SCS 15: Continuous Lattices and Universal Algebra

Also accessible at https://www2.mathematik.tu-darmstadt.de/~logik/keimel/scs.htm

SCS 4: Note on Continuous Lattices

Also accessible at https://www2.mathematik.tu-darmstadt.de/~logik/keimel/scs.htm

Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

Computer-supported Exploration of a Categorical Axiomatization of Modeloids

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fra\"iss\'e games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fra\"iss\'e games.Comment: 24 pages; accepted for conference: Relational and Algebraic Methods in Computer Science (RAMICS 2020

Category Theory in Isabelle/HOL as a Basis for Meta-logical Investigation

This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing the denotational semantic model of intuitionistic multiplicative linear logic. Next to these meta-logical-investigations, we contribute to building an Isabelle category theory library, with a focus on ease of use in the formalization beyond category theory itself. This work paves the way for future formalizations based on category theory and demonstrates the power of automated reasoning in investigating meta-logical questions.Comment: 15 pages. Preprint of paper accepted for CICM 2023 conferenc

Towards a Java Subtyping Operad

The subtyping relation in Java exhibits self-similarity. The self-similarity in Java subtyping is interesting and intricate due to the existence of wildcard types and, accordingly, the existence of three subtyping rules for generic types: covariant subtyping, contravariant subtyping and invariant subtyping. Supporting bounded type variables also adds to the complexity of the subtyping relation in Java and in other generic nominally-typed OO languages such as C# and Scala. In this paper we explore defining an operad to model the construction of the subtyping relation in Java and in similar generic nominally-typed OO programming languages. Operads, from category theory, are frequently used to model self-similar phenomena. The Java subtyping operad, we hope, will shed more light on understanding the type systems of generic nominally-typed OO languages.Comment: 13 page