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29
Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
Functional Programming in Sublinear Space
"... Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot comp ..."
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Abstract. We consider the problem of functional programming with data in external memory, in particular as it appears in sublinear space computation. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead compute and recompute small parts of the intermediate result on demand. In this paper, we study how the implementation of such techniques can be supported by functional programming languages. Our approach is based on modelling computation by interaction using the Int construction of Joyal, Street & Verity. We derive functional programming constructs from the structure obtained by applying the Int construction to a term model of a given functional language. The thus derived functional language is formulated by means of a type system inspired Baillot & Terui’s Dual Light Affine Logic. We assess its expressiveness by showing that it captures LOGSPACE. 1
Reduction in a linear lambdacalculus with applications to operational semantics
 In RTA
, 2005
"... Abstract. We study betareduction in a linear lambdacalculus derived from Abramsky’s linear combinatory algebras. Reductions are classified depending on whether the redex is in the computationally active part of a term (“surface” reductions) or whether it is suspended within the body of a thunk (“i ..."
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Abstract. We study betareduction in a linear lambdacalculus derived from Abramsky’s linear combinatory algebras. Reductions are classified depending on whether the redex is in the computationally active part of a term (“surface” reductions) or whether it is suspended within the body of a thunk (“internal” reductions). If surface reduction is considered on its own then any normalizing term is strongly normalizing. More generally, if a term can be reduced to surface normal form by a combined sequence of surface and internal reductions then every combined reduction sequence from the term contains only finitely many surface reductions. We apply these results to the operational semantics of Lily, a secondorder linear lambdacalculus with recursion, introduced by Bierman, Pitts and Russo, for which we give simple proofs that callbyvalue, callbyname and callbyneed contextual equivalences coincide. 1
Proofs as Polynomials
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
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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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...
Int Construction and Semibiproducts
, 2009
"... We study a relationship between the Int construction of Joyal et al. and a weakening of biproducts called semibiproducts. We then provide an application of geometry of interaction interpretation for the multiplicative additive linear logic (MALL for short) of Girard. We consider not biproducts but s ..."
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We study a relationship between the Int construction of Joyal et al. and a weakening of biproducts called semibiproducts. We then provide an application of geometry of interaction interpretation for the multiplicative additive linear logic (MALL for short) of Girard. We consider not biproducts but semibiproducts because in general the Int construction does not preserve biproducts. We show that Int construction is left biadjoint to the forgetful functor from the 2category of compact closed categories with semibiproducts to the 2category of traced symmetric monoidal categories with semibiproducts. We then illustrate a traced distributive symmetric monoidal category with biproducts B(Pfn) and relate the interpretation of MALL in Int(B(Pfn)) to token machines defined over weighted MALL proofs.
Linear realizability
, 2007
"... Abstract. We define a notion of relational linear combinatory algebra (rLCA) which is a generalization of a linear combinatory algebra defined by Abramsky, Haghverdi and Scott. We also define a category of assemblies as well as a category of modest sets which are realized by rLCA. This is a linear s ..."
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Abstract. We define a notion of relational linear combinatory algebra (rLCA) which is a generalization of a linear combinatory algebra defined by Abramsky, Haghverdi and Scott. We also define a category of assemblies as well as a category of modest sets which are realized by rLCA. This is a linear style of realizability in a way that duplicating and discarding of realizers is allowed in a controlled way. Both categories form linearnonlinear models and their coKleisli categories have a natural number object. We construct some examples of rLCA’s which have some relations to well known PCA’s. 1
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.