Equations Over ω-Automata

Abstract

An infinite word over an alphabet A, or ω-word, is an infinite sequence of symbols of A. A ^ω is the set of ω-words on A. An ω-language on A is a subset of A ^ω. Moreover, A ^∞ = A ^⋆ ∪A ^ω. An ω-word may be written as $\alpha = \alpha (0)\alpha (1)\ldots$, where α(i) ∈ A for every i ≥ 0; if n ≤ m, $\alpha (n,m) = \alpha (n)\ldots \alpha (m - 1)\alpha (m)$ and $\alpha (n,\infty ) = \alpha (n)\alpha (n + 1)\ldots$. The notations ∃^ω n stands for ’there are infinitely many n’ and ∃^{< ω} n stands for ’there are finitely many n’.