3,450 research outputs found

    On Second-Order Monadic Monoidal and Groupoidal Quantifiers

    Get PDF
    We study logics defined in terms of second-order monadic monoidal and groupoidal quantifiers. These are generalized quantifiers defined by monoid and groupoid word-problems, equivalently, by regular and context-free languages. We give a computational classification of the expressive power of these logics over strings with varying built-in predicates. In particular, we show that ATIME(n) can be logically characterized in terms of second-order monadic monoidal quantifiers

    Characterizing Quantifier Extensions of Dependence Logic

    Full text link
    We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.Comment: 9 page

    Logic Meets Algebra: the Case of Regular Languages

    Full text link
    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

    Turing jumps through provability

    Full text link
    Fixing some computably enumerable theory TT, the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each Σ1\Sigma_1 formula is equivalent to some formula of the form □Tφ\Box_T \varphi provided that TT is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for n>1n>1 we relate Σn\Sigma_{n} formulas to provability statements [n]TTrueφ[n]_T^{\sf True}\varphi which are a formalization of "provable in TT together with all true Σn+1\Sigma_{n+1} sentences". As a corollary we conclude that each [n]TTrue[n]_T^{\sf True} is Σn+1\Sigma_{n+1}-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates [n+1]T□[n+1]^\Box_T which look a lot like [n+1]TTrue[n+1]_T^{\sf True} except that where [n+1]TTrue[n+1]_T^{\sf True} calls upon the oracle of all true Σn+2\Sigma_{n+2} sentences, the [n+1]T□[n+1]^\Box_T recursively calls upon the oracle of all true sentences of the form ⟨n⟩T□ϕ\langle n \rangle_T^\Box\phi. As such we obtain a `syntax-light' characterization of Σn+1\Sigma_{n+1} definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates [n+1]T□[n+1]_T^\Box are well behaved in that together they provide a sound interpretation of the polymodal provability logic GLPω{\sf GLP}_\omega

    A Fragment of Dependence Logic Capturing Polynomial Time

    Get PDF
    In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D-Horn* and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore we study the question which of our results can ge generalized to the case of open formulae of D-Horn* and so-called downwards monotone polynomial time properties of teams
    • …
    corecore