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Decision Problems for Propositional Linear Logic
, 1990
"... Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, ..."
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Cited by 90 (17 self)
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Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspacecomplete. We also establish membership in np for the multiplicative fragment, npcompleteness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzenstyle sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...
The Method of Hypersequents in the Proof Theory of Propositional NonClassical Logics
 IN LOGIC: FROM FOUNDATIONS TO APPLICATIONS, EUROPEAN LOGIC COLLOQUIUM
, 1994
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The Semantics and Proof Theory of Linear Logic
 THEORETICAL COMPUTER SCIENCE
, 1988
"... Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in [Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and sem ..."
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Cited by 31 (7 self)
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Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in [Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding strong completeness theorems. Finally, we shall investigate the relation between Linear Logic and previously known systems, especially Relevance logics.
Linear Logic
, 1992
"... this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may ..."
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Cited by 24 (1 self)
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this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics
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
A cutfree proof theory for Boolean BI (via display logic). Unpublished note; available from http://www.doc.ic.ac. uk/~jbrother
, 2009
"... We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then show how to constrain proof search in the system (wit ..."
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Cited by 4 (1 self)
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We give a display calculus proof system for Boolean BI (BBI) based on Belnap’s general display logic. We show that cutelimination holds in our system and that it is sound and complete with respect to the usual notion of validity for BBI. We then show how to constrain proof search in the system (without loss of generality) by means of a series of proof transformations. By doing so, we gain some insight into the problem of decidability for BBI. 1
CLASSICAL BI: ITS SEMANTICS AND PROOF THEORY
"... Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including ..."
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Cited by 2 (2 self)
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Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBIformulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the prooftheoretic level, a very natural formalism for CBI is provided by a display calculus à la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics. 1.
An O(n log n)Space Decision Procedure for the Relevance Logic B+
, 2000
"... In previous work we gave a new prooftheoretical method for establishing upperbounds on the space complexity of the provability problem of modal and other propositional nonclassical logics. Here we extend and rene these results to give an O(n log n)space decision procedure for the basic posit ..."
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Cited by 1 (0 self)
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In previous work we gave a new prooftheoretical method for establishing upperbounds on the space complexity of the provability problem of modal and other propositional nonclassical logics. Here we extend and rene these results to give an O(n log n)space decision procedure for the basic positive relevance logic B + . We compute this upperbound by rst giving a sound and complete, cutfree, labelled sequent system for B + , and then establishing bounds on the application of the rules of this system. Keywords: Relevance Logics, Computational Complexity, Labelled Deduction Systems, Sequent Systems. 1
RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS
"... Abstract. Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete ..."
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Abstract. Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations. 1.