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Confluence of Curried TermRewriting Systems
 Journal of Symbolic Computation
, 1995
"... Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we ..."
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Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we write A j= P Q iff A j= P and A j= Q. An ARS A = (A; !) has the diamond property , A j= \Sigma, iff /;! ` !;/. It has the ChurchRosser property (is confluent), A j= CR, iff (A; !!) j= \Sigma. Given an ARS A = (A; !), we write CR(t) as shorthand for (fu j t !! ug; !) j= CR. 4 Stefan Kahrs Under most circumstances, confluence is a useful property of ARSs, mainly because: if (A; !) j= CR, and if two elements x; y 2 A are equivalent w.r.t. the smallest equivalence containing !, then there is a z 2 A such that x!! z //y. Roughly: the ARS decides the equivalence. An ARS A = (A; ! a ; ! b ) commutes directly , A j= CD, iff / a ; ! b ` ! b ; / a . To prove confluence of an ARS, it is sometimes...
Typing untyped λterms, or Reducibility strikes again!
, 1995
"... It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes ..."
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It was observed by Curry that when (untyped) λterms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having headnormal forms, can be characterized in some systems D and D. The proofs use variants of the method of reducibility. In this paper, we presenta uniform approach for proving several metatheorems relating properties ofterms and their typability in the systems D and D. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We alsocharacterize the terms that have weak headnormal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.
Strong Normalization in a NonDeterministic Typed LambdaCalculus
, 1994
"... In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen's cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, i ..."
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In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen's cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of λLK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate prooftheoretic notion of proof reduction. This reduction relation satisøes several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argument à la Tait.
Typing Untyped LambdaTerms, or Reducibility Strikes Again!
, 1995
"... . It was observed by Curry that when (untyped) terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing) . Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classe ..."
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. It was observed by Curry that when (untyped) terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing) . Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having headnormal forms, can be characterized in some systems D and D \Omega\Gamma The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several metatheorems relating properties of terms and their typability in the systems D and D Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easi...
Interpreting ChurchStyle Typed λCalculus in CurryStyle Type Assignment
, 1997
"... It is well known that there are problems with the labelled syntax in Churchstyle type assignment to lambdaterms, the syntax in which the types of bound variables are indicated, as in λx : # . M , since if #reduction is added then the ChurchRosser Theorem fails in general (although it has been p ..."
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It is well known that there are problems with the labelled syntax in Churchstyle type assignment to lambdaterms, the syntax in which the types of bound variables are indicated, as in λx : # . M , since if #reduction is added then the ChurchRosser Theorem fails in general (although it has been proved for some common systems of type assignment) . In this paper, the labelled syntax is interpreted in the standard syntax of Currystyle type assignment by means of a constant Label, so that λx : # . M is taken as an abbreviation for Label#(λx . M ). The constant Label can be defined as a closed term, so that the labelled syntax is ultimately interpreted in a syntax for which the ChurchRosser Theorem is known to hold for both #reduction and #reduction. This interpretation is carried through for three well known systems of type assignment: ordinary type assignment, the secondorder polymorphic typed lambdacalculus, and the calculus of constructions. These cases illustrate the general ...
Proving Properties of Typed Lambda Terms Using Realizability, Covers, and Sheaves
 Theoretical Computer Science
, 1995
"... . The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possib ..."
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. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called preapplicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...
Normalization of FirstOrder Logic Proofs in Isabelle
, 1999
"... Types are important in functional programming. First order logic can be viewed as, using the CurryHoward isomorphism, a very strict type theory. Proven rst order logic propositions are theorems. Hereby theorems are types and, by the CurryHoward isomorphism, a ilinearized proofj of a theorem can be ..."
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Types are important in functional programming. First order logic can be viewed as, using the CurryHoward isomorphism, a very strict type theory. Proven rst order logic propositions are theorems. Hereby theorems are types and, by the CurryHoward isomorphism, a ilinearized proofj of a theorem can be viewed as a program having the theorem as its type. A proof of a rst order logic theorem can be created automatically using some strategy for proof search. A generated proof can be huge 1 . Such a generated proof of the theorem can be reduced to be a minimal proof using techniques adopted from the theory of calculus. Using this technique a program can be specied as a rst order logic theorem, and the automatically derived program hereof can be made eOEcient. Hereby proof search tools implementing the proof reduction can be used to help writing eOEcient programs. To show how this will be done in an environment that can be used for proof search, the proof reduction is implemented in the Is...
Proving Properties of Typed λTerms Using Realizability, Covers, and Sheaves
, 1995
"... The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible ..."
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The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called preapplicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modi ed realizability both t into this framework. We are then able to prove a metatheorem which shows that if a property of realizers satis es some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typedterms, in particular, strong normalization and con uence. This approach clari es the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simplytypedcalculus (with types!,,+,and?), and to the secondorder (polymorphic)calculus (with types! and 82), for which it yields a new theorem.