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142
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
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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.
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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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 .
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 18 (6 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.
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).
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 12 (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
Simple easy terms
 Intersection Types and Related Systems, volume 70 of Electronic Notes in Computer Science
, 2002
"... Dipartimento di Informatica Universit`a di Venezia ..."
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Dipartimento di Informatica Universit`a di Venezia
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, ...
Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
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 11 (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.