This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behavioral properties of concurrent systems, especially renements; some proofs are given using OBJ3. We also discuss where modularization, bisimulation, transition systems and combinations of the object, logic, constraint and functional paradigms t into our hidden agenda. 1 Introduction Algebra can be useful in many dierent ways in software engineering, including specication, validation, language design, and underlying theory. Specication and validation can help in the practical production of reliable programs, advances in language design can help improve the state of the art, and theory can help with building new tools to increase automation, as well as with showing correctness of the whole e...

A formal language ccsl is introduced for describing specifications of classes in object-oriented languages. We show how class specifications in ccsl can be translated into higher order logic. This allows us to reason about these specifications. In particular, it allows us (1) to describe (various) implementations of a particular class specification, (2) to develop the logical theory of a specific class specification, and (3) to establish refinements between two class specifications. We use the (dependently typed) higher order logic of the proof-assistant pvs, so that we have extensive tool support for reasoning about class specifications. Moreover, we describe our own front-end tool to pvs, which generates from ccsl class specifications appropriate pvs theories and proofs of some elementary results.

Incompleteness results for behavioral logics are investigated. We show that there is a basic finite behavioral specification for which the behavioral satisfaction problem is not recursively enumerable, which means that there are no automatic methods for proving all true statements; in particular, behavioral logics do not admit complete deduction systems. This holds for all of the behavioral logics of which we are aware. We also prove that the behavioral satisfaction problem is not co-recursively enumerable, which means that there is no automatic way to refute false statements in behavioral logics. In fact we show stronger results, that all behavioral logics are # 0 2 -hard, and that, for some data algebras, the complexity of behavioral satisfaction is not even arithmetic; matching upper bounds are established for some behavioral logics. In addition, we show for the fixed-data case that if operations mayhave more than one hidden argument, then final models need not exist, so that the coalgebraic flavor of behavioral logic is lost.

: The benefits of the object, logic (or relational), functional, and constraint paradigms

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. The benefits of the object, logic (or relational), functional, and constraint paradigms can be combined, by providing existential queries over objects and their attributes, subject to constraints. This paper provides a precise mathematical foundation for this novel programming paradigm, and shows that it is computationally feasible by reducing it to familiar problems over term algebras (i.e., Herbrand universes) . We use the formalism of hidden logic, and our main result is a version of Herbrand's Theorem for that setting. By extending a result of Diaconescu, we lift our results from equational logic to Horn clause logic with equality. ? The research reported in this paper has been supported in part by the Science and Engineering Research Council, the EC under ESPRIT-2 BRA Working Groups 6071, IS-CORE and 6112, COMPASS, Fujitsu Laboratories Limited, and a contract under the management of the Information Technology Promotion Agency (IPA), Japan, as part of the Industrial Sc...

Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...

Information and Computation journal homepage:www.elsevier.com/locate/ic Complete sets of cooperations

The structure map turning a set into the carrier of a final coalgebra is not unique. This fact is well-known but commonly elided. In this paper we argue that any such concrete representation of a set as a final coalgebra is potentially interesting on its own. We discuss several examples, in particular, we consider different coalgebra structures that turn the set of infinite streams into the carrier of a final coalgebra. After that we focus on coalgebra structures that are made up using so-called cooperations. We say that a collection of cooperations is complete for a given set X if it gives rise to a coalgebra structure that turns X into the carrier set of a subcoalgebra of a final coalgebra. Any complete set of cooperations yields a coalgebraic proof and definition principle. We exploit this fact and devise a general definition scheme for constants and functions on a set X that is parametrically in the choice of the complete set of cooperations for X. Key words: Coalgebra, coinduction, infinite data structures, hidden algebra. 1

In this paper we introduce the notion of an observational coalgebra structure and of a complete set of co-operations. We demonstrate in various example the usefulness of these notions, in particular, we show how they give rise to coalgebraic proof and definition principles. Keywords: Coalgebra, Coinduction, infinite data structures, Hidden Algebra.