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Primitive Recursion for HigherOrder Abstract Syntax
 Theoretical Computer Science
, 1997
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Encoding Modal Logics in Logical Frameworks
 Studia Logica
, 1997
"... We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce severa ..."
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Cited by 15 (8 self)
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We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed calculus metalanguage of the Logical Frameworks. These formalizations yield readily proofeditors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and NaturalDeduction proof systems for Modal Logics, Logical Frameworks, Typed calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...
Primitive recursion for higher order abstract syntax with dependent types
 In International Workshop on Intuitionistic Modal Logics and Applications (IMLA
, 1999
"... Higherorder abstract syntax is a central representation technique in logical frameworks which maps variables of the object language into variables in the metalanguage. It leads to concise encodings, but is incompatible with functions dened by primitive recursion or proofs by induction. In this pap ..."
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Cited by 7 (0 self)
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Higherorder abstract syntax is a central representation technique in logical frameworks which maps variables of the object language into variables in the metalanguage. It leads to concise encodings, but is incompatible with functions dened by primitive recursion or proofs by induction. In this paper we propose an extension of the simplytyped lambdacalculus with iteration and case constructs which preserves the adequacy of higherorder abstract syntax encodings. The wellknown paradoxes are avoided through the use of a modal operator which obeys the laws of S4. In the resulting calculus many functions over higherorder representations can be expressed elegantly. Our central technical result, namely that our calculus is conservative over the simplytyped lambdacalculus, is proved by a rather complex argument using logical relations. We view our system as an important rst step towards allowing the methodology of LF to be employed eectively in systems based on induction principles such as ALF, Coq, or Nuprl, leading to a synthesis of currently incompatible paradigms.
Modal Typing for Specifying Runtime Code Generation
, 2001
"... Syntax Initially the translated code made use of the same abstract syntax modules in representing code as the did compiler/translator. However, eventually due to the complication of reasoning about embedding the abstract syntax within itself, and the desire to implement "runtime" optimiz ..."
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Cited by 2 (2 self)
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Syntax Initially the translated code made use of the same abstract syntax modules in representing code as the did compiler/translator. However, eventually due to the complication of reasoning about embedding the abstract syntax within itself, and the desire to implement "runtime" optimizations without modifying the translator, we were led to develop a cleaner and more abstract method of manipulating code fragments at run time.
Lecture Notes on Categorical Judgments 15816: Modal Logic
, 2010
"... The main basic judgments we have considered so far are: A true A↑ ..."
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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 ..."
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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.
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
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.
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