1,496 research outputs found

    Reasoning about embedded dependencies using inclusion dependencies

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    The implication problem for the class of embedded dependencies is undecidable. However, this does not imply lackness of a proof procedure as exemplified by the chase algorithm. In this paper we present a complete axiomatization of embedded dependencies that is based on the chase and uses inclusion dependencies and implicit existential quantification in the intermediate steps of deductions

    Non-Deterministic Kleene Coalgebras

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    In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines

    A simple sequent calculus for nominal logic

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    Nominal logic is a variant of first-order logic that provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the new-quantifier, but we argue that none of these attempts is completely satisfactory. In this article we develop a new sequent calculus for nominal logic in which the rules for the new- quantifier are much simpler than in previous attempts. We also prove several structural and metatheoretic properties, including cut-elimination, consistency, and equivalence to Pitts' axiomatization of nominal logic

    CCS Dynamic Bisimulation is Progressing

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    Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g.\ α.τ.β.nil\alpha.\tau.\beta.nil and α.β.nil\alpha.\beta.nil are woc but τ.β.nil\tau.\beta.nil and β.nil\beta.nil are not. This fact prevents us from characterizing CCS semantics (when τ\tau is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e.\ run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulation, which forces processes to simulate each other performing explicit steps. We provide an algebraic characterization of it in terms of finality, two characterizations via modal logic in the style of HML, and a complete axiomatization for finite agents. Finally, we prove that Dynamic Congruence and Progressing Bisimulation coincide for CCS agents. Thus the title of the paper

    Realms: A Structure for Consolidating Knowledge about Mathematical Theories

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    Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm as a structure for consolidating knowledge about a mathematical theory. A realm contains several axiomatizations of a theory that are separately developed. Views interconnect these developments and establish that the axiomatizations are equivalent in the sense of being mutually interpretable. A realm also contains an external interface that is convenient for users of the library who want to apply the concepts and facts of the theory without delving into the details of how the concepts and facts were developed. We illustrate the utility of realms through a series of examples. We also give an outline of the mechanisms that are needed to create and maintain realms.Comment: As accepted for CICM 201

    Axioms and Decidability for Type Isomorphism in the Presence of Sums

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    We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about polynomials with exponentiation that have not been used in our context, we derive: that the usual finite axiomatization known as High-School Identities (HSI) is complete for a significant subclass of types; that it is decidable for that subclass when two types are isomorphic; that, for the whole of the set of types, a recursive extension of the axioms of HSI exists that is complete; and that, for the whole of the set of types, the question as to whether two types are isomorphic is decidable when base types are to be interpreted as finite sets. We also point out certain related open problems

    Using the coral System to Discover Attacks on Security Protocols

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    The exp-log normal form of types

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    Lambda calculi with algebraic data types lie at the core of functional programming languages and proof assistants, but conceal at least two fundamental theoretical problems already in the presence of the simplest non-trivial data type, the sum type. First, we do not know of an explicit and implemented algorithm for deciding the beta-eta-equality of terms---and this in spite of the first decidability results proven two decades ago. Second, it is not clear how to decide when two types are essentially the same, i.e. isomorphic, in spite of the meta-theoretic results on decidability of the isomorphism. In this paper, we present the exp-log normal form of types---derived from the representation of exponential polynomials via the unary exponential and logarithmic functions---that any type built from arrows, products, and sums, can be isomorphically mapped to. The type normal form can be used as a simple heuristic for deciding type isomorphism, thanks to the fact that it is a systematic application of the high-school identities. We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specialized version of itself, while preserving the completeness of equality on terms. We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coq-implemented converter into/from our new term calculus. The difference with the only other previously implemented heuristic for deciding interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we exploit the type information of terms substantially and this often allows us to obtain a canonical representation of terms without performing sophisticated term analyses
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