The area of coalgebra has emerged within theoretical computer science with a unifying claim: to be the mathematics of computational dynamics. It combines ideas from the theory of dynamical systems and from the theory of state-based computation. Although still in its infancy, it is an active area of research that generates wide interest. Written by one of the founders of the field, this book acts as the first mature and accessible introduction to coalgebra. It provides clear mathematical explanations, with many examples and exercises involving deterministic and non-deterministic automata, transition systems, streams, Markov chains and weighted automata. The theory is expressed in the language of category theory, which provides the right abstraction to make the similarity and duality between algebra and coalgebra explicit, and which the reader is introduced to in a hands-on manner. The book will be useful to mathematicians and (theoretical) computer scientists and will also be of interest to mathematical physicists, biologists and economists.

Introduction to Coalgebra: Towards Mathematics of States and Observation

This paper describes a scenario-based methodology called the NeOn Methodology framework. The aim of this framework is to speed up the construction of ontologies and ontology networks by reusing available knowledge resources (ontologies, non-ontological resources and ontology design patterns). The methodology is founded on four pillars: (1) a glossary of processes and activities; (2) a set of nine scenarios for building ontologies and ontology networks; (3) two ontology life-cycle models; and (4) a set of prescriptive methodological guidelines for performing specific processes and activities. The paper also shows how to apply this methodology framework to develop an ontology network within the multimedia domain. Finally, it presents an empirical evaluation of the NeOn Methodology framework.

The NeOn Methodology framework: A scenario-based methodology for ontology development

In this paper, we present a systematic way of deriving (1) languages of (gener- alised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and de- terministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.

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Non-Deterministic Kleene Coalgebras

This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects.

Carnap, Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design

Reasoning about cardinal directions between extended objects

For finitary set functors preserving inverse images, recursive coalgebras A of Paul Taylor are proved to be precisely those for which the system described by A always halts in finitely many steps.

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Recursive coalgebras of finitary functors

A condensed semantics for qualitative spatial reasoning about oriented straight line segments

In this paper, we show how the web ontology language OWL can be accommodated within the larger framework of the heterogeneous common algebraic speciﬁcation language HETCASL. Through this change in perspective, OWL can
beneﬁt from various useful HETCASL features concerning structuring, modularity, and heterogeneity. This tackles a major problem area in ontology engineering: re-use of ontologies and re-combination of ontological modules. We discuss in particular: 
(1) the extension of the Manchester syntax for OWL with structuring mechanisms of CASL, allowing for explicit modularisation
(2) automatic translations between ontology languages to support ontology design across different ontology languages
(heterogeneity)
(3) heterogeneous ontology reﬁnements, and corresponding automated reasoning support for different logics

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The OWL in the CASL: designing ontologies across logics

Various calculi have been designed for qualitative constraint-based representation and reasoning. Especially for orientation calculi, it happens that the well-known method of algebraic closure cannot decide consistency of constraint networks, even when considering networks over base relations (= scenarios) only. We show that this is the case for all relative orientation calculi capable of distinguishing between "left of" and "right of". Indeed, for these calculi, it is not clear whether efficient (i.e. polynomial) algorithms deciding scenario-consistency exist.

As a partial solution of this problem, we present a technique to decide global consistency in qualitative calculi. It is applicable to all calculi that employ convex base relations over the real-valued space i¾?nand it can be performed in polynomial time when dealing with convex relations only. Since global consistency implies consistency, this can be an efficient aid for identifying consistent scenarios. This complements the method of algebraic closure which can identify a subset of inconsistent scenarios.

Dominik Lücke

Papers

Carnap, Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design

Recursive coalgebras of finitary functors

A condensed semantics for qualitative spatial reasoning about oriented straight line segments

The OWL in the CASL: designing ontologies across logics

Qualitative Reasoning about Convex Relations