A Turing Machine
1/45 Turing Machines and General Grammars Alan Turing (1912 – 1954) 2/45 Turing Machines (TM) Gist: The most powerful computational model. Finite State Control Tape: Read-write head a1 a2 … ai … an … moves Note: = blank 3/45 Turing Machines: Definition Definition: A Turing machine (TM) is a 6-tuple M = (Q, , , R, s, F), where • Q is a finite set of states • is an input alphabet • is a tape alphabet; ; • R is a finite set of rules of the form: pa qbt, where p, q Q, a, b , t {S, R, L} • s Q is the start state • F Q is a set of final states Mathematical note: • Mathematically, R is a relation from Q to Q {S, R, L} • Instead of (pa, qbt), we write pa qbt 4/45 Interpretation of Rules • pa qbS: If the current state and tape symbol are p and a, respectively, then replace a with b, change p to q, and keep the head Stationary. p q x a y x b y • pa qbR: If the current state and tape symbol are p and a, respectively, then replace a with b, shift the head a square Right, and change p to q. p q x a y x b y • pa qbL: If the current state and tape symbol are p and a, respectively, then replace a with b, shift the head a square Left, and change p to q. p q x a y x b y 5/45 Graphical Representation q represents q Q s represents the initial state s Q f represents a final state f F p a/b, S q denotes pa qbS R p a/b, R q denotes pa qbR R p a/b, L q denotes pa qbL R 6/45 Turing Machine: Example 1/2 M = (Q, , , R, s, F) where: 6/45 Turing Machine: Example 1/2 M = (Q, , , R, s, F) where: • Q = {s, p, q, f}; s f p q 6/45 Turing
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