This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OBJ3 is described in detail, with many examples. OBJ is a wide spectrum first-order functional language that is rigorously based on (order sorted) equational logic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.

Abstract. We present an institution of observational logic suited for state-based systems specifications. The institution is based on the notion of an observational signature (which incorporates the declaration of a distinguished set of observers) and on observational algebras whose operations are required to be compatible with the indistinguishability relation determined by the given observers. In particular, we introduce a homomorphism concept for observational algebras which adequately expresses observational relationships between algebras. Then we consider a flexible notion of observational signature morphism which guarantees the satisfaction condition of institutions w.r.t. observational satisfaction of arbitrary first-order sentences. From the proof theoretical point of view we construct a sound and complete proof system for the observational consequence relation. Then we consider structured observational specifications and we provide a sound and complete proof system for such specifications by using a general, institution-independent result of [6]. 1

We consider observational specifications of state-based systems which incorporate the declaration of a distinguished set of observer operations. These observers determine an indistinguishability relation for states which is called "observational equality". An important requirement for the non-observer operations is the compatibility with the observational equality. In the CafeOBJ language (and in extended hidden algebra) this property is called "behavioural coherence". In this presentation we introduce the notion of an "observer complete definition" and we show that any (non-observer) operation which is defined using this pattern is behaviourally coherent. We also discuss some consequences of this result for relating observational logic and extended hidden algebra semantics and for proving the correctness of observational implementations.

We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphism map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of logical multiparadigm languages, especially structuring specifications (module systems) a la OBJ-Clear. They include model reducts, free constructions (liberality), co-limits, model amalgamation (exactness), and inclusion systems. We outline a general logical semantics for languages whose semantics satisfy certain "logical" principles by extending the institutional semantics developed within the Clear-OBJ tradition. Finally, in the Appendix, we briefly illustrate it with the concrete example of CafeOBJ. Keywords Algebraic specification, Institutions, Theory morphism. AMS Classifications 68Q65, 18C10, 03G30, 08A70 2 1 Introduction Computing Motivation This work belongs to the research are...

. Ontologies allow the abstract conceptualisation of domains, but a given domain can be conceptualised through many different ontologies, which can be problematic when ontologies are used to support knowledge sharing. We present a formal account of ontologies that is intended to support knowledge sharing through precise characterisations of relationships such as compatibility and refinement. We take an algebraic approach, in which ontologies are presented as logical theories. This allows us to characterise relations between ontologies as relations between their classes of models. A major result is cocompleteness of specifications, which supports merging of ontologies across shared sub-ontologies. 1 Introduction Over the last decade ontologies --- best characterised as explicit specifications of a conceptualisation of a domain [17] --- have become increasingly important in the design and development of knowledge based systems, and for knowledge representations generally. They...

Abstract. The theory of hidden algebras combines standard algebraic techniques with coalgebraic techniques to provide a semantic foundation for the object paradigm. This paper focuses on the coalgebraic aspect of hidden algebra, concerned with signatures of destructors at the syntac-tic level and with finality and coffee constructions at the semantic level. Our main result shows the existence of cofree constructions induced by maps between coalgebraic hidden specifications. Their use in giving a se-mantics to parameterised objects and inheritance is then illustrated. The cofreeness result for hidden algebra is generalised to abstract coalgebra and a universal construction for building object systems over existing subsystems is obtained. Finally, existence of final/cofree constructions for arbitrary hidden specifications is discussed. 1

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An approach to information visualization based on algebraic semiotics is introduced, and illustrated with examples. Semiotics is the general theory of signs, and algebraic semiotics is a new approach combining algebraic abstract data type theory with a grounding in social reality. The most important new ideas are to use semiotic spaces for systems of related signs, semiotic morphisms for representations of signs, and preservation orderings for the quality of representations.

We argue for an algorithmic approach to behavioral proofs, review the hidden algebra approach, develop circular coinductive rewriting for conditional goals, extend it with case analysis, and give some examples.

This paper studies the composition of modules that can hide information, over a very general class of logical systems called inclusive institutions. Two semantics are given for composition of such modules using five familiar operations, and a property called conservativity is shown necessary and sufficient for these semantics to agree. The first semantics extracts the visible properties of the result of composing the visible and hidden parts of modules, while the second uses only the visible properties of the components; the semantics agree when the visible consequences of hidden information are enough to determine the result of the composition. A number of "laws of software composition" are proved relating the composition operations. Inclusive institutions simplify many proofs.