The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from proof-theoretic or categorical concerns and, on the other, from a possible-worlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's proof-theoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.

Abstract In this paper, we propose a method for automated web service composition by applying Linear Logic (LL) theorem proving. We distinguish value-added web services and core service by assuming that the core service is already selected by the user, but its functionality does not completely match the user’s requirement. Our method enables automation for combining the core service together with a set of value-added services to solve the problem. The method uses web service languages for external presentation of atomic web services (WSDL) or composite web services (BPEL4WS), while the services are internally presented by extralogical axioms and proofs in LL. In this paper, we are focused on the internal presentation and proof. LL, as the internal representation language, enables us to define some issues required by web service composition formally, such as qualitative and quantitative constraints plus subsumption reasoning on concepts. In addition, LL guarantees the correctness and completeness of service composition process.

The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough, for example, to form the logical basis for “pointer logic ” and “separation logic” semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI’s tableaux to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, , the challenge consists in dealing with BI’s Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete. Such a semantics, based on partiality, is closely related to the semantics of BI’s (intuitionistic) pointer and separation logics. Returning to the tableaux calculus, we propose a new version with liberalized rules for which the countermodels are closely related to the topological Kripke semantics of BI. As consequences of the relationships between semantics of BI and resource tableaux, we prove two strong new results for propositional BI: its decidability and the finite model property with respect to topological semantics.

this paper, we will give a short survey of results on decision problems and the finite model property of substructural logics. The paper is far from a complete list of these results, since a lot of results have been obtained already in some restricted classes of substructural logics, like relevant logics, and therefore it is impossible to cover all of them. ( As for surveys of decision problems and the finite model property of relevant logics, see e.g. [1, 2, 7]. Also, see [16] for a survey of decision problems of logics related to linear logic. ) Our aim of the present paper is to try to compare results from different classes of substructural logics with each other and discuss them as a whole, in order to get a perspective of them.

Abstract. Given two proofs in a logical system with a confluent cutelimination procedure, the cut-elimination problem (CEP) is to decide whether these proofs reduce to the same normal form. This decision problem has been shown to be ptime-complete for Multiplicative Linear Logic (Mairson 2003). The latter result depends upon a restricted simulation of weakening and contraction for boolean values in MLL; in this paper, we analyze how and when this technique can be generalized to other MLL formulas, and then consider CEP for other subsystems of Linear Logic. We also show that while additives play the role of nondeterminism in cut-elimination, they are not needed to express deterministic ptime computation. As a consequence, affine features are irrelevant to expressing ptime computation. In particular, Multiplicative Light Linear Logic (MLLL) and Multiplicative Soft Linear Logic (MSLL) capture ptime even without additives nor unrestricted weakening. We establish hierarchical results on the cut-elimination problem for MLL(ptime-complete), MALL(coNP-complete), MSLL(EXPTIME-complete), and for MLLL (2EXPTIME-complete). 1

The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough to provide a "pointer logic" semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI's tableaux to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, ?, the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. As consequences, we have two strong new results for BI: the decidability of propositional BI and the finite model property with respect to Grothendieck topological semantics. In addition, we propose, by considering partially defined monoids, a new semantics which generalizes the semantics of BI's pointer logic and for which BI is complete Keywords: BI; resources; semantics; tableaux; decidability; finite model property.

In this paper we present a model of cooperative problem solving (CPS). Linear Logic (LL) is used for encoding agents' states, goals and capabilities. LL theorem proving is applied by each agent to determine whether the particular agent is capable of solving the problem alone. If no individual solution can be constructed, then the agent may start negotiation with other agents in order to nd a cooperative solution.

We propose a new formulation for full (weakening and constants included) multiplicative and exponential (MELL) proof nets, allowing a complete set of rewriting rules to parse them. The recognizing grammar de ned by such a rewriting system (con uent and strong normalizing on the new proof nets) gives a correctness criterion that we show equivalent to the Danos-Regnier one. 1

We show that linear logic can serve as an expressive framework in which to model a rich variety of combinatorial auction mechanisms. Due to its resource-sensitive nature, linear logic can easily represent bids in combinatorial auctions in which goods may be sold in multiple units, and we show how it naturally generalises several bidding languages familiar from the literature. Moreover, the winner determination problem, i.e., the problem of computing an allocation of goods to bidders producing a certain amount of revenue for the auctioneer, can be modelled as the problem of finding a proof for a particular linear logic sequent.

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for specifications based on first order linear logic, a logic that can be used to naturally model infinite state systems with internal structured data. The fragment we consider in this paper is based on the linear logic programming language called LO [4] enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of concurrent systems.