Alternation Hierarchies of First Order Logic with Regular Predicates

Abstract

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with regular numerical predicates. In this paper, we focus on the quantifier alternation hierarchies of first order logic. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its twovariable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order. Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragment while in the case of the first order logic the question remains open for levels greater than two. The main ingredients of the proofs are syntactic transformations of firstorder formulas as well as the infinitely testable property, a new algebraic notion on varieties that we define.

One quantifier alternation in first-order logic with modular predicates

“…The following corollary is a consequence of Corollary 4. We note that Corollary 5.10 does not immediately follow from the fact that FO 2 [<] and ∆ 2 [<] define the same languages over finite words since, in general, we have (V * MOD)∩ (W * MOD) = (V ∩ W) * MOD for varieties V and W. The following example is due to Dartois and Paperman [5].…”

The regular languages of wire linear AC$$^0$$

“…As we hinted, the L operator intuitively adds the local predicates and the Q operator the modulo predicates, it is thus not surprising that we will end up showing QLDA = FO 2 [reg]. This result was announced without proof in [7] and we provide a proof here through the chain of inclusions that will be concluded in Section 3.4. We start with:…”

Logical and Algebraic Characterizations of Rational Transductions

“…Extensions of the above fragments with additional predicates can easily be shown to satisfy assumptions (1)-(4). A table from [DP15] sums up many results of decidability concerning fragments of FO[<] with additional predicates. For all these fragments, our Theorem 6.18 should carry over.…”