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Subtyping with Power Types
 of Lecture Notes in Computer Science
, 2000
"... This paper introduces a typed #calculus called # Power , a predicative reformulation of part of Cardelli's power type system. Power types integrate subtyping into the typing judgement, allowing bounded abstraction and bounded quantification over both types and terms. This gives a powerful a ..."
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This paper introduces a typed #calculus called # Power , a predicative reformulation of part of Cardelli's power type system. Power types integrate subtyping into the typing judgement, allowing bounded abstraction and bounded quantification over both types and terms. This gives a powerful and concise system of dependent types, but leads to di#culty in the metatheory and semantics which has impeded the application of power types so far. Basic properties of # Power are proved here, and it is given a model definition using a form of applicative structures. A particular novelty is the auxiliary system for rough typing, which assigns simple types to terms in # Power . These "rough" types are used to prove strong normalization of the calculus and to structure models, allowing a novel form of containment semantics without a universal domain.
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...
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
A Reflective Framework for Formal Interoperability
, 1998
"... In practice we find ourselves in constant need of moving back and forth between different formalizations capturing different aspects of a system. For example, in a large software system we typically have very different requirements, such as functional correctness, performance, realtime behavior, co ..."
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In practice we find ourselves in constant need of moving back and forth between different formalizations capturing different aspects of a system. For example, in a large software system we typically have very different requirements, such as functional correctness, performance, realtime behavior, concurrency, security, and fault tolerance, which correspond to different views of the system and that are typically expressed in different formal systems. Often these requirements affect each other, but it can be extremely difficult to reason about their mutual interaction, and no tools exist to support such reasoning. This situation is very unsatisfactory, and presents one of the biggest obstacles to the use of formal methods in software engineering because, given the complexity of large software systems, it is a fact of life that no single perspective, no single formalization or level of abstraction suffices to represent a system and reason about its behavior. We need (meta)formal methods and tools to achieve Formal Interoperability, that is, the capacity to move in a mathematically rigorous way across the different formalizations of a system, and to use in a rigorously integrated way the different tools supporting these formalizations. We will develop new, formal interoperability methodologies and generic metatools that are expected to achieve dramatic advances in software technology and formal methods:
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...
A Theory of Inductive Definitions With αequivalence: Semantics, Implementation, Programming Language.
"... This document was compiled from L ATEX source on 10 August 2000. Copies will be printed, bound, and submitted for the title of PhD in Mathematics from Cambridge University, England. Other copies will be passed to those interested. Those interested are invited to write to me at Trinity College, Camb ..."
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This document was compiled from L ATEX source on 10 August 2000. Copies will be printed, bound, and submitted for the title of PhD in Mathematics from Cambridge University, England. Other copies will be passed to those interested. Those interested are invited to write to me at Trinity College, Cambridge, or email m.j.gabbay@dpmms.cam.ac.uk. I remind the reader that my examiners may well suggest corrections to this document so it need not necessarily be the final version of my thesis. If the reader is wondering, DPMMS stands for the “Department of Pure Maths