In the first of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets.

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 (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded 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

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?

In recent years, there has been a considerable amount of work on using continuous domains in real analysis. Most notably are the development of the generalized Riemann integral with applications in fractal geometry, several extensions of the programming language PCF with a real number data type, and a framework and an implementation of a package for exact real number arithmetic. Based on recursion theory we present here a precise and direct formulation of effective representation of real numbers by continuous domains, which is equivalent to the representation of real numbers by algebraic domains as in the work of Stoltenberg-Hansen and Tucker. We use basic ingredients of an effective theory of continuous domains to spell out notions of computability for the reals and for functions on the real line. We prove directly that our approach is equivalent to the established Turing-machine based approach which dates back to Grzegorczyk and Lacombe, is used by Pour-El & Richards in their found...

Abstract not found

Abstract not found

Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological many-sorted algebras. In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations. With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ‘while ’ programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability. The theory of abstract models is stable: there are many models of computation, and

. 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...

Observing that networks are ubiquitous in applications for spatial databases, we define a new data model and query language that especially supports graph structures. This model integrates concepts of functional data modeling with order-sorted algebra. Besides object and data type hierar-chies graphs are available as an explicit modeling tool, and graph operations are part of the query lan-guage. Graphs have three classes of components, namely nodes, edges, and explicit paths. These are at the same time object types within the object type hierarchy and can be used like any other type. Explicit paths are useful because “real world ” objects often correspond to paths in a network. Further-more, a dynamic generalization concept is introduced to handle heterogeneous collections of objects in a query. In connection with spatial data types this leads to powerful modeling and querying capa-bilities for spatial databases, in particular for spatially embedded networks such as highways, rivers, public transport, and so forth. We use multi-level order-sorted algebra as a formal framework for the specification of our model. Roughly spoken, the first level algebra defines types and operations of the query language whereas the second level algebra defines kinds (collections of types) and type con-structors as functions between kinds and so provides the types that can be used at the first level.

We generate a native pattern language from a context-free grammar. So if we have the underlying grammar of code that needs to be analyzed, or renovated the pattern language comes for free. We use native patterns for recognition and renovation of code. The pattern language is global in the sense that patterns can match entire programs. We illustrate native patterns by discussing a tool that remediates a notoriously difficult Year 2000 problem using native patterns. Categories and Subject Description: D.2.6 [Software Engineering ]: Programming Environments---Interactive; D.2.7 [Software Engineering]: Distribution and Maintenance--- Restructuring; D.3.4. [Processors]: Parsing. Additional Key Words and Phrases: Reengineering, System renovation, Patterns, Plans, Clich'es, Native pattern language, Year 2000 problem, Leap year problem. 1 Introduction One key task in reengineering software is the recognition of problematic code. For example, for Year 2000 related remediation, it is necessa...