Introduction to CS Theory Lecture 15 –Turing Machines Piotr Faliszewski

Our Models of Computation So Far  Models of computation Finite automata Pushdown automata  Restrictions FA  Finite memory  Left-to-right access to input PDA  Infinite memory…  … but accessible only in a stack form  Left-to-right access to input  Unrestricted model of computation? Turing machine δ, q …

Church-Turing Thesis  Church-Turing Thesis A language can be solved by an algorithm if and only if it can be accepted by a Turing machine that terminates on every output.  How to validate Church- Turing Thesis? Many models of algorithms Unrestricted grammars Lambda calculus RAM machines … all equivalent to Turing machines  Can Church-Turgin thesis be proven? Turing machines were, at the time, the most convincing model

Definition  Turing machine M is a quintuple M = (Q, Σ, Γ, q 0, δ) where: Q – finite set of states, assumed not to contain the two halting states, h a and h r. Σ is the input alphabet and Γ is the tape alphabet. Σ  Γ. Γ does not contain the special blank symbol Δ q 0 – the initial state δ – the transition function δ: Q  (Γ  {Δ})  (Q  {h a, h r })  (Γ  {Δ})  {R, L, S} δ is a total funciton (i.e., your machine should not get stuck in a non-halting state without knowing what to do)

Transition of a TM  δ(q, X) = (p, Y, D) means: If the machine is in state q, scanning symbol X, then: Change state to p Replace symbol X with symbol Y Move the tape according to the direction D (L – left, R – right, S – stay)  Questions What happens if the machine “falls off” the left side of the tape? Do we really need the “S” move?  Example Strings of length 2 n Turing machine for the palindromes language

Configurations  Configuration of a Turing machine Contents of the tape Position of the head Current state  Notation (q, xay) means that  Machine in state q  The head is at the last symbol of the string xa  The head is followed by string y, followed by an infinte number of blank symbols  Initial configuration (q0, Δx) x – machines’s input string  Given a Turing machine T, we define relation (q,xay) (p,zbw) If the latter can be obtained in one step from the former

Languages Accepted by TMs  Definition. Let M = (Q, Σ, Γ, q 0, δ) and let x  Σ * x is accepted by M if (q 0, Δx) * (h a, yaz) for some y,a, z  A string can be rejected if: M ends up in state h r, or M runs forever on x  Notation L(M) – language accepted by a Turing machine M If a language is accepted by a Turing machine then we say that it is recursively enumerable The set of all languages accetped by Turing machines is called RE If a language is accepted by a Turing machine that halts on all inputs then we say that it is recursive

Examples  Give a Turing machine for the following language { a i b i c i | i ≥ 0 }

Computing a Function on a TM  Definition Let M = (Q, Σ, Γ, q 0, δ) be a TM Let f: Σ *  Γ * be a total function We say that M computes f if for each x in Σ * it holds that (q 0, Δx) * (h a,Δf(x))  Example Give a TM that computes f(1 n ) = 1 2n

Power of Turing Machines  We can add extra features to a Turing machine 2-way infinite tape More than one tape Random access to the memory Nondeterminisms 2-dimensional tape  Example: Try to simulate these features (or prove that simulation is possible) None of the extra features strengthens the model!

Languages and Decision Problems  Decision problem Name Input  the mathematical entities about whose properties we ask Question  A well-defined yes/no question  Language A set of strings Languages encode decision problems  Example Name: Connectivity Input: Graph G, vertices s and t Question: Are vertices s and t connected in G

Languages and Decision Problems  Our two fundamental questions What decision problems can and cannot be solved? What decision problem can and cannot be solved efficiently  We focus on languages  Chomsky Hierarchy  Models of computation Regular languages / finite automata Context-free languages / push-down automata Turing machines

Hierarchy of Languages Finite languages Context-free languages Recursive languages All languages

Undecidable Languages  Can everything be solved on a computer? Recursive languages Recursively enumerable languages  do they really exist?  Two proofs… Counting… And an example of a language