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**Deterministic Turing Machines**

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**Unique accept state. Also no arrows are coming out.**

Constituents Infinite tape Head Transition diagram … … q1 q0 qf Unique accept state. Also no arrows are coming out.

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Formal definition A Deterministic Turing Machine (DTM) is a sextuple (Q, Σ, Γ, δ, q0, qf) where: Q is a finite set of states Σ is the input alphabet Γ is the tape alphabet, Σ ⊆ Γ There is a special blank symbol □ in Γ q0 is the start state qf is the final state δ: Q x Γ → Q x Γ x {L,R,S}, where L stands for Left, R stands for Right and S stands for Stay.

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**Formal definition (cont.)**

δ(q , a) = (q’ , b, L) means: “if you are in state q and the head in the tape points to symbol a then move to state q’, replace symbol a with symbol b in the tape and move the head one position to the left” Illustration: … … … … a b head head q‘ q‘ a→b,L … a→b,L … … q … q

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**Determinism Determinism means that I have no choices!**

The transition function δ sends pair (q, a) to at most one triple (q’, b, x) (in other words there is at most one arrow from state q reading symbol a –maybe there are no such arrows). No ε-moves are allowed (I must read something in the tape in order to change state).

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**Machine’s special status**

Start: Be at initial state , the tape contains only the input and the head points to the first (leftmost symbol of the input) Accept: Reach the accept state. The machine stops the computation and accepts (notice that part of the input might be unread). Reject: Be in a state q (other than the accept state), read symbol a and find no outgoing arrows under symbol a. Loop for ever: Enter a subset of states which repeat for ever (different than the “reject case”)

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**Machine’s special status**

Start: … … a b a a b head q1 a→b,R q0 …

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**Machine’s special status**

Accept: … … b b a a b head … qf

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**Machine’s special status**

Reject: … … b b a a b head q’ … a→b,R q …

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**Machine’s special status**

Loop: … … b b a a b head … q b→b,S

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Recursive Languages A language is recursive (or decidable, or computable) if there is a Turing Machine which: Accepts for every string in the language Rejects for every string not in the language

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**Example Find a DTM that computes the language L = {anbn : n ≥ 0}.**

High level program: Repeat Erase an a. Pass along the rest of as. Erase a b Pass along the rest of bs. Go to the beginning of the input. Until either the whole input is erased (accept) or you find unmatched as or bs (reject).

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**Example Find a DTM that computes the language L = {anbn : n ≥ 0}.**

Answer: a → a , R x → x , R x → x , R b → b , R a → x , R b → x , R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

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Testing Test the machine for several possible inputs to see if it works as it should. Test inputs: ε (it should accept) aaabbb (it should accept) aaabb (it should reject) aabbb (it should reject) aabba (it should reject) (See file dtm_example.pptx).

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Introduction to Computability Theory

Introduction to Computability Theory

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