Erwin Engeler

Logic of Programs

On the duality of dynamic algebras and kripke models.- The deducibility problem in propositional dynamic logic.- PAL - Propositional algorithmic logic.- Propositional dynamic logics of programs: A survey.- On the algorithmic theory of dictionaries.- On the algorithmic properties of concurrent programs.- A survey of the logic of effective definitions.

Generalized Galois theory and its application to complexity

Abstract We generalize those aspects of classical Galois theory that have to do with the discussion of solvability of problems (namely polynomial equations) relative to auxiliary procedures (e.g. radicals). The underlying structures need no longer be fields, and the problems and procedures more typically arise as algorithmic (e.g. combinatorial) problems. Some of the classical notions and results, e.g. resolvents and discriminants have their natural counterparts. We extend the classical theory mainly in the direction of relations between the group of a problem and the structure and complexity of its solution algorithm. The present paper gives a connected and detailed exposition of this theory, improving and considerably expanding our earlier reports [3, 4]. It now represents a tool for the systematic discussion of the solvability of algorithmic problems, their dependence on structural settings, and the relative merits of solution strategies.

On the Solvability of Algorithmic Problems

Publisher Summary This chapter discusses the basic concepts of a generalized Galois theory to make the paradigm of Galois available for the discussion of the solvability of algorithmic problems.. The chapter presents this development as an abstract theory in the framework of the theory of models of a modest extension of first-order logic. The chapter also discusses some relations to computing. Other attempts to generalize Galois theory in a universal algebra setting started from the concept of an algebraic element.

Combinatory differential fields

Combinatory differential fields arise if differential fields are augmented by operations which allow functions that are programmable in the usual recursive sense to be denoted. The present paper defines this concept. It is shown that every differential field whose field of constants is ordered can be extended to a combinatory field. We generalize the basic notions of the Liouville-Ritt-Risch theory of closed-form solvability to combinatory field extensions and present some explorative examples of problems and solutions. —Authors Abstract

Representation of varieties in combinatory algebras

It is shown that the set of completions of algebras in a variety can be represented as the set of solutions of a single equation of the formA · X=B · X in the authors model of combinatory algebra.A andB are determined directly from the equations which present the variety. Conversely, the individual structures are realized as retracts and the algebraic operations as combinatory objects; these are reclaimable by fixed combinators from the individual solutionsX. These results can be extended to universal classes and to algorithmic classes.

Cumulative Logic Programs and Modelling

Abstract The formation process of pure logic programs over a first-order language is iterated to give rise to cumulative logic programs. Such programs turn out to be objects in a combinatory model and are therefore amenable to algebraic manipulation including equation solving. It is pointed out how this fact can be employed in a discipline of modelling for cooperative processes. Other results concern the representability of equational classes, universal classes and algorithmic classes as solution sets of a set of combinatory equations.

Scientific Computation: The Integration of Symbolic, Numeric and Graphic Computation

It is a commonplace, that the use of mathematical software is in the process of influencing, if not shaping, the working style of the mathematician, physicist and engineer. Considerable effort has been put into the creation of integrated systems by various groups. On the other hand, relatively small progress has been seen in making a telling impact on the larger scientific community with respect to widespread use of such systems. The computer workplace for the scientist has not quite happened yet.

Algorithmic Properties of Structures: Selected Papers of E. Engeler

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Categories of Mapping Filters

The subject of this paper is a topic in that branch of universal algebra called the theory of models. The main trait that distinguishes model theory from other approaches to algebra is the fact that in model theory the language in which theorems and definitions are to be coined is explicitly, indeed formally, specified. This gives, of course, a peculiar slant to the type of problems that are of immediate interest to model theorists. The generalities in which we are interested concern the exact extent of definability of mathematical notions and the characterizability of types of mathematical structures in various formal languages, the existence of structures with particular properties in formally characterizable classes of structures, formal descriptions of types of properties preserved under various mathematical constructions, and the like. In a nutshell, the difference between an algebraist and a model theorist is the following: To an algebraist two mathematical structures A, ℬ are “essentially the same” if they are isomorphic,