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Orderincompleteness and finite lambda reduction models
 Theoretical Computer Science
, 2003
"... Abstract Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi or fijconversion, the pa ..."
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Cited by 23 (0 self)
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Abstract Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi or fijconversion, the partial order is trivial on term denotations. Equivalently, theopen and closed term algebras of the untyped lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal'cev operators, of those lambda theorieswhich cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the untyped lambda calculus, or more precisely, finite models of reduction. We demonstrate how such models can beused as practical tools for giving finitary proofs of term inequalities. 1 Introduction Perhaps the most important contribution in the area of mathematical programming semantics was the discovery, byD. Scott in the late 1960's, that models for the untyped lambda calculus could be obtained by a combination of ordertheoretic and topological methods. A long tradition of research in domain theory ensued, and Scott's methods havebeen successfully applied to many aspects of programming semantics.
Filter Models for ConjunctiveDisjunctive λcalculi
, 1996
"... The distinction between the conjunctive nature of nondeterminism as opposed to the disjunctive character of parallelism constitutes the motivation and the starting point of the present work. λcalculus is extended with both a nondeterministic choice and a parallel operator; a notion of reduction i ..."
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Cited by 14 (6 self)
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The distinction between the conjunctive nature of nondeterminism as opposed to the disjunctive character of parallelism constitutes the motivation and the starting point of the present work. λcalculus is extended with both a nondeterministic choice and a parallel operator; a notion of reduction is introduced, extending fireduction of the classical calculus. We study type assignment systems for this calculus, together with a denotational semantics which is initially defined constructing a set semimodel via simple types. We enrich the type system with intersection and union types, dually reflecting the disjunctive and conjunctive behaviour of the operators, and we build a filter model. The theory of this model is compared both with a Morrisstyle operational semantics and with a semantics based on a notion of capabilities.
Orderincompleteness and finite lambda models (Extended Abstract)
 PROCEEDINGS OF THE 11TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE, IEEE COMPUTER SOCIETY
, 1996
"... Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is ..."
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Cited by 10 (2 self)
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Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the typefree lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal’cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the typefree lambda calculus. We introduce partial syntactical lambda models, which are derived from Plotkin’s syntactical models of reduction, and we investigate how these models can be used as practical tools for giving finitary proofs of term inequalities. We give a 3element model as an example.
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
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Intersection Types, λmodels, and Böhm Trees
"... This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of λcalculus. We start by describing the wellknown results showing the deep connection between intersection type systems and normalization properties, i.e. ..."
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This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of λcalculus. We start by describing the wellknown results showing the deep connection between intersection type systems and normalization properties, i.e., their power of naturally characterizing solvable, normalizing, and strongly normalizing pure λterms. We then explain the importance of intersection types for the semantics of λcalculus, through the construction of filter models and the representation of algebraic lattices. We end with an original result that shows how intersection types also allow to naturally characterize tree representations of unfoldings of λterms (Böhm trees).
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.
Kripke Models and the (in)equational Logic of the SecondOrder LambdaCalculus
, 1995
"... . We define a new class of Kripke structures for the secondorder calculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an ..."
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. We define a new class of Kripke structures for the secondorder calculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: W ! Preor equipped with certain natural transformations corresponding to application and abstraction (where W is a preorder, the set of worlds, and Preor is the category of preorders). We make use of an explicit construction of the exponential of functors in the Cartesianclosed category Preor W , and we also define a kind of exponential Q \Phi (A s ) s2T to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in thi...
Intersection Types, *models, and B"ohm Trees Mariangiola DezaniCiancaglini, Elio Giovannetti, Ugo de'Liguoro
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Preliminary Version
, 1993
"... Kripke Models for the SecondOrder lambdaCalculus We define a new class of Kripke structures for the secondorder λcalculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) or equations. The Kripke structures under consideration are equ ..."
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Kripke Models for the SecondOrder lambdaCalculus We define a new class of Kripke structures for the secondorder λcalculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) or equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A:W → Preor equipped with certain natural transformations corresponding to application and abstraction (where is a preorder, the set of worlds, and Preor is the category of preorders). We make use of an explicit construction of the exponential of functors in the Cartesianclosed category PreorW, and we also define a kind of exponential ∏Φ(As)s∈Τ to take care of type abstraction. We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the secondorder λcalculus, and