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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
Linear Logic by Levels and Bounded Time Complexity
, 2009
"... This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity b ..."
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This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity bound on the cutelimination procedure. Here, we propose a more flexible control principle, that of indexing, which allows us to extend Girard’s systems while keeping the same complexity properties. A consequence of the higher flexibility of indexing with respect to stratification is the absence of boxes for handling the § modality. We finally propose a variant of our polytime system in which the § modality is only allowed on atoms, and which may thus serve as a basis for developing λcalculus type assignment systems with more efficient typing algorithms than existing ones.
Bounded Linear Logic, Revisited
, 904
"... We present QBAL, an extension of Girard, Scedrov and Scott’s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for poly ..."
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We present QBAL, an extension of Girard, Scedrov and Scott’s bounded linear logic. The main novelty of the system is the possibility of quantifying over resource variables. This generalization makes bounded linear logic considerably more flexible, while preserving soundness and completeness for polynomial time. In particular, we provide compositional embeddings of Leivant’s RRW and Hofmann’s LFPL into QBAL. 1
ML with PTIME complexity guarantees ∗
"... Implicit Computational Complexity is a line of research where the possibility to inference a valid property for a program implies that the program runs in particular complexity class. Soft type systems are one of the research threads within the field. We present here a soft type system with MLlike ..."
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Implicit Computational Complexity is a line of research where the possibility to inference a valid property for a program implies that the program runs in particular complexity class. Soft type systems are one of the research threads within the field. We present here a soft type system with MLlike polymorphism that enjoys decidable typechecking, type inference and typability problems and gives polynomial time computational guarantees for the running time of typed programs.
1.1. Linear Logic and Implicit Computational Complexity
"... In a recent work, Girard [1] proposed a new and innovative approach to computational complexity based on the proofsasprograms correspondence. In a previous paper [2], the authors showed how Girard proposal succeeds in obtaining a new characterization of coNL languages as a set of operators acting ..."
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In a recent work, Girard [1] proposed a new and innovative approach to computational complexity based on the proofsasprograms correspondence. In a previous paper [2], the authors showed how Girard proposal succeeds in obtaining a new characterization of coNL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages.
Under consideration for publication in Math. Struct. in Comp. Science Characterizing coNL by a Group Action
, 2013
"... In a recent paper (Girard 2012), Girard proposes to use his recent construction of a geometry of interaction in the hyperfinite factor (Girard 2011) in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard’s ..."
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In a recent paper (Girard 2012), Girard proposes to use his recent construction of a geometry of interaction in the hyperfinite factor (Girard 2011) in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard’s definitions. We then provide a complete proof that the complexity class coNL can be characterized using this new approach. We introduce as a technical tool the nondeterministic pointer machine, a concrete model to computes algorithms. 1.