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28
The Logic of Bunched Implications
 BULLETIN OF SYMBOLIC LOGIC
, 1999
"... We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication; it can be viewed as a merging of intuition ..."
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Cited by 190 (38 self)
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We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers # new and # new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predic...
Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
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Cited by 52 (17 self)
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this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is MartinLo# f 's proof of admissibility [ML68]. In MartinLo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, ext ..."
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Cited by 49 (27 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
A modal walk through space
 JOURNAL OF APPLIED NONCLASSICAL LOGICS
, 2002
"... We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and ..."
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Cited by 31 (5 self)
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We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.
From truth to computability I
 Theoretical Computer Science
"... The recently initiated approach called computability logic is a formal theory of interactive computation. It understands computational problems as games played by a machine against the environment, and uses logical formalism to describe valid principles of computability, with formulas representing c ..."
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Cited by 14 (14 self)
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The recently initiated approach called computability logic is a formal theory of interactive computation. It understands computational problems as games played by a machine against the environment, and uses logical formalism to describe valid principles of computability, with formulas representing computational problems and logical operators standing for operations on computational problems. The concept of computability that lies under this approach is a nontrivial generalization of ChurchTuring computability from simple, twostep (question/answer, input/output) problems to problems of arbitrary degrees of interactivity. Restricting this concept to predicates, which are understood as computational problems of zero degree of interactivity, yields exactly classical truth. This makes computability logic a generalization and refinement of classical logic. The foundational paper “Introduction to computability logic ” [Annals of Pure and Applied Logic 123 (2003), pp. 199] was focused on semantics rather than syntax, and certain axiomatizability assertions in it were only stated as conjectures. The present contribution contains a verification of one of such conjectures: a soundness and completeness proof for the deductive system CL3 which axiomatizes the most basic firstorder fragment of computability logic called the finitedepth, elementarybase fragment.
On traced monoidal closed categories
, 2008
"... ... focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal coreflexive full subcategory of a tortile monoidal category. From this, we der ..."
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Cited by 12 (2 self)
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... focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal coreflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixedpoint computation. To make the paper more selfcontained, we also include various background results for traced monoidal categories.
Logical Predicates for Intuitionistic Linear Type Theories
 In Typed Lambda Calculi and Applications (TLCA'99), Lecture Notes in Computer Science 1581
, 1999
"... We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal co ..."
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Cited by 11 (4 self)
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We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal cocompletion. As applications, we obtain full completeness results of translations between linear type theories.
Linear logic as a framework for specifying sequent calculus
 Lecture Notes in Logic 17, Logic Colloquium’99
, 2004
"... Abstract. In recent years, intuitionistic logic and type systems have been used in numerous computational systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be used similarly to specify the more general setting of sequent cal ..."
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Cited by 11 (3 self)
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Abstract. In recent years, intuitionistic logic and type systems have been used in numerous computational systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be used similarly to specify the more general setting of sequent calculus proof systems. Linear logic’s meta theory can be used also to analyze properties of a specified objectlevel proof system. We shall present several example encodings of sequent calculus proof systems using the Forum presentation of linear logic. Since the objectlevel encodings result in logic programs (in the sense of Forum), various aspects of objectlevel proof systems can be automated. §1. Introduction. Logics and type systems have been exploited in recent years as frameworks for the specification of deduction in a number of logics. Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (nonpredicate) higherorder types is available. Identifying a framework that allows the specification of a wide range of logics has
Classical linear logic of implications
 In Proc. Computer Science Logic (CSL'02), Springer Lecture Notes in Comp. Sci. 2471
, 2002
"... Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technica ..."
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Cited by 10 (4 self)
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Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for categorytheoretic models given by ∗autonomous categories with linear exponential comonads. 1