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## Turing Machines

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**Formal Defn of a Turing Machine**A TM is a 7-Tuple M = (Q, , , , q0, B, F) where • Q – Set of states • – Input alphabet (B ) • – Tape alphabet ( and B ) • – Transition Function (partial function) • : Q x Q x x {L, R} • q0 – Start state (q0 Q ) • B Blank symbol • F Q set of final states**Turing Machine Configurations**• String uqv such that • u,v * • uv is information on the tape • q is the current state • tape head is pointing to first symbol in v**Standard Turing Machine**• No transitions out of accepting states • Halts when no transitions defined • Tape unbounded in both directions • Deterministic: ≤ 1 transition per configuration • Input: content of tape at start • Output: content of tape when halted**Taking Steps in a Turing Machine**• For a,b,c and u,v * and qi,qj Q • uaqibv ├ uacqjv if (qi,b) = (qj,c,R) • uaqibv ├ uqjacv if (qi,b) = (qj,c,L)**Starting, Accepting and Rejecting**• For M = (Q, , , , q0, , F) • Start configuration of input w * • q0w • Accepting configuration • Halt in configuration containing q F • Rejecting configuration • Halt in configuration containing q F**Computation**• Sequence from a start configuration to a halting configuration**Infinite Loop**• Never reach a halting configuration for some start configuration C**Language of a Turing Machine**• M accepts a string w iff there exists sequence of configuration C1, C2,...,Ck • C1 is the start configuration of M on input w • each Ci yields Ci+1 • Ck is an accepting configuration • L(M) = { w * | M accepts w } • A language L is recursively-enumerable if L = L(M) for some Turing Machine M**Accepting, Rejecting and Looping**• On input w TM M may • accept by halting in accept state • reject by halting in reject state • fail to accept by looping • Deciders – TMs that never loop**Turing-decidable Languages**• L is recursive (or decidable) • if some TM decides it • if there exists TM M such that L = L(M) and M halts on all input.**Sets of Languages**Recursively-enumerable Recursive Context-Free Regular