B w dfa b accepts string w
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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

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

MAdfa

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

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

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)


A a is a dfa l a

{<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

{<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

{<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


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