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25
Semantic Domains
, 1990
"... this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype fu ..."
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Cited by 148 (3 self)
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this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as welland that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the socalled untyped calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary wayavoiding heavy prerequisites (such as category theory) but still maintaining some level of abstractionwith the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...
Topological Incompleteness and Order Incompleteness of the Lambda Calculus
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2001
"... A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In th ..."
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Cited by 23 (15 self)
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A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λterms closed under ff and ficonversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by BastoneroGouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is nontrivial and metrizable.
On The Algebraic Models Of Lambda Calculus
 Theoretical Computer Science
, 1997
"... . The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory ..."
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Cited by 20 (11 self)
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. The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a firstorder algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus...
A Continuum of Theories of Lambda Calculus Without Semantics
 16TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2001), IEEE COMPUTER
, 2001
"... In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we ..."
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Cited by 16 (11 self)
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In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by BastoneroGouy [6, 7] and by Berline [10], that the strongly stable semantics is incomplete. 1
A Complete Characterization of Complete IntersectionType Theories (Extended Abstract)
 ACM TOCL
, 2000
"... M. DEZANICIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersectiontype theories which yield complete intersectiontype assignment systems for lcalculi, with respect to the three canonical ..."
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Cited by 12 (5 self)
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M. DEZANICIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersectiontype theories which yield complete intersectiontype assignment systems for lcalculi, with respect to the three canonical settheoretical semantics for intersectiontypes: the inference semantics, the simple semantics and the Fsemantics. Keywords Lambda Calculus, Intersection Types, Semantic Completeness, Filter Structures. 1 Introduction Intersectiontypes disciplines originated in [6] to overcome the limitations of Curry 's type assignment system and to provide a characterization of strongly normalizing terms of the lcalculus. But very early on, the issue of completeness became crucial. Intersectiontype theories and filter lmodels have been introduced, in [5], precisely to achieve the completeness for the type assignment system l" BCD W , with respect to Scott's simple semantics. And this result, ...
Practical Foundations for Programming Languages
 In Dynamic Languages Symposium (DLS
, 2007
"... Types are the central organizing principle of the theory of programming languages. Language features are manifestations of type structure. The syntax of a language is governed by the constructs that define its types, and its semantics is determined by the interactions among those constructs. The sou ..."
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Cited by 11 (4 self)
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Types are the central organizing principle of the theory of programming languages. Language features are manifestations of type structure. The syntax of a language is governed by the constructs that define its types, and its semantics is determined by the interactions among those constructs. The soundness of a language design—the absence of illdefined programs— follows naturally. The purpose of this book is to explain this remark. A variety of programming language features are analyzed in the unifying framework of type theory. A language feature is defined by its statics, the rules governing the use of the feature in a program, and its dynamics, the rules defining how programs using this feature are to be executed. The concept of safety emerges as the coherence of the statics and the dynamics of a language. In this way we establish a foundation for the study of programming languages. But why these particular methods? Though it would require a book in itself to substantiate this assertion, the typetheoretic approach
The Meaning of Types  From Intrinsic to Extrinsic Semantics
 Department of Computer Science, University of Aarhus
, 2000
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typing ..."
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Cited by 10 (1 self)
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings. For a simply typed lambda calculus, extended with recursion, subtypes, and named products, we give an intrinsic denotational semantics and a denotational semantics of the underlying untyped language. We then establish a logical relations theorem between these two semantics, and show that the logical relations can be "bracketed" by retractions between the domains of the two semantics. From these results, we derive an extrinsic semantics that uses partial equivalence relations.
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Cited by 7 (0 self)
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
What do Types Mean?  From Intrinsic to Extrinsic Semantics
, 2001
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typ ..."
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Cited by 7 (0 self)
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings.
Predicative algebraic set theory
 Theory Appl. of Categ
"... Abstract. In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certa ..."
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Cited by 7 (1 self)
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Abstract. In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of ideals. The purpose of this paper is to generalize the machinery and results developed by Awodey, Butz, Simpson and Streicher in [Awodey et al, 2004] to the predicative case. Specifically, in ibid. it was shown that: