# Chapter 4: Decidability

## For Your Enjoyment

- The Liar's Paradox
is an everyday example of an undecidable statement.

## Chapter 4.1, Decidable Languages

### Decidable Languages Concerning Regular Languages

- A
_{DFA} = {<B, w> | B is a DFA that accepts w} is
a decidable language.
- A
_{NFA} is a decidable language.
- A
_{REX} = {<R, w> | R is a regular expression that
generates string w} is a decidable language.
- E
_{DFA} = {<A> | A is a DFA and L(A) = ∅} is a
decidable language.
- EQ
_{DFA} = {<A, B> | A and B are DFAs and
L(A) = L(B)} is a decidable language.

### Decidable Languages Concerning Context-Free Languages

- A
_{CFG} = {<G, w> | G is a CFG that generates
string w} is a decidable language.
- E
_{CFG} = {<G> | G is a CFG and L(G) = ∅}
is a decidable language.
- Every context-free language is decidable.

Note: EQ_{CFG} = {<G, H> | G and H are CFGs
and L(G) = L(H)} is NOT decidable.

### Language Hierarchy

- Regular Languages ⊂ Context-Free Languages
- Context-Free Languages ⊂ Decidable Languages
- Decidable Languages ⊂ Turing-Recognizable Languages