Results 11  20
of
40
Intersection Types and Domain Operators
, 2003
"... We use intersection types as a tool for obtaining λmodels. Relying on the notion of easy intersection type theory we successfully build a λmodel in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This a ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We use intersection types as a tool for obtaining &lambda;models. Relying on the notion of easy intersection type theory we successfully build a &lambda;model in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the &lambda;theory where the &lambda;term (&lambda;x.xx)(&lambda;x.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of &lambda;theories. The second result is that for any simple easy term there is a &lambda;model where the interpretation of the term is the minimal fixed point operator.
A Lambda Model Characterizing Computational Behaviours of Terms
 PROCEEDINGS OF THE AND LIKAVEC INTERNATIONAL WORKSHOP REWRITING IN PROOF AND COMPUTATION
, 2001
"... We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms.
Discrimination by Parallel Observers: the Algorithm
 LICS '97 , IEEE COMP. SOC
, 1998
"... The main result of the paper is a constructive proof of the following equivalence: two pure λterms are observationally equivalent in the lazy concurrent λcalculus iff they have the same LévyLongo trees. An algorithm which allows to build a context discriminating any two pure λterms with diffe ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
The main result of the paper is a constructive proof of the following equivalence: two pure λterms are observationally equivalent in the lazy concurrent λcalculus iff they have the same LévyLongo trees. An algorithm which allows to build a context discriminating any two pure λterms with different LévyLongo trees is described. It follows that contextual equivalence coincides with behavioural equivalence (bisimulation) as considered by Sangiorgi. Another consequence is that the discriminating power of concurrent lambda contexts is the same as that of BoudolLaneve's contexts with multiplicities.
Two behavioural lambda models
 Types for Proofs and Programs
, 2003
"... Abstract. We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. This is proved by using the finitary logical description of the model obtained by defining a suitable intersection type assign ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. This is proved by using the finitary logical description of the model obtained by defining a suitable intersection type assignment system.
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Intersection Types and Lambda Theories
 International Workshop on Isomorphisms of Types
, 2002
"... We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of ltheories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation of an arbitrary simple easy term is any filter which can be desc ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of ltheories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation of an arbitrary simple easy term is any filter which can be described in an uniform way by a recursive predicate. This allows us to prove the consistency of a wellknow ltheory: this consistency has interesting consequences on the algebraic structure of the lattice of ltheories.
Towards Lambda Calculus OrderIncompleteness
 Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science
"... After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially orde ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially ordered model (orderincompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is orderincomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the orderincompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).
Lambda calculus: models and theories
 Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture
"... In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories in ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories induced by graph models of lambda calculus.
Applying Universal Algebra to Lambda Calculus
, 2007
"... The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to se ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the longstanding open questions concerning the representability of λtheories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λabstraction algebras. In every combinatory and λabstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λabstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λtheories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.
Reflexive Scott domains are not complete for the extensional lambda calculus
"... A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
A longstanding open problem is whether there exists a model of the untyped λcalculus in the category Cpo of complete partial orderings and Scott continuous functions, whose theory is exactly the least λtheory λβ or the least extensional λtheory λβη. In this paper we analyze the class of reflexive Scott domains, the models of λcalculus living in the category of Scott domains (a full subcategory of Cpo). The following are the main results of the paper: (i) Extensional reflexive Scott domains are not complete for the λβηcalculus, i.e., there are equations not in λβη which hold in all extensional reflexive Scott domains. (ii) The order theory of an extensional reflexive Scott domain is never recursively enumerable. These results have been obtained by isolating among the reflexive Scott domains a class of webbed models arising from Scott’s information systems, called iwebmodels. The class of iwebmodels includes all extensional reflexive Scott domains, all preordered coherent models and all filter models living in Cpo. Based on a finegrained study of an “effective” version of Scott’s information systems, we have shown that there are equations not in λβ (resp. λβη) which hold in all (extensional) iwebmodels.