Results 1  10
of
38
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
Abstract

Cited by 298 (31 self)
 Add to MetaCart
In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
Abstract

Cited by 139 (16 self)
 Add to MetaCart
Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Towards a Mathematical Operational Semantics
 In Proc. 12 th LICS Conf
, 1997
"... We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation ..."
Abstract

Cited by 134 (9 self)
 Add to MetaCart
We present a categorical theory of `wellbehaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets both an operational model and a canonical, internally fully abstract denotational model for free; moreover, both models satisfy the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of wellbehaved rules for structural operational semantics, such as GSOS.
Automata and coinduction (an exercise in coalgebra
 LNCS
, 1998
"... The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which ..."
Abstract

Cited by 62 (16 self)
 Add to MetaCart
The classical theory of deterministic automata is presented in terms of the notions of homomorphism and bisimulation, which are the cornerstones of the theory of (universal) coalgebra. This leads to a transparent and uniform presentation of automata theory and yields some new insights, amongst which coinduction proof methods for language equality and language inclusion. At the same time, the present treatment of automata theory may serve as an introduction to coalgebra.
Initial Algebra and Final Coalgebra Semantics for Concurrency
, 1994
"... The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial ..."
Abstract

Cited by 55 (9 self)
 Add to MetaCart
The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.
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.
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
"... ..."
Mongruences and Cofree Coalgebras
 Algebraic Methods and Software Technology, number 936 in Lect. Notes Comp. Sci
, 1995
"... . A coalgebra is introduced here as a model of a certain signature consisting of a type X with various "destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in objectoriented programming: they provide access to the type (or st ..."
Abstract

Cited by 30 (10 self)
 Add to MetaCart
. A coalgebra is introduced here as a model of a certain signature consisting of a type X with various "destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in objectoriented programming: they provide access to the type (or state) X. We show that the category of such coalgebras and structure preserving functions is comonadic over sets. Therefore we introduce the notion of a `mongruence' (predicate) on a coalgebra. It plays the dual role of a congrence (relation) on an algebra. An algebra is a set together with a number of operations on this set which tell how to form (derived) elements in this set, possibly satisfying some equations. A typical example is a monoid, given by a set M with operations 1 ! M , M \Theta M ! M . Here 1 = f;g is a singleton set. In mathematics one usually considers only singletyped algebras, but in computer science one more naturally uses manytyped algebras like 1 ! list(A), A \Theta l...
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
A Calculus of Transition Systems (towards Universal Coalgebra)
 In Alban Ponse, Maarten de Rijke, and Yde Venema, editors, Modal Logic and Process Algebra, CSLI Lecture Notes No
, 1995
"... By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). In this paper, some standard results from universal algebra (such as the three isomorphism theorems and facts on the lattices of subalgebras and congruences) are reformulated (using the afore mentioned correspondence) and proved for transition systems. AMS Subject Classification (1991): 68Q10, 68Q55 CR Subject Classification (1991): D.3.1, F.1.2, F.3.2 Keywords & Phrases: Transition system, bisimulation, universal coalgebra, universal algebra, congruence, homomorphism. Note: This paper will appear in `Modal Logic and Process Algebra', edited by Ponse, De Rijke and Venema [PRV95]. 2 Table of Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...