Notes-SecG - COT 4210 Turing Machines Section G Spring 2001...

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COT 4210 Section G Spring 2001 Turing Machines We complete our study of the Chomsky Hierarchy by introducing the definition of a Turing Machine. You will recall that at the beginning of this course we defined Chomsky’s notion of Phrase Structure Grammar, or Type-0 grammar. Although we will not have time to do so here, it can be shown that a Turing Machine accepts exactly the same family of languages that can be generated by PSGs. DEFINITION 32. A Turing Machine(TM) is a 7-tuple, M = (Q, Σ , Γ , δ , q 0 , B, F), where Q denotes a finite set of states , Γ denotes the tape alphabet , Σ⊆Γ denotes the input alphabet , δ denotes the transition relation , q 0 is the initial state , B Γ denotes the blank tape symbol , and F the set of final states . M is said to be deterministic if and only if the mapping, δ : Q × Γ Q × Γ × {L, R} is a partial function . A transition of M consists of reading the tape symbol, “a”, under its read/write head in a given state, q. Based on the symbol read (and the current state), M decides on three actions: (a) what symbol, “b”, to write on the tape replacing “a”, (b) what direction {Left, Right}to move its read/write head, and (c) what its next state, q’, shall be. We write, (q’,b,R or L) δ (q,a). Note the before and after configurations in the figure below. a 1 a 2 a 3 a 4 … a n B B B . .. input/work tape q finite control right move left move (a) before a transition a 1 a 2 X a 4 … a n B B B . .. input/work tape q’ finite control right move (a) after right move Initially, the TM is presented with an input tape containing some string x ∈Σ * with its read/write head positioned over the leftmost symbol of x, if x is non-null. If x is null, then the read/write head will be positioned over the special symbol, B. If M is positioned over its leftmost tape symbol and attempts to move left (L), then M halts. M also halts in any configuration for which

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Notes-SecG - COT 4210 Turing Machines Section G Spring 2001...

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