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14
Semantic evaluation, intersection types and complexity of simply typed lambda calculus
 Schloss Dagstuhl  LeibnizZentrum fuer Informatik
, 2012
"... Consider the following problem: given a simply typed lambda term of Boolean type and of order r, does it normalize to “true”? A related problem is: given a term M of word type and of order r together with a finite automaton D, does D accept the word represented by the normal form of M? We prove that ..."
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Consider the following problem: given a simply typed lambda term of Boolean type and of order r, does it normalize to “true”? A related problem is: given a term M of word type and of order r together with a finite automaton D, does D accept the word represented by the normal form of M? We prove that these problems are nEXPTIME complete for r = 2n + 2, and nEXPSPACE complete for r = 2n + 3. While the hardness part is relatively easy, the membership part is not so obvious; in particular, simply applying β reduction does not work. Some preceding works employ semantic evaluation in the category of sets and functions, but it is not efficient enough for our purpose. We present an algorithm for the above type of problem that is a fine blend of β reduction, Krivine abstract machine and semantic evaluation in a category based on preorders and order ideals, also known as the Scott model of linear logic. The semantic evaluation can also be presented as intersection type checking.
A categorical semantics for polarized mall
 Ann. Pure Appl. Logic
"... In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of ..."
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In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of an ambient ∗autonomous category C (with products). Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and to present various completeness theorems. As concrete examples, we present (i) a hypercoherence model, using Ehrhard’s hereditary/antihereditary objects, (ii) a Chuspace model, (iii) a double gluing model over our categorical framework, and (iv) a model based on iterated double gluing over a ∗autonomous category. For the multiplicative fragment MLLP of MALLP, we present both weakly full (Läuchlistyle) as well as full completeness theorems, using a polarized version of functorial
Functional interpretations of intuitionistic linear logic
 Computer Science Logic
, 2009
"... Abstract. We present three functional interpretations of intuitionistic linear logic and show how these correspond to wellknown functional interpretations of intuitionistic logic via embeddings of ILω into ILLω. The main difference from previous work of the second author is that in intuitionistic l ..."
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Abstract. We present three functional interpretations of intuitionistic linear logic and show how these correspond to wellknown functional interpretations of intuitionistic logic via embeddings of ILω into ILLω. The main difference from previous work of the second author is that in intuitionistic linear logic the interpretations of!A are simpler (at the cost of an asymmetric interpretation of pure ILLω) and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. 1
An Indexed System for Multiplicative Additive Polarized Linear Logic
 Proc. of 17th Annual Conference on Computer Science Logic (CSL'08), Lecture
"... Abstract. We present an indexed logical system MALLP(I) for Laurent’s multiplicative additive polarized linear logic (MALLP) [14]. The system is a polarized variant of BucciarelliEhrhard’s indexed system for multiplicative additive linear logic [4]. Our system is derived from a webbased instance ..."
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Abstract. We present an indexed logical system MALLP(I) for Laurent’s multiplicative additive polarized linear logic (MALLP) [14]. The system is a polarized variant of BucciarelliEhrhard’s indexed system for multiplicative additive linear logic [4]. Our system is derived from a webbased instance of HamanoScott’s denotational semantics [12] for MALLP. The instance is given by an adjoint pair of right and left multipointed relations. In the polarized indexed system, subsets of indexes for I work as syntactical counterparts of families of points in webs. The rules of MALLP(I) describe (in a prooftheoretical manner) the denotational construction of the corresponding rules of MALLP. We show that MALLP(I) faithfully describes a denotational model of MALLP by establishing a correspondence between the provability of indexed formulas and relations that can be extended to (nonindexed) proofdenotations. 1
Linear logical reasoning on programming
 In: Acta Electrotechnica et Informatica
"... In our paper we follow the development of our approach of regarding programming as logical reasoning in intuitionistic linear logic. We present basic notions of linear logic and its deduction system and we define categorical semantics of linear logic as a symmetric monoidal closed category. Then we ..."
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In our paper we follow the development of our approach of regarding programming as logical reasoning in intuitionistic linear logic. We present basic notions of linear logic and its deduction system and we define categorical semantics of linear logic as a symmetric monoidal closed category. Then we construct linear type theory over linear Church’s types involving linear calculus with equational axioms. We conclude with the interpretation of the linear type theory in symmetric monoidal closed category. Defined entities included in our whole linear logical system give us a possible mean for deduction and reduction of problem solving in the framework of mathematics and computer science.
On Geometry of Interaction for Polarized Linear Logic
, 2014
"... We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multi ..."
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We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multipoints are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as playing a fundamental role in allowing us to construct concrete polarized GoI models. Our approach to polarized GoI involves two independent studies, based on different categorical perspectives of GoI. (i) Inspired by the work of Abramsky, Haghverdi, and Scott, a polarized GoI situation is dened which considers multipoints added to a traced monoidal category with an appropriate re
exive object U. Categorical versions of Girard's Execution formula (taking into account the multipoints) are dened, as well as the GoI
Similarity as Nearness: Information Quanta, Approximation Spaces and Nearness Structures
"... Abstract. The present paper investigates approximation spaces in the context of topological structures which axiomatise the notion of nearness. Starting with the framework of information quanta which distinguishes two levels of information structures, namely property systems (the first level) and in ..."
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Abstract. The present paper investigates approximation spaces in the context of topological structures which axiomatise the notion of nearness. Starting with the framework of information quanta which distinguishes two levels of information structures, namely property systems (the first level) and information quantum relational systems (the second level), we shall introduce the notion of Pawlak’s property system. These systems correspond bijectively to finite approximation spaces, i.e. their respective information quantum relational systems. Then we characerise Pawlak’s property systems in terms of symmetric topological spaces. In the second part of the paper, these systems are defined by means of topo
1 Physics, Topology, Logic and Computation:
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning
Categorical Semantics of Linear Logic For All
"... This paper is a survey of results on categorical modeling of linear ..."