# { | DFA B accepts string w} - PowerPoint PPT Presentation

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{<B,w> | DFA B accepts string w}. M Adfa. accept. B. accept. <B,w>. S. w. reject. reject. S simulates B with input w M Adfa halts because simulation only runs |w| steps. {<N,y> | NFA N accepts string y}. M Anfa. <N,y>. accept. C. <P,y>. accept. M Adfa. reject. reject.

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{ | DFA B accepts string w}

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### {<B,w> | DFA B accepts string w}

accept

B

accept

<B,w>

S

w

reject

reject

• S simulates B with input w

• MAdfa halts because simulation only runs |w| steps

### {<N,y> | NFA N accepts string y}

MAnfa

<N,y>

accept

C

<P,y>

accept

reject

reject

• C converts NFA N to DFA P (known algorithm)

• MAnfa halts because C and MAdfa are decidable and are run a finite number of times (once each, actually)

### {<A> | A is a DFA, L(A) = { } }

MEdfa

<A>

accept

< q >

accept

GM

Z

reject

reject

• GM (graph marker) marks all accepts states reachable from init state and produces that list as < q >

• Z accepts if input is empty; otherwise rejects

• MEdfa halts because GM, Z are decidable and are run a finite number of times (once each)

### {<A,B> | DFAs A & B, L(A)=L(B)}

MEQdfa

<A,B>

accept

< C >

accept

SD

MEdfa

reject

reject

• SD creates the DFA C as symmetric difference of L(A), L(B)

• MEQdfa halts because SD, MEdfa are decidable and are run a finite number of times (once each)

### {<M,w> | TM M accepts string w}

MU

<M,w>

accept

accept

S

reject

reject

• S simulates M on input w

• MU doesn’t always halt because M could loop forever on w

• But, problem is even worse than that: some problems can’t even be encoded using our formal system