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Deciding Provability of Linear Logic Formulas
 Advances in Linear Logic
, 1994
"... 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 ..."
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Cited by 21 (0 self)
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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 CCR9224858. 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
Interpreting Strands in Linear Logic
, 2000
"... The adoption of the DolevYao 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 assumptio ..."
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Cited by 21 (10 self)
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The adoption of the DolevYao 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 DolevYao 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 firstorder 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 DolevYao 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 socalled "DolevYao" 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.
Foundations of Proof Search Strategies Design in Linear Logic
 In Symposium on Logical Foundations of Computer Science
, 1994
"... 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 topdown, bottomup and mixed proof search procedures with a systematic formalization of s ..."
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Cited by 20 (11 self)
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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 topdown, bottomup 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.
Linear Logic and Computation: A Survey
 Proof and Computation, Proceedings Marktoberdorf Summer School
, 1993
"... . 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 ..."
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. 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...
Linearizing Intuitionistic Implication
 In Proc. 6th Annual IEEE Symposium on Logic in Computer Science
, 1993
"... An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of t ..."
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An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspacehardness of imall. It exploits several prooftheoretic 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 Gentzenstyle 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...
Decidability of Linear Affine Logic
 10th Annual IEEE Symposium on Logic in Computer Science
, 1995
"... 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 sp ..."
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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".
First Order Linear Logic without Modalities Is NEXPTIMEHard
 Theoretical Computer Science
, 1994
"... The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace ..."
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Cited by 15 (11 self)
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The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace hardness of quantified boolean formula validity, utilizing some of the surprisingly powerful and expressive machinery of linear logic. 1 Introduction Linear logic, introduced by Girard, is a resourcesensitive refinement of classical logic [10, 29]. Linear logic gains its expressive power by restricting the "structural" proof rules of contraction (copying) and weakening (erasing). The contraction rule makes it possible to reuse any stated assumption as often as desired. The weakening rule makes it possible to use dummy assumptions, i.e., it allows a deduction to be carried out without using all of the hypotheses. Because contraction and weakening together make it possible to use an assu...
ILC: A Foundation for Automated Reasoning About Pointer Programs
, 2005
"... This paper shows how to use Girard’s intuitionistic linear logic extended with arithmetic or other constraints to reason about pointer programs. More specifically, first, the paper defines the proof theory for ILC (Intuitionistic Linear logic with Constraints) and shows it is consistent via a proof ..."
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This paper shows how to use Girard’s intuitionistic linear logic extended with arithmetic or other constraints to reason about pointer programs. More specifically, first, the paper defines the proof theory for ILC (Intuitionistic Linear logic with Constraints) and shows it is consistent via a proof of cut elimination. Second, inspired by prior work of O’Hearn, Reynolds and Yang, the paper explains how to interpret linear logical formulas as descriptions of a program store. Third, we define a simple imperative programming language with mutable references and arrays and give verification condition generation rules that produce assertions in ILC. Finally, we identify a fragment of ILC, ILC − , that is both decidable and closed under generation of verification conditions. In other words, if loop invariants are specified in ILC − , then the resulting verification conditions are also in ILC −. Since verification condition generation is syntaxdirected, we obtain a decidable procedure for checking properties of pointer programs.
Constructive Logics. Part II: Linear Logic and Proof Nets
 Proceedings of the International Joint Conference and Symposium on Logic Programming
, 1997
"... . 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 ..."
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Cited by 14 (0 self)
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. 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 DanosRegnier criterion for proof nets. This proof is based on a purely graphtheoretic 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 NOOO1488K0593. Contents 1 Core Linear Logic and Propositional Linear Logic 3 2 Representing I...