Algebraic Characterization of Forest Logics

In this paper we define future-time branching temporal logics evaluated over forests, that is, ordered tuples of ordered, but unranked, finite trees. We associate a rich class FL[$\mathcal{L}$] of temporal logics to each set L of (regular) modalities. Then, we define an algebraic product operation which we call the Moore product, which operates on forest automata, algebraic devices recognizing forest languages. We show a lattice isomorphism between the pseudovarieties of finite forest automata, closed under the Moore product, and the classes of languages of the form FL[$\mathcal{L}$]. We demonstrate the usefulness of the algebraic approach by showing the decidability of the membership problem of a specific pseudovariety of finite forest automata, implying the decidability of the definability problem of the FL[EF] fragment of the logic CTL. Then, using the same approach, we also formulate a conjecture regarding a decidable characterization of the FL[AF] fragment which has currently an unknown decidability status (also in the setting of ranked trees)

Some connections between universal algebra and logics for trees

One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this problem and similar ones using the language of universal algebra, highlighting potential connections to the structural theory of finite algebras, including Tame Congruence Theory

Algebraic characterization of logically defined tree languages

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindstr\"om quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.Comment: 46 pages. Version 3: various local improvements (more typos corrected, and "intuitive" explanations added