Nesting negations in FO2 over infinite words

Abstract

We consider two-variable first-order logic FO 2 over infinite words. Restricting the number of nested negations defines an infinite hierarchy; its levels are often called the half-levels of the FO 2 quantifier alternation hierarchy. For every level of this hierarchy, we give an effective characterization. For the lower levels, this characterization is a combination of an algebraic and a topological property. For the higher levels, algebraic properties turn out to be sufficient. Within two-variable first-order logic, each algebraic property is a single ordered identity of omega-terms. The topological properties are the same as for the lower half-levels of the quantifier alternation hierarchy without the two-variable restriction (i.e., the Cantor topology and the alphabetic topology).Our result generalizes the corresponding result for finite words. The proof uses novel techniques and is based on a refinement of Mal'cev products for ordered monoids.

Deciding FO2 Alternation for Automata over Finite and Infinite Words

“…However, for certain varieties, a more direct approach using the relations ∼ K and ∼ D is sufficient. This approach was refined in [11] to define a chain of ordered monoids. Let s, t ∈ M , then:…”

“…Each of these logical fragments defines a language variety. These varieties have decidability characterizations for both finite words [17,9] and infinite words [1,11]. These criteria are presented in Table 1.…”

“…[6,12,16]. In particular, this approach was used in yet unpublished work by Boussidan and the second author for the characterization of the join levels of the alternation hierarchies, and by the authors for the characterization of the half-levels [11].…”