B,w DFA B accepts string w

1
ltB,wgt DFA B accepts string w
MAdfa
accept
B
accept
ltB,wgt
S
w
reject
reject

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

2
ltN,ygt NFA N accepts string y
MAnfa
ltN,ygt
accept
accept
ltP,ygt
C
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
ltAgt A is a DFA, L(A)
MEdfa
ltAgt
accept
accept
lt q gt
GM
Z
reject
reject

• GM (graph marker) marks all accepts states
  reachable from init state and produces that list
  as lt q gt
• 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
ltA,Bgt DFAs A B, L(A)L(B)
MEQdfa
ltA,Bgt
accept
accept
lt C gt
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
ltM,wgt TM M accepts string w
MU
ltM,wgt
accept
accept
S
reject
reject

• S simulates M on input w
• MU doesnt always halt because M could loop
  forever on w
• But, problem is even worse than that some
  problems cant even be encoded using our formal
  system