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16
The Lazy Lambda Calculus in a Concurrency Scenario (Extended Abstract)
 Information and Computation
, 1994
"... ) Davide Sangiorgi LFCS  Department of Computer Science Edinburgh University Edinburgh  EH9 3JZ  UK Abstract The use of lambda calculus in richer settings, possibly involving parallelism, is examined in terms of its effect on the equivalence between lambda terms. We concentrate here on Abra ..."
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) Davide Sangiorgi LFCS  Department of Computer Science Edinburgh University Edinburgh  EH9 3JZ  UK Abstract The use of lambda calculus in richer settings, possibly involving parallelism, is examined in terms of its effect on the equivalence between lambda terms. We concentrate here on Abramsky's lazy lambda calculus and we follow two directions. First, the lambda calculus is studied within a process calculus by examining the equivalence $ induced by Milner's encoding into the calculus. We give exact operational and denotational characterizations for $. Secondly, we examine Abramsky's applicative bisimulation when the lambda calculus is augmented with (wellformed) operators, i.e. symbols equipped with reduction rules describing their behaviour. Then, maximal discrimination is obtained when all operators are considered; we show that this discrimination coincides with the one given by $ and that the adoption of certain nondeterministic operators is sufficient and necessary...
SetTheoretical and Other Elementary Models of the lambdacalculus
 Theoretical Computer Science
, 1993
"... Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. F ..."
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Cited by 40 (0 self)
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Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. Functions are modeled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the fijcalculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the fijcalculus which are continuous complete lattices. The second part begins with a brief discussion of models of the calculus in set theories with antifoundation axioms. Next the model of the fi calculus of Part 1 and also the closely relatedbut different!models of Scott [53, 54] and of Engeler [21, 22] are reviewed....
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 resul ..."
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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
Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pc ..."
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Cited by 13 (2 self)
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Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 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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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, ...
Intersection Types and Lambda Models
, 2005
"... Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of t ..."
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Invariance of interpretation by #conversion is one of the minimal requirements for any standard model for the #calculus. With the intersection type systems being a general framework for the study of semantic domains for the #calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.
A Semantics for Static Type Inference
 Information and Computation
, 1993
"... Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model cl ..."
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Curry's system for Fdeducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry's system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry's system itself, relative to an extended notion of model that validates reduction but not conversion.
A settheoretical definition of application
 University of Edinburgh
, 1972
"... [41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the λβcalculus. Functions are modelled in a similar way to that normally employed in ..."
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[41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the λβcalculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβηcalculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the λβηcalculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λcalculus in set theories with antifoundation axioms. Next we review the model of the λβcalculus of Part 1 and also the closely related—but different!—models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can be given. Following Longo [36], one can employ certain ScottEngeler algebras.
Lazy Lambda Calculus: Theories, Models and Local Structure Characterisation
 AUTOMATA, LANGUAGES AND PROGRAMMING, LNCS 623
, 1994
"... Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programm ..."
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Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programming. This paper proposes modification of a number of key notions in the sensible theory along the lines of laziness. Starting from the strongly unsolvables as the meaningless terms, we define and investigate properties of lazy (or weakly sensible) λtheories, lazy λmodels and a number of lazy behavioural preorders on λterms. In the second part, we show that all these notions have a natural place in a class of lazy psemodels. A major result of this paper is a new local structure theorem for lazy psemodels. This characterizes the ordering between denotations of λterms in the model by a new lazy behavioural preorder.
Finite Type Structures within Combinatory Algebras
 Annals of Pure and Applied Logic
, 1999
"... Inside a combinatory algebra, there are 'internal' versions of the finite type structure over !, which form models of various systems of finite type arithmetic. This paper compares internal representations of the intensional and extensional functionals. If these classes coincide, the algeb ..."
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Inside a combinatory algebra, there are 'internal' versions of the finite type structure over !, which form models of various systems of finite type arithmetic. This paper compares internal representations of the intensional and extensional functionals. If these classes coincide, the algebra is called ftextensional. Some criteria for ftextensionality are given and a number of wellknown ca's are shown to be ftextensional, regardless of the particular choice of representation for !. In particular, DA , P! , T ! , H! and certain D1models all share the property of ftextensionality. It is also shown that ftextensionality is by no means an intrinsic property of ca's, i.e. that there exists a very concrete class of ca's  the class of reflexive coherence spaces  no member of which has this property. This leads to a comparison of ftextensionality with the wellstudied notions of extensionality and weak extensionality. Ftextensionality turns out to be completely independent.