Ω-regular language

The ω-regular languages are a class of ω-languages that generalize the definition of regular languages to infinite words. Büchi showed in 1962 that ω-regular languages are precisely the ones definable in a particular monadic second-order logic called S1S.

An ω-language L is ω-regular if it has the form

The elements of A^ω are obtained by concatenating words from A infinitely many times. Note that if A is regular, A^ω is not necessarily ω-regular, since A could be {ε}, the set containing only the empty string, in which case A^ω=A, which is not an ω-language and therefore not an ω-regular language.

Theorem: An ω-language is recognized by a Büchi automaton if and only if it is an ω-regular language.

Proof: Every ω-regular language is recognized by a nondeterministic Büchi automaton; the translation is constructive. Using the closure properties of Büchi automata and structural induction over the definition of ω-regular language, it can be easily shown that a Büchi automaton can be constructed for any given ω-regular language.

Conversely, for a given Büchi automaton A = (Q,Σ,Δ,I,F), we construct an ω-regular language and then we will show that this language is recognized by A. For an ω-word w=a₁,a₂,... , let w(i,j) be the finite segment a_i+1,...,a_j-1,a_j of w. For every q,q'∈ Q, we define a regular language L_q,q' that is accepted by the finite automaton (Q,Σ,Δ,{q},{q'}).

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