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Elements Of The General Theory Of Coalgebras
, 1999
"... . Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intend ..."
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Cited by 30 (7 self)
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. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and coinduction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...
Finality Regained  A Coalgebraic Study of Scottsets and Multisets
 Arch. Math. Logic
, 1999
"... In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of suchsets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the ..."
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Cited by 21 (1 self)
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In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of suchsets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFAuniverse. Wewillhave a closer look into the connection of the iterated circular multisets and arbitrary trees. Key words: multiset, nonwellfounded set, Scottuniverse, AFA, coalgebra, modal logic, graded modalities MSC2000 codes: 03B45, 03E65, 03E70, 18A15, 18A22, 18B05, 68Q85 1 Contents 1 Introduction 3 1.1 Multisets on a Given Domain . . . . . . . . . . . . . . . . . . . . 3 1.2 Iterated and Circular Multisets . . . . . . . . . . . . . . . . . . . 6 1.3 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . 7 2 Prerequisites 8 2.1 Coalgebras and Morphisms . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 A Prototype: Pow . . . . . . . . . . . . . . . ...
A Transition System Semantics for the ControlDriven Coordination Language MANIFOLD
 THEORETICAL COMPUTER SCIENCE
, 1998
"... Coordination languages are a new class of parallel programming languages which manage the interactions among concurrent programs. Basically, coordination is achieved either by manipulating data values shared among all active processes or by dynamically evolving the interconnections among the process ..."
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Cited by 17 (9 self)
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Coordination languages are a new class of parallel programming languages which manage the interactions among concurrent programs. Basically, coordination is achieved either by manipulating data values shared among all active processes or by dynamically evolving the interconnections among the processes as a consequence of observations of their state changes. The latter, also called controldriven coordination, is supported by MANIFOLD. We present the formal semantics of a kernel of MANIFOLD, based on a twolevel transition system model: the first level is used to specify the ideal behavior of each single component in a MANIFOLD system, whereas the second level captures their interactions. Although we apply our twolevel model in this paper to define the semantics of a controloriented coordination language, this approach is useful for the formal studies of other coordination models and languages as well.
A Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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Cited by 13 (5 self)
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkho# [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
Covarieties and Complete Covarieties
, 1999
"... We present two ways to de ne covarieties and complete covarieties, i.e. covarieties that are closed under total bisimulation: by closure operators and by subcoalgebras of coalgebras. ..."
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Cited by 13 (3 self)
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We present two ways to de ne covarieties and complete covarieties, i.e. covarieties that are closed under total bisimulation: by closure operators and by subcoalgebras of coalgebras.
Coalgebraic Structure From Weak Limit Preserving Functors
, 1999
"... Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and ..."
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Cited by 13 (7 self)
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Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and subcoalgebras. We give a characterization of monos in Set F in terms of congruences and bisimulations, which explains, under which conditions monos must be injective maps.
Equational And Implicational Classes Of CoAlgebras (Extended Abstract)
 In The 4th International Seminar on Relational Methods in Logic, Algebra and Computer Science
"... ) H. PETER GUMM Abstract. If T : Set ! Set is a functor which is bounded and preserves weak pullbacks then a class of T coalgebras is a covariety, i.e closed under H (homomorphic images), S (subcoalgebras) and \Sigma (sums), if and only if it can be defined by a set of "coequations". Similarly, c ..."
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Cited by 3 (0 self)
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) H. PETER GUMM Abstract. If T : Set ! Set is a functor which is bounded and preserves weak pullbacks then a class of T coalgebras is a covariety, i.e closed under H (homomorphic images), S (subcoalgebras) and \Sigma (sums), if and only if it can be defined by a set of "coequations". Similarly, classes closed under H and \Sigma can be characterized by implications of coequations. These results are analogous to the theorems of G.Birkhoff and of A.I.Mal'cev in classical universal algebra. 1. Introduction The recently developed theory of coalgebras under a functor T provides a highly attractive framework for describing the semantics and the logic of various types of transition systems. In contrast to the algebraic semantics of abstract data types where data objects are constructed recursively and equality is proven by induction, coalgebras support definitions by corecursion and define equivalence by coinduction. This view is appropriate in many contexts, prominently when modelling o...
Analyzing Transaction Logs for Building Coordination Models
, 1999
"... Tasks like supply chain management, design of interorganizational workflow and design of virtual organizations or consortia, require mechanisms to analyze interaction requirements spanning across autonomous organizations. While existing means of analysis would help in identifying pertinent actors ..."
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Cited by 1 (1 self)
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Tasks like supply chain management, design of interorganizational workflow and design of virtual organizations or consortia, require mechanisms to analyze interaction requirements spanning across autonomous organizations. While existing means of analysis would help in identifying pertinent actors and interactions among them, properties which could manifest by virtue of the interactions themselves may go undetected. In this paper, we present an approach that looks for properties of a given problem domain which manifest due to the interactions that take place in the domain. We use data mining to look for knowledge from a set of transaction data collected from past transactions that have taken place. A specific kind of problem is considered involving interactions among a set of autonomous actors, and an interaction model is built based on the results of the data mining process. Keywords: Coordination, Interaction Properties, Data Mining, Modeling Interactive Behavior. This d...
Reasoning about Knowledge and ContextAwareness
"... In this paper we propose a knowledgebased approach for the support of adaptive and contextaware behavior in multiagent systems. We identify the agents knowledge which is present in a system as a central factor for contextawareness and for the systems ability to adapt to changing environments. We ..."
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In this paper we propose a knowledgebased approach for the support of adaptive and contextaware behavior in multiagent systems. We identify the agents knowledge which is present in a system as a central factor for contextawareness and for the systems ability to adapt to changing environments. We provide a framework which supports simple and intuitive specification of complex knowledge configurations. We discuss the semantics and formal backgrounds of the central notion of knowledge conformance. Finally we describe an algorithm for reasoning about knowledge conformance, i.e. for deciding whether a knowledge specification holds for an agent (or a group of agents). Our algorithm heavily relies on the features from membrane computing and is especially wellsuited for reasoning about incomplete knowledge. In addition it supports the incremental introduction of situational knowledge and thus supports the dynamics of contextawareness.