Turing machines

Sipser 2.3 and 3.1 (pages 123-144)

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A Context-free Grammar for {anbncn| n ≥ 0}?

• Theorem 2.34 (Pumping lemma for CFLs): If A is a CFL, then there is a number p where, if s is any string in A of length ≥ p, then s = uvxyz such that: 1. For each i ≥ 0, uvixyiz A∈ , 2. |vy| > 0, and 3. |vxy| ≤ p

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Proof idea

• Surgery on parse trees

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So…

• Is {anbncn| n ≥ 0} a CFL?

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Chomsky hierarchy

Context-free languages

Regular languages

0n1n

anbncn

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Introducing… Turing machines

a b a b ⨆ ⨆ ⨆

Finite control

Bi-directional read/write head

Infinite tape

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Formally…

• A Turing machine is a 7-tuple (Q, Σ, Γ, δ, q0, qaccept, qreject), where– Q is a finite set called the states– Σ is a finite set not containing the blank symbol ⨆

called the input alphabet– Γ is a finite set called the tape alphabet with Γ⨆∈

and Σ Γ ⊆– δ:Q ×Γ → Q ×Γ ×{L,R}

– q0 Q∈ is the start state

– qaccept Q∈ is the accept state

– qaccept Q ∈ is the reject state

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Recognizing {anbncn| n ≥ 0}

b b c c ⨆ ⨆ ⨆

Finite control

Infinite tape

a a

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Configurations

• A configuration is– Current state– Current tape contents– Current head location

• u q v means– Current state is q– Current tape contents is uv– Current head points at first symbol of v

• Example– âaq1bbcc

– In state q1

– Tape contents are âabbcc– Tape head is on first b

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Yields

• A configuration C1 yields configuration C2 if the Turing machine can legally go from C1 to C2 in a single step

• yields • Written ⊢

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Turing-recognizable languages

• A Turing machine accepts input w if a sequence of configurations C1,C2,...,Ck exists where 1. C1 is the start configuration of M on input w

2. Each Ci yields Ci+1

3. Ck is an accepting configuration

• Defn 3.5: A language is Turing-recognizable if it is accepted by some Turing machine.

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Recognizing .