15,070 research outputs found

    Bipolar Proof Nets for MALL

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    In this work we present a computation paradigm based on a concurrent and incremental construction of proof nets (de-sequentialized or graphical proofs) of the pure multiplicative and additive fragment of Linear Logic, a resources conscious refinement of Classical Logic. Moreover, we set a correspon- dence between this paradigm and those more pragmatic ones inspired to transactional or distributed systems. In particular we show that the construction of additive proof nets can be interpreted as a model for super-ACID (or co-operative) transactions over distributed transactional systems (typi- cally, multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International Workshop, 17-18 August 2012, University of Copenhagen, Denmar

    Modules and Logic Programming

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    We study conditions for a concurrent construction of proof-nets in the framework developed by Andreoli in recent papers. We define specific correctness criteria for that purpose. We first study closed modules (i.e. validity of the execution of a logic program), then extend the criterion to open modules (i.e. validity during the execution) distinguishing criteria for acyclicity and connectability in order to allow incremental verification

    Rewritings for Polarized Multiplicative and Exponential Proof Structures

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    13 pagesInternational audienceWe study conditions for a concurrent construction of proof-nets in the framework of linear logic following Andreoli's works. We define specific correctness criteria for that purpose. We first study the multiplicative case and show how the correctness criterion given by Danos and decidable in linear time, may be extended to closed modules (i.e. validity of polarized proof structures). We then study the exponential case. This has natural applications in (concurrent) logic programming as validity of partial proof structures may be interpreted in terms of validity of a concurrent execution of clauses in an environment

    Rewritings in Polarized (Partial) Proof Structures

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    This paper is a first step towards a study for a concurrent construction of proof-nets in the framework of linear logic after Andreoli's works, by taking care of the properties of the structures. We limit here to multiplicative linear logic. We first give a criterion for closed modules (i.e. validity of polarized proof structures), then extend it to open modules (i.e. validity of partial proof structures) distinguishing criteria for acyclicity and connectability. The keypoint is an extensive use of the fundamental structural properties of the logics. We consider proof structures as built from n-ary bipolar objects and we show that strongly confluent (local) reductions on such objects are an elegant answer to the correctness problem. This has natural applications in (concurrent) logic programming

    Event structures for Petri nets with persistence

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    Event structures are a well-accepted model of concurrency. In a seminal paper by Nielsen, Plotkin and Winskel, they are used to establish a bridge between the theory of domains and the approach to concurrency proposed by Petri. A basic role is played by an unfolding construction that maps (safe) Petri nets into a subclass of event structures, called prime event structures, where each event has a uniquely determined set of causes. Prime event structures, in turn, can be identified with their domain of configurations. At a categorical level, this is nicely formalised by Winskel as a chain of coreflections. Contrary to prime event structures, general event structures allow for the presence of disjunctive causes, i.e., events can be enabled by distinct minimal sets of events. In this paper, we extend the connection between Petri nets and event structures in order to include disjunctive causes. In particular, we show that, at the level of nets, disjunctive causes are well accounted for by persistent places. These are places where tokens, once generated, can be used several times without being consumed and where multiple tokens are interpreted collectively, i.e., their histories are inessential. Generalising the work on ordinary nets, Petri nets with persistence are related to a new subclass of general event structures, called locally connected, by means of a chain of coreflections relying on an unfolding construction

    Two Algebraic Process Semantics for Contextual Nets

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    We show that the so-called 'Petri nets are monoids' approach initiated by Meseguer and Montanari can be extended from ordinary place/transition Petri nets to contextual nets by considering suitable non-free monoids of places. The algebraic characterizations of net concurrent computations we provide cover both the collective and the individual token philosophy, uniformly along the two interpretations, and coincide with the classical proposals for place/transition Petri nets in the absence of read-arcs

    A Comparison of Petri Net Semantics under the Collective Token Philosophy

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    In recent years, several semantics for place/transition Petri nets have been proposed that adopt the collective token philosophy. We investigate distinctions and similarities between three such models, namely configuration structures, concurrent transition systems, and (strictly) symmetric (strict) monoidal categories. We use the notion of adjunction to express each connection. We also present a purely logical description of the collective token interpretation of net behaviours in terms of theories and theory morphisms in partial membership equational logic

    Algebraic Models for Contextual Nets

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    We extend the algebraic approach of Meseguer and Montanari from ordinary place/transition Petri nets to contextual nets, covering both the collective and the individual token philosophy uniformly along the two interpretations of net behaviors

    Process versus Unfolding Semantics for Place/Transition Petri Nets

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    In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game," one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc, a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification

    Between quantum logic and concurrency

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    We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842
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