Results 1 
5 of
5
On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
Abstract

Cited by 48 (10 self)
 Add to MetaCart
Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages  concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.
A Structural CoInduction Theorem
 PROC. MFPS '93, SPRINGER LNCS 802
, 1993
"... The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final c ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order stronglyextensional (sometimes called internal full abstractness): the order is the union of all (ordered) Fbisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar coinduction theorem is given for a category of complete metric spaces and locally contracting functors.
Processes and Hyperuniverses
 Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science 1994, volume 841 of LNCS
, 1994
"... . We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses ar ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard ZermeloFraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten. Introduction In the Semantics of ...
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 ...
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...