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

In-Class Notes