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The Impact of the Lambda Calculus in Logic and Computer Science
 Bulletin of Symbolic Logic
, 1997
"... One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the represent ..."
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One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. Acknowledgement. The following persons provided help in various ways. Erik Barendsen, Jon Barwise, Johan van Benthem, Andreas Blass, Olivier Danvy, Wil Dekkers, Marko van Eekelen, Sol Feferman, Andrzej Filinski, Twan Laan, Jan Kuper, Pierre Lescanne, Hans Mooij, Robert Maron, Rinus Plasmeijer, Randy Pollack, Kristoffer Rose, Richard Shore, Rick Statman and Simon Thompson. Partial support came from the European HCM project Typed lambda calculus (CHRXCT920046), the Esprit Working Group Types (21900) and the Dutch NWO project WINST (612316607). 1. Introduction This paper is written to honor Church's gr...
Predicate logic with sequence variables and sequence function symbols
 Proc. of the 3rd Int. Conference on Mathematical Knowledge Management. Vol. 3119 of LNCS
, 2004
"... Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded ..."
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Abstract. We describe an extension of firstorder logic with sequence variables and sequence functions. We define syntax, semantics and inference system for the extension so that Completeness, Compactness, LöwenheimSkolem, and Model Existence theorems remain valid. The obtained logic can be encoded as a special ordersorted firstorder theory. We also define an inductive theory with sequence variables and formulate induction rules. The calculus forms a basis for the topdown systematic theory exploration paradigm. 1
The GirardReynolds isomorphism
 Proc. of 4th Int. Symp. on Theoretical Aspects of Computer Science, TACS 2001
, 2001
"... Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented ..."
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Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented in F2. Reynolds additionally proved an abstraction theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. The Girard projection discards all firstorder quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embedding should also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions corresponding to inductive definitions of naturals, products, sums, and fixpoint types. 1
From Constructivism to Computer Science
, 1996
"... This paper is an expanded version of a lecture presented november 15, 1996, at the Technische Universitat Munchen, on the occassion of receiving the F.L. BauerPreis. I am indebted to K.R. Apt for detailed comments on an earlier draft of this paper. heated debate, but attracted few actual followers ..."
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This paper is an expanded version of a lecture presented november 15, 1996, at the Technische Universitat Munchen, on the occassion of receiving the F.L. BauerPreis. I am indebted to K.R. Apt for detailed comments on an earlier draft of this paper. heated debate, but attracted few actual followers. But even those who did not agree with Brouwer's ideas, such as D. Hilbert, were influenced by them and by the debate generated by Brouwer's views. Ideas developed in the context of intuitionism turned out to have a relevance which transcends the original setting. Brouwer's ideas on the foundations of mathematics were embedded in a highly personal and rather extreme version of idealistic philosophy, but it would carry us too far to go into this here. The first exposition of these ideas on the foundations of mathematics is found in his thesis from 1907 ("On the Foundations of Mathematics"). Briefly, Brouwer viewed mathematics as the activity of building constructions in the mind (of an ideal mathematician); mathematics is about such mental constructions, not about objects in some outside reality. For Brouwer, there is no platonistic universe of abstract ideas existing somewhere quite independently of human cognition. In his thesis, Brouwer had not yet realized the effect of his views on logic; but in a paper which appeared a year later, in 1908 ("On the unreliability of the logical principles") he did see the consequences. He demonstrated that from an intuitionistic point of view, we cannot assume that a mathematical statement is either true or false, independently of human knowledge; we can assert "A or not A" only in case we either have a proof of A or an argument showing that any attempt at constructing a proof of A must fail. As a result, if A represents an open mathemat...
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.
Mechanical Verification of Invariant Properties of DisCo Specifications
 in Information Engineering, and M.E. degrees in Information Sciences from Tohoku University, Sendai,Japa in
, 1997
"... This thesis describes how invariant properties of specifications written in the DisCo specification language can be mechanically verified. DisCo is an objectoriented specification language based on joint actions and stepwise refinement. The PVS theorem prover is used for the actual verification, an ..."
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This thesis describes how invariant properties of specifications written in the DisCo specification language can be mechanically verified. DisCo is an objectoriented specification language based on joint actions and stepwise refinement. The PVS theorem prover is used for the actual verification, and an existing compiler for the DisCo language has been modified to produce specifications in the PVS logic. The chain from a DisCo specification to an invariance proof has thus been completely mechanized. The contribution of the thesis consists of three parts: a mapping of the DisCo language to the PVS logic, a verification methodology suggesting how to use the mapping effectively, and automated support for invariance proofs. The DisCo language is mapped to the higher order logic of the PVS theorem prover. This is apparently the first mapping of an objectoriented specification language to the logic of a theorem prover. To facilitate the mapping, a subset of the Temporal Logic of Actions has...
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
The SAD System: Deductive Assistance in an Intelligent Linguistic Environment
"... Abstract — Formal methods are widely used in the computer science community. Formal verification and certification is an important component of any formal approach. Such a work can not be done by hand, hence the software that can do a part of it is rather required. The verification methods are often ..."
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Abstract — Formal methods are widely used in the computer science community. Formal verification and certification is an important component of any formal approach. Such a work can not be done by hand, hence the software that can do a part of it is rather required. The verification methods are often based on a deductive system and “verify” means “prove”. Corresponding software is called proof assistant. We describe in this paper the System for Automated Deduction (SAD): its architecture, input language, and reasoning facilities. We show how to use SAD as a proof assistant. We outline specific features of SAD — a handy input language, powerful reasoning strategy, opportunity to use various low level inference engines. Examples and results of some experiments are also given. I.