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157
Not enough points is enough
 IN: COMPUTER SCIENCE LOGIC. VOLUME 4646 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... Models of the untyped λcalculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as “λmodels”, or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: ..."
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Models of the untyped λcalculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as “λmodels”, or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: given a λmodel A, one may define a ccc in which A (the carrier set) is a reflexive object; conversely, if U is a reflexive object in a ccc C, having enough points, then C ( , U) may be turned into a λmodel. It is well known that, if C does not have enough points, then the applicative structure C ( , U) is not a λmodel in general. This paper: (i) shows that this mismatch can be avoided by choosing appropriately the carrier set of the λmodel associated with U; (ii) provides an example of an extensional reflexive object D in a ccc without enough points: the Kleislicategory of the comonad “finite multisets ” on Rel; (iii) presents some algebraic properties of the λmodel associated with D by (i) which make it suitable for dealing with nondeterministic extensions of the untyped λcalculus.
Algebraic Approaches to Nondeterminism  an Overview
 ACM Computing Surveys
, 1997
"... this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University ..."
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Cited by 24 (3 self)
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this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University
A Theory of Recursive Domains with Applications to Concurrency
 In Proc. of LICS ’98
, 1997
"... Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains. ..."
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Cited by 24 (14 self)
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Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains.
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation
, 1996
"... ..."
Compositional Characterizations of λterms using Intersection Types (Extended Abstract)
, 2000
"... We show how to characterize compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the ..."
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Cited by 19 (5 self)
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We show how to characterize compositionally a number of evaluation properties of λterms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalization, normalization, head normalization, and weak head normalization. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterization results are new, to our knowledge, or else they streamline, strengthen, or generalize earlier results in the literature. The completeness parts of the characterizations are proved uniformly for all the properties, using a settheoretical semantics of intersection types over suitable kinds of stable sets. This technique generalizes Krivine 's and Mitchell's methods for strong normalization to other evaluation properties.
Galois connections and fixed point calculus
 In Algebraic and Coalgebraic Methods in the Mathematics of Program Construction
, 2002
"... Fixed point calculus is about the solution of recursive equations de˛ned by a monotonic endofunction on a partially ordered set. This tutorial presents the basic theory of ˛xed point calculus together with a number of applications of direct relevance to the construction of computer programs. The tut ..."
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Cited by 19 (0 self)
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Fixed point calculus is about the solution of recursive equations de˛ned by a monotonic endofunction on a partially ordered set. This tutorial presents the basic theory of ˛xed point calculus together with a number of applications of direct relevance to the construction of computer programs. The tutorial also presents the theory and application of Galois connections between partially ordered sets. In particular, the intimate relation between Galois connections and ˛xed point equations
A Categorical Axiomatics for Bisimulation
 In Proc. of CONCUR’98, LNCS 1466
, 1998
"... We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in ..."
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Cited by 18 (8 self)
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We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2monad T .
The waybelow relation of function spaces over semantic domains
 TOPOLOGY APPL
, 1998
"... For partially ordered sets that are continuous in the sense of D. S. Scott, the waybelow relation is crucial. It expresses the approximation of an ideal element by its finite parts. We present explicit characterizations of the waybelow relation on spaces of continuous functions from topological spa ..."
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Cited by 15 (3 self)
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For partially ordered sets that are continuous in the sense of D. S. Scott, the waybelow relation is crucial. It expresses the approximation of an ideal element by its finite parts. We present explicit characterizations of the waybelow relation on spaces of continuous functions from topological spaces into continuous posets. Although it is wellknown in which cases these function spaces are continuous posets, such characterizations were lacking until now.
Injective spaces via the filter monad
 Topology Proceedings
, 1997
"... An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein typ ..."
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Cited by 14 (3 self)
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An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein type and apply this to obtain a simple proof of the fact that the algebras are the continuous lattices (Alan Day, 1975, Oswald Wyler, 1976). In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras. For the filter monad, the class turns out to consist precisely of the subspace embeddings. We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972). Similar results are obtained for continuous Scott domains, which are characterized as the injective spaces over dense subspace embeddings. Keywords: Extension of maps, injective space, continuous lattice, continuous Scott domain, domain theory, KockZöberlein monad. AMS classification: 54C20, 06B35, 18C20. 1
The sensible graph theories of lambda calculus
 IN: 19TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS’04), IEEE COMPUTER
, 2004
"... Sensible λtheories are equational extensions of the untyped lambda calculus that equate all the unsolvable λterms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a nonsyntactic model whose equational theory is the least sensible λtheory H (g ..."
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Sensible λtheories are equational extensions of the untyped lambda calculus that equate all the unsolvable λterms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a nonsyntactic model whose equational theory is the least sensible λtheory H (generated by equating all the unsolvable terms). A related question is whether, given a class of models, there exist a minimal and maximal sensible λtheory represented by it. In this paper we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models. We conjecture that the least sensible graph theory, where “graph theory ” means “λtheory of a graph model”, is equal to H, while in the main result of the paper we characterize the greatest sensible graph theory as the λtheory B generated by equating λterms with the same Böhm tree. This result is a consequence of the fact that all the equations between solvable λterms, which have different Böhm trees, fail in every sensible graph model. Further results of the paper are: (i) the existence of a continuum of different sensible graph theories strictly included in B (this result positively answers Question 2 in [7, Section 6.3]); (ii) the nonexistence of a graph model whose equational theory is exactly the minimal lambda theory λβ (this result negatively answers Question 1 in [7, Section 6.2] for the restricted class of graph models).