The half-levels of the FO2 alternation hierarchy

© 2016, Springer Science+Business Media New York. The alternation hierarchy in two-variable first-order logic FO 2 [ < ] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment Σm2 of FO 2 is defined by disallowing universal quantifiers and having at most m−1 nested negations. The Boolean closure of Σm2 yields the m th level of the FO 2 -alternation hierarchy. We give an effective characterization of Σm2, i.e., for every integer m one can decide whether a given regular language is definable in Σm2. Among other techniques, the proof relies on an extension of block products to ordered monoids

One Quantifier Alternation in First-Order Logic with Modular Predicates

Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison $x<y$ and predicates for $x \equiv i \mod n$ has been investigated by Barrington, Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO2[<,MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD] which is closed under negation, has the same expressive power as two-variable logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of FO2[<,MOD]

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

Block products and nesting negations in FO2

The alternation hierarchy in two-variable first-order logic FO 2 [∈ < ∈] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. In this paper we consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment of FO 2 is defined by disallowing universal quantifiers and having at most m∈-∈1 nested negations. One can view as the formulas in FO 2 which have at most m blocks of quantifiers on every path of their parse tree, and the first block is existential. Thus, the m th level of the FO 2 -alternation hierarchy is the Boolean closure of. We give an effective characterization of, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give ω-terms U m and V m such that an FO 2 -definable language is in if and only if its ordered syntactic monoid satisfies the identity U m ∈V m. Among other techniques, the proof relies on an extension of block products to ordered monoids. © 2014 Springer International Publishing Switzerland