6,745 research outputs found

    Two-variable first order logic with modular predicates over words

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    We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO 2 [<, MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO 2 [<, MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Fraïssé games) and combinatorics

    Adding modular predicates to first-order fragments

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    We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with modular predicates. Our approach aims toward the most generic statements that we could achieve, which successfully covers the quantifier alternation hierarchy of first order logic and some of its fragments. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order and the successor. For two-variable fragments we also treat the case of the signature containing only the order and modular predicates.Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragmentwhile 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 first order formulas as well as the algebraic framework of finite categories

    One Quantifier Alternation in First-Order Logic with Modular Predicates

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    Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison x<yx<y and predicates for ximodnx \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]

    Covering and separation for logical fragments with modular predicates

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    For every class C\mathscr{C} of word languages, one may associate a decision problem called C\mathscr{C}-separation. Given two regular languages, it asks whether there exists a third language in C\mathscr{C} containing the first language, while being disjoint from the second one. Usually, finding an algorithm deciding C\mathscr{C}-separation yields a deep insight on C\mathscr{C}. We consider classes defined by fragments of first-order logic. Given such a fragment, one may often build a larger class by adding more predicates to its signature. In the paper, we investigate the operation of enriching signatures with modular predicates. Our main theorem is a generic transfer result for this construction. Informally, we show that when a logical fragment is equipped with a signature containing the successor predicate, separation for the stronger logic enriched with modular predicates reduces to separation for the original logic. This result actually applies to a more general decision problem, called the covering problem

    Logic Meets Algebra: the Case of Regular Languages

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    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

    Monadic Second-Order Logic with Arbitrary Monadic Predicates

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    We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent characterizations. We consider the regularity question: given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates, and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer Science Journal version: ToCL'17, Transactions on Computational Logi

    The ciao modular, standalone compiler and its generic program processing library

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    Ciao Prolog incorporates a module system which allows sepárate compilation and sensible creation of standalone executables. We describe some of the main aspects of the Ciao modular compiler, ciaoc, which takes advantage of the characteristics of the Ciao Prolog module system to automatically perform sepárate and incremental compilation and efficiently build small, standalone executables with competitive run-time performance, ciaoc can also detect statically a larger number of programming errors. We also present a generic code processing library for handling modular programs, which provides an important part of the functionality of ciaoc. This library allows the development of program analysis and transformation tools in a way that is to some extent orthogonal to the details of module system design, and has been used in the implementation of ciaoc and other Ciao system tools. We also describe the different types of executables which can be generated by the Ciao compiler, which offer different tradeoffs between executable size, startup time, and portability, depending, among other factors, on the linking regime used (static, dynamic, lazy, etc.). Finally, we provide experimental data which illustrate these tradeoffs

    On the logical definability of certain graph and poset languages

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    We show that it is equivalent, for certain sets of finite graphs, to be definable in CMS (counting monadic second-order logic, a natural extension of monadic second-order logic), and to be recognizable in an algebraic framework induced by the notion of modular decomposition of a finite graph. More precisely, we consider the set F_F\_\infty of composition operations on graphs which occur in the modular decomposition of finite graphs. If FF is a subset of F_F\_{\infty}, we say that a graph is an \calF-graph if it can be decomposed using only operations in FF. A set of FF-graphs is recognizable if it is a union of classes in a finite-index equivalence relation which is preserved by the operations in FF. We show that if FF is finite and its elements enjoy only a limited amount of commutativity -- a property which we call weak rigidity, then recognizability is equivalent to CMS-definability. This requirement is weak enough to be satisfied whenever all FF-graphs are posets, that is, transitive dags. In particular, our result generalizes Kuske's recent result on series-parallel poset languages
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