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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 194 (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...
On Bunched Predicate Logic
 Proceedings of the IEEE Symposium on Logic in Computer Science
, 1999
"... We present the logic of bunched implications, 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, and may be viewe ..."
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Cited by 29 (17 self)
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We present the logic of bunched implications, 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, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predicate version of BI includes, in addition to usual additive quantifiers, multiplicative (or intensional) quantifiers 8new and 9new , which arise from observing restrictions on structural rules on the level of terms as well as propositions. Moreover, these restrictions naturally allow the distinction between additive predication and multiplicative predication for each propositional connective. We provide a natural deduction system, a sequent calculus, a Kripke semantics and a BHK semantics for BI. We mention computational interpretations, based on locality and sharing, at both the propositiona...
Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pair ..."
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Cited by 21 (0 self)
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Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
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.
Categorical Glueing and Logical Predicates for Models of Linear Logic
, 1999
"... We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality ..."
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Cited by 5 (3 self)
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We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) autonomous categories (models of Multiplicative Linear Logic); the glueing construction for autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan. Each of the glueing techniques can be used for creating interesting models of linear logic. In particular, we use them, together with the free symmetric monoidal cocompletion, for deriving Kripkelike parameterized logical predicates (logical relations) for the fragments of linear logic. As an application, we show full completeness results for translations between linear type theories. Contents 1 Introduction 3 2 Preliminaries 4 2.1 Symmetric Monoidal Structures . . . . . . . ....
Categorical Models Of FirstOrder Classical Proofs
, 2006
"... This thesis introduces the notion of a classical doctrine: a semantics for proofs in firstorder classical logic derived from the classical categories of Fuhrmann and Pym, using Lawvere's notion of hyperdoctrine. We introduce a hierarchy of classes of model, increasing in the strength of cutreductio ..."
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Cited by 2 (0 self)
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This thesis introduces the notion of a classical doctrine: a semantics for proofs in firstorder classical logic derived from the classical categories of Fuhrmann and Pym, using Lawvere's notion of hyperdoctrine. We introduce a hierarchy of classes of model, increasing in the strength of cutreduction theory they model; the weakest captures cut reduction, and the strongest gives De Morgan duality between quantifiers as an isomorphism. Whereas classical categories admit the elimination of logical cuts as equalities, (and cuts against structural rules as inequalities), classical doctrines admit certain logical cuts as inequalities only. This is a result of the additive character of the quantifier introduction rules, as is illustrated by a concrete model based on families of sets and relations, using an abstract Geometry of Interaction construction. We establish that each class of models is sound and complete with respect to the relevant cutreduction theory on proof nets based on those of Robinson for propositional classical logic. We show also that classical categories and classical doctrines are not only a class of models for the sequent calculus, but also for deep inference calculi due to Brunnler for classical logic. Of particular interest are the local systems for classical logic, which we show are modelled by categorical models with an additional axiom forcing monoidality of certain functors; these categorical models correspond to multiplicative presentations of the sequent calculus with additional additive features. Acknowledgements There are many without whom this work would languish unfinished. I must begin by thanking David Pym for introducing to me the core ideas in this work, and for the guidance he has provided. In matters both sacred and profane, both integra...
Monoid Based Semantics for Linear Formulas
, 1997
"... Each Girard quantale (i.e. commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partia ..."
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Cited by 1 (0 self)
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Each Girard quantale (i.e. commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality. Linear logic was introduced by Girard in [2]. Originally this was conceived as a way of organizing the properties of certain categories of domains, but it was soon recognized that it has a wider signicance. Quantales have been around for much longer. Although their prehistory goes back to the 1930s, they were rst studied systematically by Mulvey in the paper with a silly title [4] and later papers. Initially quantales were conceived as a way of producing a topologicallike gadget in a situation were there doesn't seem to be enough points around to carry a classical topological space. But again it was soon recognized that quantales have a wider signicance. The two notions were brought together by Yetter [9] who showed that commutative quantales provide a standard semantics for linear formulas. (With hindsight we can see that there is some use of quantales in Girard's original paper, but not expressed in these terms.) Essentially, quantales are to linear logic what frames (complete Heyting algebras) are to intuitionistic logic and complete Boolean algebras are to classical logic. In the intuitionistic and classical cases these algebras can be specialized to the point based algebras of topologies and power sets, respectively. In fact, thes...