In this paper, we investigate automated proof construction in classical linear logic (CLL) by giving logical foundations for the design of proof search strategies. We propose common theoretical foundations for top-down, bottom-up and mixed proof search procedures with a systematic formalization of strategies construction using the notions of immediate or chaining composition or decomposition, deduced from permutability properties and inference movements in a proof. Thus, we have logical bases for the design of proof strategies in CLL fragments and then we can propose sketches for their design.

Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and the multiplicatives\Omega and . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR-9224858. lincoln@csl.sri.com http://www.csl.sri.com/lincoln/lincoln.html Patrick Lincoln For the most part we will consider fragments of linear logic built up using these connectives in any combination. For example, full linear logic formulas may employ any connective, while multiplic

The adoption of the Dolev-Yao model, an abstraction of security protocols that supports symbolic reasoning, is responsible for many successes in protocol analysis. In particular, it has enabled using logic effectively to reason about protocols. One recent framework for expressing the basic assumptions of the Dolev-Yao model is given by strand spaces, certain directed graphs whose structure reflects causal interactions among protocol participants. We represent strand constructions as relatively simple formulas in first-order linear logic, a refinement of traditional logic known for an intrinsic and natural accounting of process states, events, and resources. The proposed encoding is shown to be sound and complete. Interestingly, this encoding differs from the multiset rewriting definition of the Dolev-Yao model, which is also based on linear logic. This raises the possibility that the multiset rewriting framework may differ from strand spaces in some subtle way, although the two settings are known to agree on the basic secrecy property. 1 Introduction In recent years, a variety of methods have been developed for analyzing and reasoning about protocols based on cryptographic primitives. Although there are many differences among these proposals, most current formal approaches use the so-called "Dolev-Yao" model of adversary capabilities, which appears to be drawn from positions taken in [34] and from a simplified model presented in [11]. In this idealized setting, a protocol adversary is allowed to nondeterministically choose among possible actions. Messages are composed of indivisible abstract values, not sequences of bits, and encryption is modeled in an idealized way. The adversary may only send messages comprised of data it "knows" as the result of overhearing past transmissions.

An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cut-free proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspace-hardness of imall. It exploits several proof-theoretic properties of intuitionistic implication that analyze the use of resources in iil proofs. Linear logic is a refinement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard's words [12], "linear logic is a logic behind logic." A convenient way to present linear logic is by modifying the traditional Gentzen-style sequent calculus axiomatization of classical logic (see, e.g., [15, 22]). The modification may be briefly described in three steps. The first step is to remove two structural rules, contraction and weakening, which manipulate the use of hypotheses and conclusi...

. The purpose of this paper is to give an exposition of material dealing with constructive logics, typed -calculi, and linear logic. The first part of this paper gives an exposition of background material (with a few exceptions). This second part is devoted to linear logic and proof nets. Particular attention is given to the algebraic semantics (in Girard's terminology, phase semantics) of linear logic. We show how phase spaces arise as an instance of a Galois connection. We also give a direct proof of the correctness of the Danos-Regnier criterion for proof nets. This proof is based on a purely graph-theoretic decomposition lemma. As a corollary, we give an O(n 2 )-time algorithm for testing whether a proof net is correct. Although the existence of such an algorithm has been announced by Girard, our algorithm appears to be original. This research was partially supported by ONR Grant NOOO14-88-K-0593. Contents 1 Core Linear Logic and Propositional Linear Logic 3 2 Representing I...

Linear Logic is gaining momentum in computer science because it offers a unified framework and a common vocabulary for studying and analyzing different aspects of programming and computation. We focus here on models where computation is identified with proof search in the sequent system of Linear Logic. A proof normalization procedure, called "focusing", has been proposed to make the problem of proof search tractable. Correspondingly, there is a normalization procedure mapping formulae of Linear Logic into a syntactic fragment of that logic, called LinLog, and in which the focusing normalization for proofs can be most conveniently expressed. In this paper, we propose to push this compilation/normalization process further, by applying abstract interpretation and partial evaluation techniques to (focused) proofs in LinLog. These techniques provide information concerning the evolution of the computational resources (formulae) during the execution (proof construction). The practical outcome that we expect from this theoretical effort is the definition of a general tool for statically analyzing and reasoning about the runtime behavior of programs in frameworks where computations can be accounted for in terms of proof search in Linear Logic.

The propositional Linear Logic is known to be undecidable. In the current paper we prove that the full propositional Linear Affine Logic containing all the multiplicatives, additives, exponentials, and constants is decidable. The proof is based on a reduction of Linear Affine Logic to sequents of specific "normal forms", and on a generalization of Kanovich computational interpretation of Linear Logic adapted to these "normal forms".

. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...

This paper attempts to evaluate the use of a theorem prover in the multiplicative fragment of linear logic which has been shown to simulate conjunctive Strips-like planning [9]. A proof search procedure is presented that is correct, complete and only generates linear proofs (i.e. not trees). Plans that can be extracted from proofs are either totally or partially ordered. The procedure is tested against Strips-like planners and results are given. However, since linear logic is a resource-sensitive logic viewing formulas as data types, partial description of the final situation are impossible in linear logic; and shared postconditions are impossible in the fragment presented here. It is then argued that these restrictions eventually makes the presented fragment of linear logic, despite its formal framework, somewhat useless for practical planning purposes. 1 Introduction Framework The linear logic framework is that of [9] and table 1 gives its related sequent calculus. The classical St...

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