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A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatib ..."
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Cited by 77 (3 self)
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We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and firstclass continuations. Prooftheoretically the dual ¯ v constructs of naming and ¯abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...
Natural Deduction and Coherence for Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categorie ..."
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Cited by 73 (26 self)
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This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the twotensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansionreduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending simi...
A semantic view of classical proofs  typetheoretic, categorical, and denotational characterizations (Extended Abstract)
 IN PROCEEDINGS OF LICS '96
, 1996
"... Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, class ..."
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Cited by 30 (2 self)
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Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, classical logic is often held to be nonconstructive, and so, is said to admit no proof semantics. To draw an analogy in the proofsas programs paradigm, it is as if we understand well the theory of manipulation between equivalent specifications (which we do), but have comparatively little foundational insight of the process of transforming one program to another that implements the same specification. This extended abstract outlines a semantic theory of classical proofs based on a variant of Parigot's λµcalculus [24], but presented here as a type theory. After reviewing the conceptual problems in the area and the potential benefits of such a theory, we sketch the key steps of our approach in ...
The Logic of Correct Description
 Advances in Intensional Logic
, 1995
"... Austin's theory of truth is formulated in terms of a relation of correct description holding between a sentence and a situation. A recursive denition of correct description is provided for rstorder languages containing terms denoting situations and a predicate denoting correctdescription. We exam ..."
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Cited by 23 (1 self)
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Austin's theory of truth is formulated in terms of a relation of correct description holding between a sentence and a situation. A recursive denition of correct description is provided for rstorder languages containing terms denoting situations and a predicate denoting correctdescription. We examine a very strong logic of situations, by restricting our attention to situated consequence between descriptions of omniscient situations, arguing that weaker logics may be obtained using standard methods. Rules of natural deduction for the logic are introduced by way of examples of natural reasoning using spatial indexicals. Finally, a Gentzenstyle sequent calculus is oered. Keywords: Austin, truth, correct description, situation, indexical, natural deduction, sequent calculus. 1 Truth and Correct Description A basic tenet of situation semantics ([BP83, BE87, Bar89]), deriving from Austin's theory of truth ([Aus50]), is that every statement is about a situation. To make a statement by...
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Modified Realizability Toposes and Strong Normalization Proofs (Extended Abstract)
 Typed Lambda Calculi and Applications, LNCS 664
, 1993
"... ) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent appl ..."
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Cited by 14 (1 self)
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) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (cpca). Remarkably, an arbitrary rightabsorptive cpca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizability, as opposed to the standard Kleenestyle realizability. Starting from an arbitrary rightabsorptive cpca U , the tripostotopos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm (U ) of nonstandard sets equipped with an equali...
A New Method for Bounding the Complexity of Modal Logics
, 1997
"... . We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility r ..."
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Cited by 12 (2 self)
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. We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with bounded space requirements. As examples we give O(n log n) space procedures for the modal logics K and T. 1 Introduction We present a new prooftheoretic approach to bounding the complexity of the decision problem for propositional modal logics. We formalize logics in a uniform way as cutfree labelled sequent systems and then restrict the structural rules for particular systems. This, combined with an analysis of the accessibility relation of the corresponding Kripke structures, yields decision procedures with space requirements that are easily bounded. As examples we give O(n log n) space decision procedures f...
! and ?  Storage as tensorial strength
, 1996
"... this paper were produced with the help of the diagram macros of F. Borceux. Blute, Cockett, & Seely 2 ..."
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Cited by 9 (5 self)
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this paper were produced with the help of the diagram macros of F. Borceux. Blute, Cockett, & Seely 2
Structuring Metatheory on Inductive Definitions
, 2000
"... We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theorystructuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and s ..."
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Cited by 7 (0 self)
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We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theorystructuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and show how a statement about a theory can explicitly formalize the closure conditions extensions should satisfy for it to remain true. We show how metatheories based on inductive denitions allow theories and general metatheorems to be organized this way, and report on a case study using the theory FS0 . 1 Introduction Hierarchical theory structuring plays an important role in the application of theorem provers to nontrivial problems, and many systems provide support for it. For example HOL [6], Isabelle [13] and their predecessor LCF [7] support simple theory hierarchies. In these systems a theory is a specication of a language, using types and typed constants, and a collection of rules...
ProofDirected Decompilation of LowLevel Code
 Programming Languages and Systems. Proceedings, LNCS 2028
, 2001
"... We present a proof theoretical method for decompiling lowlevel code to the typed lambda calculus. We first define a proof system for a lowlevel code language based on the idea of CurryHoward isomorphism. This allows us to regard an executable code as a proof in intuitionistic propositional logi ..."
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Cited by 6 (3 self)
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We present a proof theoretical method for decompiling lowlevel code to the typed lambda calculus. We first define a proof system for a lowlevel code language based on the idea of CurryHoward isomorphism. This allows us to regard an executable code as a proof in intuitionistic propositional logic. As being a proof of intuitionistic logic, it can be translated to an equivalent proof of natural deduction proof system. This property yields an algorithm to translate a given code into a lambda term. Moreover, the resulting lambda term is not a trivial encoding of a sequence of primitive instructions, but reflects the behavior of the given program. This process therefore serves as proofdirected decompilation of a lowlevel code language to a highlevel language. We carry out this development for a subset of Java Virtual Machine instructions including most of its features such as jumps, object creation and method invocation. The proofdirected decompilation algorithm has been implemented, which demonstrates the feasibility of our approach.