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A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deductio ..."
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Cited by 194 (47 self)
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
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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Cited by 3 (0 self)
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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.
Under consideration for publication in Math. Struct. in Comp. Science A Judgmental Reconstruction of Modal Logic
, 2000
"... We reconsider the foundations of modal logic, following MartinLöf’s methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility which yields a simple and uniform system of natural deduction for intuitionistic modal logic which d ..."
Abstract
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We reconsider the foundations of modal logic, following MartinLöf’s methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility which yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi’s monadic metalanguage.
General Terms: Languages,Theory
"... 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 typed λcalculi and functionallanguages. We directly demonstrate the sense in which our λ→2 ecalculus captures staging, and also g ..."
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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 typed λcalculi and functionallanguages. We directly demonstrate the sense in which our λ→2 ecalculus captures staging, and also give a conservative embedding of Nielson & Nielson’s twolevel functional language in our functional language MiniML2, thus proving that bindingtime correctness is equivalent to modal correctness on this fragment. In addition, MiniML2 can also express immediate evaluation and sharing of code across multiple stages, thus supporting runtime code generation as well as partial evaluation.
F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—lambda calculus and
"... Abstract. 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 typed �calculi and functional languages. We directly demonstrate the sense in which our � e 3 �calculus captures stagin ..."
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Abstract. 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 typed �calculi and functional languages. We directly demonstrate the sense in which our � e 3 �calculus captures staging, and also give a conservative embedding of Nielson and Nielson’s twolevel functional language in our functional language MiniML � , thus proving that bindingtime correctness is equivalent to modal correctness on this fragment. In addition, MiniML � can also express immediate evaluation and sharing of code across multiple stages, thus supporting runtime code generation as well as partial evaluation.