Algebraic recognizability of regular tree languages

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem \`{a} la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization

Characterization of preclones by matrix collections

Preclones are described as the closed classes of the Galois connection induced by a preservation relation between operations and matrix collections. The Galois closed classes of matrix collections are also described by explicit closure conditions.Comment: 11 page

Automaták, fák és logika = Automata, trees and logic

Elemi idejű exponenciális algoritmus adtunk meg reguláris szavak ekvivalenciájának eldönthetőségére. Általánosítottuk Kleene tételét végtelen szavakat is felismerő súlyozott automatákra. Kifejlesztettünk egy algebrai módszert, amellyel a CTL logika számos szegmense estén eldönthető, hogy egy reguláris fanyelv definiálható-e a szegmensben. Vizsgáltuk a faautomaták algebrai tulajdonságait, megadtuk a felismerhetőség egy algebrai jellemzését. Definiáltunk a multi-leszálló fatranszformátort és megmutattuk, hogy ekvivalens a determinisztikus reguláris szűkítésű felszálló fatranszformátorral. Meghatároztuk a lineáris multi-leszálló osztály számítási erejét. Megmutattuk, hogy az alakmegőrző leszálló fatranszformátorok ekvivalensek az átcímkézőkkel és bebizonyítottuk, hogy az alakmegőrző tulajdonság eldönthető. Megadtuk a kavics makró fatranszformációk egy felbontását és megmutattuk, hogy a különböző cirkularitási tulajdonságok eldönthetők. Ugyancsak megadtuk a felbontást erős kavics kezelés estén is. Általánosítottuk J. Engelfriet hiararchia tételét súlyozott fatranszformátorokra. Súlyozott faautomatákra definiáltuk a termátíró szemantikát és megmutattuk, hogy ekvivalens az algebari szenmatikával. Algoritmust adtunk annak eldöntésére, hogy egy polinomiálisan súlyozott faautomata véges költségű-e. Vizsgáltuk a súlyozott faautomata különböző változatait: fuzzy faautomata, multioperátor monoid feletti faautomata, Ez utóbbi esetre általánosítottuk a Kleene tételt. | We gave an elementary algorithm for deciding the equivalence of regular words. We generalized Kleene's theorem to weighted automata processing infinite words. We developed an algebraic method that, for several segments of the CTL logic, can be applied to decide if a regular tree language can be defined in that segment. We examined algebraic properties of tree automata, and gave an algebraic characterization of recognizability. We defined multi bottom-up tree transducers and showed that they are equivalent to top-down tree transducers with regular look-ahead. We determined the computation power of the linear subclass. We showed that shape preserving bottom-up tree transducers are equivalent to relabelings. We proved that the shape preserving property is decidable. We gave a decomposition for pebble macro tree transducers and showed that certain circularity properties are decidable. We also gave a decomposition for the strong pebble handling. We have generalized the hierarchy theorem of J. Engelfriet to weighted tree transducers. We defined the term rewrite semantics of weighted tree transducers and showed that it is equivalent to the algebraic semantics. We gave a decision algorithm for the finite cost property of a polynomially weighted tree automata. We defined different versions of weighted tree automata: fuzzy tree automata, weighted tree automata over a multioperator monoid. For the latter we generalized Kleene's theorem

Regular tree languages and quasi orders

Regular languages were characterized as sets closed with respect to monotone well-quasi orders. A similar result is proved here for tree languages. Moreover, families of quasi orders that correspond to positive varieties of tree languages and varieties of finite ordered algebras are characterized

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

Wreath Products of Forest Algebras, with Applications to Tree Logics

We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics

Profinite trees, through monads and the lambda-calculus

In its simplest form, the theory of regular languages is the study of sets of finite words recognized by finite monoids. The finiteness condition on monoids gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. Yet, some aspects of the theory of regular languages are not particular to monoids and can be described in a general setting. On the one hand, Boja\'{n}czyk has shown how to use monads to generalize the theory of regular languages and has given an abstract definition of the free profinite structure, defined by codensity, given a fixed monad and a notion of finite structure. On the other hand, Salvati has introduced the notion of language of $\lambda$-terms, using denotational semantics, which generalizes the case of words and trees through the Church encoding. In recent work, the author and collaborators defined the notion of profinite $\lambda$-term using semantics in finite sets and functions, which extend the Church encoding to profinite words. In this article, we prove that these two generalizations, based on monads and denotational semantics, coincide in the case of trees. To do so, we consider the monad of abstract clones which, when applied to a ranked alphabet, gives the associated clone of ranked trees. This induces a notion of free profinite clone, and hence of profinite trees. The main contribution is a categorical proof that the free profinite clone on a ranked alphabet is isomorphic, as a Stone-enriched clone, to the clone of profinite $\lambda$-terms of Church type. Moreover, we also prove a parametricity theorem on families of semantic elements which provides another equivalent formulation of profinite trees in terms of Reynolds parametricity