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Equivalences between Logics and their Representing Type Theories
, 1992
"... We propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are abl ..."
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We propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction firstorder logic can be wellrepresented in LF + , whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between typetheoretic and categorical approaches to frameworks. 1 Introduction Much effort has been devoted to building systems for supporting the construction of f...
Pure type systems in rewriting logic: Specifying typed higherorder languages in a firstorder logical framework
 In Essays in Memory of OleJohan Dahl, volume 2635 of LNCS
, 2004
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ThirdOrder Matching in the Polymorphic Lambda Calculus
, 1995
"... We show that it is decidable whether a thirdorder matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1 Intro ..."
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We show that it is decidable whether a thirdorder matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1 Introduction This paper is a contribution to the theory of (pattern) matching in higher order type theory. The starting point is the fact that thirdorder matching is decidable in the simply typed lambda calculus with constant types (see [5]). The question we would like to answer is: what happens if we extend this calculus with the type features that are characteristic for the Calculus of Constructions [2]: dependent types, type constructors and polymorphism. In [3], Dowek showed that in lambda calculi with dependent types thirdorder matching is undecidable. In contrast, we showed in [15] that the presence of type constructors is not sufficient to make thirdorder matching undecidable. In this ...
Categorical Properties of Logical Frameworks
, 1993
"... In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a ..."
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In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a logical framework is a typed calculus which formalizes the relationship between internal typed languages and frame languages. In the second half of the paper, we demonstrate the advantage of our logical framework by showing some categorical properties of it and of encodings in it. By doing so we hope to indicate a sensible model theory of encodings. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Sc...
Proceedings Of The 1992 Workshop On Types For Proofs And Programs
, 1992
"... The aim of this note is first to set up some general theory for discussing different aspects of the notion of a logic and then to draw attention to the schematic aspects of logic and suggest a way of capturing this aspect without making any commitment to the kind of syntax a logic should have. Intro ..."
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The aim of this note is first to set up some general theory for discussing different aspects of the notion of a logic and then to draw attention to the schematic aspects of logic and suggest a way of capturing this aspect without making any commitment to the kind of syntax a logic should have. Introduction Nowadays we are well aware that there are many different logics. There are computer systems which are meant to be used to implement many logics. But there is no generally accepted account of what a logic is. Perhaps this is as it should be. We need imprecision in our vocabulary to mirror the flexible imprecision of our thinking. There are a number of related phrases that seem to have a similar imprecision; e.g. formal system, language, axiom system, theory, deductive system, logical system, etc... These are sometimes given technical meanings, often without adequate consideration of the informal notions. When a logic has been implemented in a computer system the logic has been repres...
ThirdOrder Matching in the Polymorphic Lambda Calculus
"... We show that it is decidable whether a thirdorder matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1 ..."
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We show that it is decidable whether a thirdorder matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1