An Outline of the Theory of Generalized Sketches

Abstract

Based on a new concept of first-order generalized sketches we coined lately “Logics of Statements in Context” to provide a unified view on formalisms like Algebraic Specifications, Prolog, First-Order Logic, Ehresmann Sketches, Description Logics, Generalized Sketches à la Makkai/Diskin, Diagram Predicate Framework, Graph Conditions, and others.
In the talk we present Generalized Sketches à la Makkai/Diskin as a quite natural generalization of traditional Ehresmann sketches. Generalized Sketches à la Makkai/Diskin can be defined in arbitrary categories. They built upon “atomic statements in context” and utilize sketch implications for axiomatization purposes. Going beyond atomic statements, we outline the definition of arbitrary first-order statements in arbitrary categories enabling us to enhance the expressiveness of Generalized Sketches. In analogy to first-order statements, we can also define arbitrary first-order sketch conditions generalizing thereby different kinds of “nested graph constraints and conditions”.
We intend to discuss, on the way, two essential constructions Makkai’s work on Generalized Sketches relies on: “Syntactic representation of models” and “internalization of atomic statements”.

Uwe Wolter

Professor

Uwe E. Wolter is associate professor at the University of Bergen, Norway. He received his PhD degree in 1989 from the Technical University Magdeburg, Germany. He was awarded with the Medal of Honour from the East-German Society of Mathematics for the best PhD thesis of the year. He held positions at Humboldt-University Berlin and Technical University Berlin before joining the Department of Informatics at the University of Bergen in 2000.