The work of Erwin Engeler in the logic and algebra of computer science has been influential but has become difficult to access because it has appeared in different types of publications. This collection of selected papers is therefore timely and useful. It represents an original and coherent approach to the basic interrelationships between mathematics and computer science.The volume begins with the area of enrichment of classical model theory by languages which express properties representing the outcome of hypothetical computer programs executed in a given class of mathematical structures, and is related to questions of correctness and provability of programs. This point of view allowed the generalization of classical Galois theory to the point of discussing the relation between structure and complexity of solution programs for problems posed in various mathematical theories. The algebraic approach is deepened and enlarged in the later papers by showing that the algorithmic aspects of any mathematical structure can be uniformly dealt with by expanding these structures into combinatory algebras.
Part 1 Axiomatics and logics of algorithmic properties: algorithmic properties of structures; formal languages - automata and structures; proof theory and the accuracy of computations. Part 2 Galois-connections and structure of problem classes: on the solvability of algorithmic problems; lower bounds by Galois theory; towards a Galois theory of algorithmic problems. Part 3 An algebraization of algorithmics: algebras and combinators; cumulative logic programs and modelling; formal models of computation in which data are processes - theory and applications; and others.