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Metatheoretical Results for a Modal λCalculus
, 2000
"... This paper presents the proofs of the strong normalization, subject reduction, and ChurchRosser theorems for a presentation of the intuitionistic modal λcalculus S4. It is adapted from Healfdene Goguen's thesis, where these properties are shown for the simply typed λcalculus and for Luo&a ..."
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This paper presents the proofs of the strong normalization, subject reduction, and ChurchRosser theorems for a presentation of the intuitionistic modal λcalculus S4. It is adapted from Healfdene Goguen's thesis, where these properties are shown for the simply typed λcalculus and for Luo's type theory UTT. Following this method, we introduce the notion of typed operational semantics for our system. We define a notion of typed substitution for our system, which has context stacks instead of the usual contexts. This latter peculiarity leads to the main diculties and consequently to the main original features in our proofs. The techniques elaborated in this work have already been found useful in recent works [DL98, DL99] and should be further exploited to prove the properties of other systems based on modality.
Reactive Constraint Programming
"... The tcc model is a concurrent constraint programming formalism for timed, reactive and deterministic computation. A remarkable feature of the tcc model is that programs and specifications are given in the same language. In this report we develop an extension of tcc that allows the specification o ..."
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The tcc model is a concurrent constraint programming formalism for timed, reactive and deterministic computation. A remarkable feature of the tcc model is that programs and specifications are given in the same language. In this report we develop an extension of tcc that allows the specification of nondeterministic computation. We call this extension the ntcc model. The ntcc model is based upon ideas from both concurrent constraint programming and CCSlike models. The expressiveness of ntcc is illustrated by showing derived constructs such as parameterless recursion, cells and valuebroadcasting processes, and by specifying temporal requirements such as eventuality, timebounded response and timebounded invariance. We claim the applicability of ntcc by modeling reactive system examples of RCX controllers. We present ongoing work on a denotational semantics and a proof system for the model. In the spirit of Process Algebra, we also define a behavioral equivalence for the ntcc model based upon the notion of (weak) bisimulation.
A Modal Lambda Calculus with Iteration and Case Constructs
, 1998
"... An extension of the simplytyped calculus, allowing iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax, has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the h ..."
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An extension of the simplytyped calculus, allowing iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax, has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic S4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal calculus that is better from the user's point of view because it requires fewer annotations in terms. Moreover we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the ChurchRosser and strong normalization properties and it is a conservative extension of the simplytyped calculus.
Abstract On a Modal λCalculus for S4 ⋆
"... We present λ →2, a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prov ..."
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We present λ →2, a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for welltyped terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and bindingtime analysis for functional languages [6]. 1
Abstract A Modal Analysis of Staged Computation
"... We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of functional languages. Our main technical result is a conservative embedding of Nielson & Nielson’s twolevel functional languag ..."
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We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of functional languages. Our main technical result is a conservative embedding of Nielson & Nielson’s twolevel functional language in our language MiniML 2, thus proving that bindingtime correctness is equivalent to modal correctness on this fragment. In addition MiniML 2 can also express immediate evaluation and sharing of code across multiple stages, thus supporting runtime code generation as well as partial evaluation. 1
Topological Duality for Intuitionistic Modal Algebras
, 1999
"... The paper goes on to consider those modal frames which are freely generated from modal distributive lattices, and in particular, those arising from the Lindenbaum algebras of intuitionistic propositional modal languages. These spectral modal frames are constructed by defining a modal structure on th ..."
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The paper goes on to consider those modal frames which are freely generated from modal distributive lattices, and in particular, those arising from the Lindenbaum algebras of intuitionistic propositional modal languages. These spectral modal frames are constructed by defining a modal structure on the frame of ideals, and characterised (up to isomorphism) by exactness and compactness conditions of the modal connectives. Finally, they are shown to be equivalent to the frames of open sets of relational spaces; from this a completeness theorem for intuitionistic modal logic follows. 1 Introduction The contravariant adjunction and duality between topological spaces and frames (or locales) underlies important results in lattice theory, representation theory, topos theory and logic; for a full exposition, see [5]. In particular, the completeness theorem for intuitionistic propositional logic follows immediately from the fact that the spectral (or coherent) frames &quot;have enough 1
A Model for Knowledge Representation in Distributed Systems
, 2002
"... this article, we present a simplified formalism of distributed systems in order to show some concepts which we think are important in the study of the flow of information between different parts or agents of a system. Then, we use those 1 basic concepts and generalize them to define an algebraic fr ..."
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this article, we present a simplified formalism of distributed systems in order to show some concepts which we think are important in the study of the flow of information between different parts or agents of a system. Then, we use those 1 basic concepts and generalize them to define an algebraic framework for formalizing distributed systems. Finally, we study the logical structure of this framework and show that in this formalism, the distributed systems form a model for the intuitionistic modal logic IS4+KV
A Polynomial Translation of Propositional S4 into Propositional Intuitionistic Logic
"... . We present a polynomial translation of the propositional fragment of the modal logic S4 into the propositional fragment of intuitionistic logic. The translation is performed in three main steps. Properties of intermediate translations are established by purely prooftheoretical means, i.e., by pro ..."
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. We present a polynomial translation of the propositional fragment of the modal logic S4 into the propositional fragment of intuitionistic logic. The translation is performed in three main steps. Properties of intermediate translations are established by purely prooftheoretical means, i.e., by proof transformations between dierent cutfree sequent calculi. Consequently, this approach yields eective translation procedures. 1 Introduction Embeddings of logics into other logics have a long tradition in the area of mathematical logic (see, e.g., [7, 11, 13, 10] for some classical references). Some embeddings are practically motivated, some others are of theoretical interest. The embedding of the modal logic S4 into the modal logic T (see, e.g., [2, 6]) is an example for the former, because, due to the transitivity of the S4accessibility relation, usual cutfree Gentzen systems for propositional S4 require a loopcheck for termination, whereas similar systems for T do not. In the cont...