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{** | DFA B accepts string w}PowerPoint Presentation**

{** | DFA B accepts string w}**

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{** | DFA B accepts string w} **

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MAdfa

accept

B

accept

<B,w>

S

w

reject

reject

- S simulates B with input w
- MAdfa halts because simulation only runs |w| steps

MAnfa

<N,y>

accept

C

<P,y>

accept

MAdfa

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)

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)

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)

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