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14
Program extraction from classical proofs
 Annals of Pure and Applied Logic
, 1994
"... 1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its v ..."
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Cited by 54 (9 self)
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1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its very construction is correct as well. This at least in principle opens a way to produce correct software, e.g. for safetycritical applications. Moreover, programs obtained from proofs are "commented " in a rather extreme sense. Therefore it is easy to maintain them, and also to adapt them to particular situations. We will concentrate on the question of classical versus constructive proofs. It is known that any classical proof of a specification of the form 8x9yB with B quantifierfree can be transformed into a constructive proof of the same formula. However, when it comes to extraction of a program from a proof obtained in this way, one easily ends up with a mess. Therefore, some refinements of the standard transformation are necessary.
An intuitionistic proof of Kruskal's Theorem
 Archive for Mathematical Logic
, 2000
"... this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by NashWilliams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relation ..."
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Cited by 8 (2 self)
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this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by NashWilliams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The Finite Sequence Theorem 6. Vazsonyi's Conjecture for binary trees 7. Higman's Theorem 8. Vazsonyi's Conjecture and the Tree Theorem 9. MinimalBadSequence Arguments 10. The Principle of Open Induction 11. Concluding Remarks Except for Section 9, we will argue intuitionistically. 1 1 Dickson's Lemma
Higman's Lemma in Type Theory
 PROCEEDINGS OF THE 1996 WORKSHOP ON TYPES FOR PROOFS AND PROGRAMS
, 1997
"... This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type t ..."
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Cited by 5 (0 self)
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This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the first paper, A Lambda Calculus Model of MartinLöf's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of MartinLof's logical framework with explicit substitution extended with some inductively defined sets, also given in complete detail. These inductively defined sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossib...
Ramsey's Theorem in Type Theory
, 1993
"... We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs ..."
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Cited by 4 (1 self)
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We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almostfullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almostfullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almostfullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...
The NetherlandsDempster.. Belief functions and inner measures
, 1992
"... In this note we study the relation between belief functions of DempsterShafer theory and inner measures induced by probability functions. In, [3,4] Joe Halpern and Ron Fagin claim that these classes of functions are essentially the same, or, more precisely, that.theyare exactly the same in case th ..."
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Cited by 1 (0 self)
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In this note we study the relation between belief functions of DempsterShafer theory and inner measures induced by probability functions. In, [3,4] Joe Halpern and Ron Fagin claim that these classes of functions are essentially the same, or, more precisely, that.theyare exactly the same in case the functions are defined on formulas rather than sets. We show that=,when the functions are defined on sets only a proper subclass off the belief functions over aframe S corrsponds to the class of inner measures induced by a probability measurer on.somealgebra, onS. However, belief functions, over S do correspond to inner measures induced by probability measures 'defined on algebras on refinements of S. The fact that in general refinemeats of S are needed to obtain all belief functions over S is shown to be obscured;by the particular way formulas are. assigned probabilities 'or, weights in. [3]. r4 1.
The NetherlandsNetherlands, A Modal Perspective on the Computational Complexity ofAttribute Value Grammar
, 1992
"... I Many of the formalisms; used in Attribute Value grammar are notational variants of languages of propositional modal logic,. and testing whether two Attribute Value descriptions unify amounts to testing for modal satisfiablity. In this paper we put this. observation to work. We study the complexity ..."
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I Many of the formalisms; used in Attribute Value grammar are notational variants of languages of propositional modal logic,. and testing whether two Attribute Value descriptions unify amounts to testing for modal satisfiablity. In this paper we put this. observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express reeintrancy, the ability to express generalisations, and the ability to express recursive constraints. Two mail techniques axe used: either Kripke models with desirable properties are " constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic:: Further possibilities for the application of modal logic in computational linguistics are"noted Attribute Value Structures (AVSs) are probably the most widely used means of repre_ senting linguistic structure in current computational linguistics, and the process of unifying descriptions of AVSs lies at the heart of many parsers. As a number of people have recently
3584 CS Utrecht
, 1992
"... By constructing a counter, model we show that a,.appealing certain equation E has no solution in Girard's [1972] second order lambdacalculus (the socalled polymorphic, lambda " calculus). The equation E ', = E(4i) ' (with 45 a type 3 variable) is a simple functional equation in thelkiguage of Gode ..."
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By constructing a counter, model we show that a,.appealing certain equation E has no solution in Girard's [1972] second order lambdacalculus (the socalled polymorphic, lambda " calculus). The equation E ', = E(4i) ' (with 45 a type 3 variable) is a simple functional equation in thelkiguage of Godel's [1958] system of higher order primitive recursive fanctiotals and` has an easy solution in Spector"s°[1962] system of bar recursive functionals This shows that the class of bar recursive functionals differs from the class _ of functionals definable ' in the polymorphic lambda calculus."The fact that the two calculi have different classes of definable functionals (at least of type 3), contrasts the metamathematical results from 'Spector [1962] and Girard [1972] which state that the twocalculi have the same class of definable functions, 'namely, the 'provably.total recursive:.functions. of `..analysis
The NetherlandsA tractable algorithm for the wellfounded model
, 1992
"... A tractable algorithm ..."