Turing Machines Decidability

Transcription

1 Turing Machines Decidability Master Informatique 2016 Some General Knowledge Alan Mathison Turing UK, Mathematician, computer scientist, cryptanalyst Most famous works: Computation model («Turing Machines») Work on Enigma during WWII Imitation Game («Turing Test») Father of theoretical CS and AI Turing Award Nobel Award for CS 1 Turing Machines 1.1 Introduction to Turing Machines Turing Machines Computing Machine [Turing 1936] Abstract model to compute any «calculable» decimal number i.e. any number which can be computed with a finite amount of resources Proposed years before the apparition of computer, but quite a faithful abstraction of modern machines Roughly speaking, tape (sequence of squares) browsed by a reader/writer: memory of a computer transition function: processor of a computer Definition of Turing Machines A Turing machine is a tuple Q, Γ, B, Σ, q 0,, F with: Q = {q 0, q 1,, q m }, a finite set of states Γ, the finite set of symbols used by the machine (vocabulary) B Γ, a particular symbol called blank Σ Γ, the input vocabulary q 0 Q, the initial state of the machine : Q Σ Q Γ {L, R}, the transition function 1

2 F Q, the set of final states Transition Function : Q Σ Q Γ {L, R} Input: a pair (current state, symbol on the tape) called configuration Output: a tuple (next state, symbol to write, move) Example of Turing Machines (1/2) Multiplying Integers by 2 Let us consider M = Q, Γ, B, Σ, q 0,, F with: Q = {in_progress, done} Γ = {0, 1, B} Σ = {0, 1} q 0 = in_progress as described in the table F = done Current state Current symbol Next State Write Move in_progress 0 in_progress 0 R in_progress 1 in_progress 1 R in_progress B done 0 R done STOP Example of Turing Machines (2/2) Multiplying Integers by 2 Read 1 in state in_progress: write 1, move right, state in_progress Read 0 in state in_progress: write 0, move right, state in_progress Read 1 in state in_progress: write 1, move right, state in_progress Read 0 in state in_progress: write 0, move right, state in_progress Read 1 in state in_progress: write 1, move right, state in_progress Read 0 in state in_progress: write 0, move right, state in_progress 2

3 Read B in state in_progress: write 0, move right, state done State done: stop d B 1.2 Turing Machines and Decision Problems Accept vs Reject vs Loop We consider a special class of Turing machines, with three possible «results». For an input x, we say that the Turing machine M accepts x if M reaches the final state YES after a finite number of steps (i.e. transitions) rejects x if M reaches the final state NO after a finite number of steps loops on x if M never reaches a final state Language of a Turing Machine We call language of a Turing Machine M the set of all inputs which are accepted by M L(M) = {x Σ M stops on x and M(x) = YES} Example If M returns YES on inputs P N[X] with exactly one root, then L(M) = {P N[X] P is a polynomial with exactly one root } Language Problem We associate the language L(M) of a Turing Machine M to a decision problem P x L(M) iff x is a positive instance of P 1.3 Non-Determinism (Non-)Deterministic Turing Machines Definition Given M a Turing machine and its transition function, M is deterministic (DTM) if d is a mapping from any configuration (q, x) to a single image (q, x, m) Otherwise, M is non-deterministic (NDTM) Solving a Decision Problem A DTM solves a decision problem P if the sequence of transitions leads to the answer for any instance Not so easy for NDTM: when the transition function has several images, each of them must be checked The machine solves the problem if for each positive instance, at least one sequence of transitions leads to a final state with the answer YES for each negative instance, every possible sequence of transitions leads to a final state with the answer NO 3

4 Illustration: DTM vs NDTM (q 0, x 0 ) (q 1, x 1 ) Deterministic (q n, x n ) Non-Deterministic (q m, x m ) (q 0, x 0 ) (q 1, x 1 ) (q k, x k ) (q n, x n ) (q o, x o ) (q p, x p ) 1.4 Church-Turing Thesis Other Computing Models Other models have been proposed Turing machines with several tapes/several dimensions/several readers-writers λ-calculus [Church 1936] Kolmogorov-Uspensky machines [Kolmogorov and Uspensky 1958] Schönhage machines [Schönhage 1980] Power of Turing machines: the Church-Turing Thesis Everything which can be computed with any of these models can be computed with a Turing machine Turing completeness A system (computing model, programming language, ) is called Turing complete if it can compute every possible computable function 2 Decidability 2.1 Recognizable and Decidable Problems Recognizability Recognizability of a Problem A problem P is called recognizable if there exists a Turing machine M such that, for each instance i of P, M(i) answers YES iff i is a positive instance of P Recognizability of a Language A language L is called recognizable if there exists a Turing machine M such that L = L(M) Prime Numbers Is n N a prime number? This problem/language is recognizable: we can write an algorithm which answers YES when n is prime, and doesn t care of non-prime numbers 4

