Omega-regular language

Template:Short description In computer science and formal language theory, the ω-regular languages are a class of ω-languages that generalize the definition of regular languages to infinite words. As regular languages accept finite strings (such as strings beginning in an a, or strings alternating between a and b), ω-regular languages accept infinite words (such as, infinite sequences beginning in an a, or infinite sequences alternating between a and b).

Formal definition

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

• A^ω where A is a regular language not containing the empty string
• AB, the concatenation of a regular language A and an ω-regular language B (Note that BA is not well-defined)
• A ∪ B where A and B are ω-regular languages (this rule can only be applied finitely many times)

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 for example {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.

It is a straightforward consequence of the definition that the ω-regular languages are precisely the ω-languages of the form A₁B₁^ω ∪ ... ∪ A_nB_n^ω for some n, where the A_is and B_is are regular languages and the B_is do not contain the empty string.

Equivalence to Büchi automaton

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 Template:Math, we construct an ω-regular language and then we will show that this language is recognized by A. For an ω-word w = a₁a₂... let w(i,j) be the finite segment a_i+1...a_j-1a_j of w. For every Template:Math, we define a regular language L_q,q' that is accepted by the finite automaton Template:Math.

Lemma: We claim that the Büchi automaton A recognizes the language Template:Math

Proof: Let's suppose word Template:Math and q₀,q₁,q₂,... is an accepting run of A on w. Therefore, q₀ is in Template:Mvar and there must be a state Template:Mvar in F such that Template:Mvar occurs infinitely often in the accepting run. Let's pick the strictly increasing infinite sequence of indexes i₀,i₁,i₂... such that, for all k≥0, Template:Mvar is Template:Mvar. Therefore, Template:Math and, for all Template:Math Therefore, Template:Math

Conversely, suppose Template:Math for some Template:Math and Template:Math Therefore, there is an infinite and strictly increasing sequence i₀,i₁,i₂... such that Template:Math and, for all Template:Math By definition of L_q,q', there is a finite run of Template:Mvar from Template:Mvar to Template:Mvar on word w(0,i₀). For all k≥0, there is a finite run of Template:Mvar from Template:Mvar to Template:Mvar on word w(i_k,i_k+1). By this construction, there is a run of A, which starts from Template:Mvar and in which Template:Mvar occurs infinitely often. Hence, Template:Math.

Equivalence to Monadic second-order logic

Büchi showed in 1962 that ω-regular languages are precisely the ones definable in a particular monadic second-order logic called S1S.

Bibliography

• Wolfgang Thomas, "Automata on infinite objects." In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, pages 133-192. Elsevier Science Publishers, Amsterdam, 1990.