130 research outputs found

    On factorisation forests

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    The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page

    Regular Cost Functions, Part I: Logic and Algebra over Words

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    The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page

    Automata Minimization: a Functorial Approach

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    In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: A) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. B) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. C) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization, minimization and syntactic algebras. We provide explanations of Choffrut's minimization algorithm for subsequential transducers and of Brzozowski's minimization algorithm in this setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306

    Boundedness in languages of infinite words

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    We define a new class of languages of ω\omega-words, strictly extending ω\omega-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of ω\omega-regular expressions where two new variants of the Kleene star L∗L^* are added: LBL^B and LSL^S. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression (aBb)ω(a^Bb)^\omega represents the language of infinite words over the letters a,ba,b where there is a common bound on the number of consecutive letters aa. The expression (aSb)ω(a^Sb)^\omega represents a similar language, but this time the distance between consecutive bb's is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending B\"uchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either LBL^B or LSL^S---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets

    Unambiguous Separators for Tropical Tree Automata

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    In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g. This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different

    Logics with rigidly guarded data tests

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    The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi's approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata

    The Bridge Between Regular Cost Functions and Omega-Regular Languages

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    In this paper, we exhibit a one-to-one correspondence between omega-regular languages and a subclass of regular cost functions over finite words, called omega-regular like cost functions. This bridge between the two models allows one to readily import classical results such as the last appearance record or the McNaughton-Safra constructions to the realm of regular cost functions. In combination with game theoretic techniques, this also yields a simple description of an optimal procedure of history-determinisation for cost automata, a central result in the theory of regular cost functions

    Rewriting in the partial algebra of typed terms modulo AC

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    AbstractWe study the partial algebra of typed terms with an associative commutative and idempotent operator (typed AC-terms). The originality lies in the representation of the typing policy by a graph which has a decidable monadic theory.In this paper we show on two examples that some results on AC-terms can be raised to the level of typed AC-terms. The examples are the results on rational languages (in particular their closure by complement) and the property reachability problem for ground rewrite systems (equivalently process rewrite systems)

    Transforming structures by set interpretations

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    We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.Comment: 36 page
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