5 Decidability Decidability of a Problem A problem P is called decidable if there exists a Turing machine M such that, for each instance i of P, M(i) answers YES when i is a positive instance of P M(i) answers NO when i is a negative instance of P Decidability of a Problem A language L is called decidable if there exists a Turing machine M which accepts each x L and rejects each x / L Equivalently: L is decidable iff both L and L are recognizable Prime Numbers Is n N a prime number? This problem/language is decidable: we can write an algorithm which answers YES when n is prime, and NO otherwise Complement of a Decision Problem Definition A problem P 1 is called the complement of P 2 L(P 1 ) = L(P 2 ). Equivalently: P 1 and P 2 are defined on the same set of instances Positive instances of P 1 are negative instances of P 2 Negative instances of P 1 are positive instances of P 2 Notation: P 1 = P 2 Prime Numbers Given n N, is there m N, m 1, m n, s.t. m divides n? This problem is the complement of the prime number problem. 2.2 Halting Problem and Reductions Halting Problem Is there a Turing machine H which has two parameters: A Turing machine M An input i of M and which returns YES when M stops on i NO when M loops on i If there is such a machine, then Halting problem is decidable; otherwise, it is undecidable Result from [Turing 1936] Halting is undecidable Reducing a Problem to Another Functional Reduction A functional reduction f is a total computable function from a problem P 1 to a problem P 2 such that, for any instance i of P 1, i is a positive instance iff f(i) is a positive instance of P 2 Notation: P 1 f P 2 Theorem If P 1 f P 2 and P 1 is undecidable, then P 2 is undecidable 5

6 (Trivial) Example of Functional Reduction P 1 : Given n N, is n even? P 2 : Given n N, is n odd? f : N N is defined by f(n) = n + 1 For a positive instance i of P 1, f(i) is a positive instance of P 2. For a negative instance i of P 1, f(i) is a negative instance of P 2. So P 1 f P 2 Here we also have P 2 f P 1 ; this is trivial since these are decidable problems. A Less Trivial Example Language with ɛ the empty word. Is L 1 decidable? L 1 = {M M halts on ɛ} Proof of Undecidability We need a mapping f : (M, x) M s.t. (M, x) Halting iff M L 1 M x(y) = M(x.y) with x.y the concatenation of words M x(ɛ) = M(x), so obviously: If M halts on x, then M x halts on ɛ If M loops on x, then M x loops on ɛ Other Method to Prove Undecidability «Algorithmic» Proof If P 1 is known to be undecidable, we can prove that P 2 is undecidable with an algorithm which computes P 1 using only «simple» steps and calls to P 2 Remember: L 1 = {M M halts on ɛ}. Suppose that M L1 is a Turing Machine which decides L 1. Algorithm 1 Halting Input: M, x = x 0 x 1 x n if x = ɛ then return M L1 (M) else x = x 1 x n M = M with q 0 replaced by (q 0, x 0 ) return Halting(M, x ) end if How to Prove Decidability? Prove P 1 f P 2 with P 2 decidable, then P 1 is decidable Write an algorithm which solves your problem with only «simple» steps! Of course, the more complex is the algorithm, the less easy it is to be sure that it is correct :( Example: Is n N a prime number? Algorithm 2 Prime Input: n N for x {2,, n } do if y N s.t. n = x y then return false end if end for return true 6

7 Références [Turing 1936] A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, [Church 1936] A. Church, An unsolvable problem of elementary number theory. American Journal of Mathematics, [Kolmogorov and Uspensky 1958] A. N. Kolmogorov and V. Uspensky, On the definition of an algorithm. Uspekhi Mat. Naut, [Schönhage 1980] A. Schönhage, Storage modification machines. SIAM Journal on Computing,