1. {<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

2. {<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)

3. {<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)

4. {<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)

5. {<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