Decidability (Section 4.1)

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Decidability(Section 4.1)
Héctor Muñoz-Avila
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Theorem 4.1 ltD,wgt D is a DFA accepting w is
decidable Proof. Go forward from starting state
Theorem 4.2 ltN,wgt N is a NFA accepting w is
decidable Proof. Use Theorem 4.1
Theorem 4.3 lte,wgt e is a reg. exp. generating
w is decidable Proof. Use Theorem 4.1 or 4.2
Theorem 4.4 ltDgt D is a DFA and L(D) ? is
decidable Proof. Example of one. Go forward from
starting state. Mark states.
Theorem 4.5 ltD,Dgt D and D are DFAs and L(D)
L(D) is decidable Proof. Use Theorem 4.4 and
construct a DFA C such that L(C) (L(D) n
L(D)C)? (L(D)C n L(D))
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Theorem 2.9. If L is a CFL, there is a a CFG in
Chomsky normal form generating L Proof. We did
this in the class of Context-free Languages
Problem 2.26. If w is generated by a CFG G in
Chomsky normal form, then derivations is
bounded by a factor of n. Proof. Size of parse
tree. For direct proof with exact see this
problem.
Theorem 4.7 ltG,wgt G is a CFG generating w is
decidable Proof. Use Theorem 2.9 and Problem 2.26
Theorem 4.8 ltGgt D is a CFG and L(G) ? is
decidable Proof. From terminals go backwards in
rules. Mark variables.
Problem 2.19 If L is a CFL and L is a regular
language then L n L is a CFL Proof. Take PDA P
for L and DFA D for L. Construct P?D PDA.