, there have been no conclusive proofs about whether they are more expressive than a-machines.

Conclusions Turing machines are a minimal extension over PDAs which provide greater expressiveness. TMs are at a level that is much below the assembly language of any typical microprocessor. So in the practical world, TMs are more useful in what they cannot do rather than in what they can.