Results 1 
9 of
9
From Settheoretic Coinduction to Coalgebraic Coinduction: some results, some problems
, 1999
"... ..."
Themes in Final Semantics
 Dipartimento di Informatica, Università di
, 1998
"... C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e
Axiomatic Characterizations of Hyperuniverses and Applications
 University of Southern
, 1996
"... Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not a ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not abide by the "limitation of size" principle. We present correspondences between settheoretic properties and topological properties of hyperuniverses. We give existence theorems and discuss applications and generalizations to the non compact case. Work partially supported by 40% and 60% MURST grants, CNR grants, and EEC Science MASK, and BRA Types 6453 contracts. y Member of GNSAGA of CNR. z The main results of this paper have been communicated by this author at the "11 th Summer Conference on General Topology and Applications" August 1995, Portland, Maine. Introduction Natural frameworks for dicussing Selfreference and other circular phenomena are extremely useful in areas such ...
Coinductive Characterizations of Applicative Structures
 MATH. STRUCTURES IN COMP. SCI. 9(4):403–435
, 1998
"... We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss in particular, what we call, the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor (P( \Theta )) \Phi (P( \Theta )). In this paper, we present two general methods for showing the soundenss of this principle. The first applies to approximable applicative structures. Many c.p.o. models in...
Coalgebraic Semantics of an Imperative Class Based Language
 Dipartimento di Matematica e Infomatica, Universita’ di
, 2003
"... We study two observational equivalences of Fickle programs. Fickle is a classbased object oriented imperative language... ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We study two observational equivalences of Fickle programs. Fickle is a classbased object oriented imperative language...
Coalgebraic Semantics and Observational Equivalences of an Imperative Classbased OOLanguage
 University of Nijmegen, The Netherlands
"... Fickle is a classbased object oriented imperative language, which extends Java with object reclassification. In this paper, we introduce a natural observational equivalence over Fickle programs. This is a contextual equivalence over main methods with respect to a given sequence of class definitio ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Fickle is a classbased object oriented imperative language, which extends Java with object reclassification. In this paper, we introduce a natural observational equivalence over Fickle programs. This is a contextual equivalence over main methods with respect to a given sequence of class definitions, i.e. a program. In order to study it, we utilize the formal computational model for OOprogramming based on coalgebras, which has recently emerged, whereby objects are taken to be equal when the actions of methods on them yield the same observations and equivalent next states. However, in order to deal with imperative features, we need to extend the original approach of H.Reichel and B.Jacobs in various ways. In particular, we introduce a coalgebraic description of objects (states of a class), which induces a coinductive behavioural equivalence on programs. For simplicity, we focus on Fickle objects whose methods do not take more than one object parameter as argument. Completeness results as well as problematic issues arising from binary methods are also discussed.
A Complete Coinductive Logical System for Bisimulation Equivalence on Circular Objects
 in FoSSaCS'99 (ETAPS) Conf. Proc., W.Thomas ed., Springer LNCS 1578
, 1983
"... We introduce a coinductive logical system à la Gentzen for establishing bisimulation equivalences on circular nonwellfounded regular objects, inspired by work of Coquand, and of Brandt and Henglein. In order to describe circular objects, we utilize a typed language, whose coinductive types involve ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We introduce a coinductive logical system à la Gentzen for establishing bisimulation equivalences on circular nonwellfounded regular objects, inspired by work of Coquand, and of Brandt and Henglein. In order to describe circular objects, we utilize a typed language, whose coinductive types involve disjoint sum, cartesian product, and finite powerset constructors. Our system is shown to be complete with respect to a maximal fixed point semantics. It is shown to be complete also with respect to an equivalent final semantics. In this latter semantics, terms are viewed as points of a coalgebra for a suitable endofunctor on the category Set of nonwellfounded sets. Our system subsumes an axiomatization of regular processes, alternative to the classical one given by Milner.
Coalgebraic Coinduction in (Hyper)settheoretic Categories
, 2000
"... This paper is a contribution to the foundations of coinductive types and coiterative functions, in (Hyper)settheoretical Categories, in terms of coalgebras. We consider atoms as first class citizens. First of all, we give a sharpening, in the way of cardinality, of Aczel's Special Final Coalgebra ..."
Abstract
 Add to MetaCart
This paper is a contribution to the foundations of coinductive types and coiterative functions, in (Hyper)settheoretical Categories, in terms of coalgebras. We consider atoms as first class citizens. First of all, we give a sharpening, in the way of cardinality, of Aczel's Special Final Coalgebra Theorem, which allows for good estimates of the cardinality of the final coalgebra. To these end, we introduce the notion of Y uniform functor, which subsumes Aczel's original notion. We give also an nary version of it, and we show that the resulting class of functors is closed under many interesting operations used in Final Semantics. We define also canonical wellfounded versions of the final coalgebras of functors uniform on maps. This leads to a reduction of coiteration to ordinal induction, giving a possible answer to a question raised by Moss and Danner. Finally, we introduce a generalization of the notion of F bisimulation inspired by Aczel's notion of precongruence, and we show t...
Coinductive Axiomatization of Recursive Type Equality and Subtyping ⋆
"... Abstract. We present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respective ..."
Abstract
 Add to MetaCart
Abstract. We present new sound and complete axiomatizations of type equality and subtype inequality for a firstorder type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The main novelty of the axiomatization is the fixpoint rule (or coinduction principle), which has the form A, P ⊢ P A ⊢ P where P is either a type equality τ = τ ′ or type containment τ ≤ τ ′. We define what it means for a proof (formal derivation) to be formally contractive and show that the fixpoint rule is sound if the proof of the premise A, P ⊢ P is contractive. (A proof of A,P ⊢ P using the assumption axiom is, of course, not contractive.) The fixpoint rule thus allows us to capture a coinductive relation in the fundamentally inductive framework of inference systems. The new axiomatizations are “leaner ” than previous axiomatizations, particularly so for type containment since no separate axiomatization of type equality is required, as in Amadio and Cardelli’s axiomatization. They give rise to a natural operational interpretation of proofs as coercions. In particular, the fixpoint rule corresponds to definition by recursion. Finally, the axiomatization is closely related to (known) efficient algorithms for deciding type equality and type containment. These can be modified to not only decide type equality and type containment, but also construct proofs in our axiomatizations efficiently. In connection with the operational interpretation of proofs as coercions this gives efficient (O(n 2) time) algorithms for constructing efficient coercions from a type to any of its supertypes or isomorphic types. 1