arXiv subjects

Lukas Mulder

Publications and source records attributed to Lukas Mulder.

3 recordsLinked to original sources

arXiv2026

Continuous Algebras with Hypotheses

In the literature on Kleene algebra (KA), a number of variants have been proposed such as Kleene algebra with tests, commutative KA, bi-KA, and concurrent KA. The equational theories of some of these structures have then been studied in the presence of additional assumptions, called hypotheses. We propose a unifying framework encompassing all the previous structures, as well as regular tree languages. This is done by considering algebras ordered by complete lattices, where least fixpoints can be computed. We provide a canonical model consisting of closed languages, which we prove sound and complete with respect to all continuous models. Then we study quasi-equational axiomatisations. It is illusory to hope for a generic axiomatisation which would be sound and complete for all instances. Instead, we provide a generic axiomatisation which we prove sound and we setup tools that make it possible to get complete ones in a modular way, building on previous works from the literature. We showcase these tools by proving new completeness results for commutative KA, bi-KA, and regular tree languages, in each case extended with various hypotheses.

cs.LO↗

arXiv2025

Functoriality of Enriched Data Types

In previous work, categories of algebras of endofunctors were shown to be enriched in categories of coalgebras of the same endofunctor, and the extra structure of that enrichment was used to define a generalization of inductive data types. These generalized inductive data types are parametrized by a coalgebra $C$, so we call them $C$-inductive data types; we call the morphisms induced by their universal property $C$-inductive functions. We extend that work by incorporating natural transformations into the theory: given a suitable natural transformation between endofunctors, we show that this induces enriched functors between their categories of algebras which preserve $C$-inductive data types and $C$-inductive functions. Such $C$-inductive data types are often finite versions of the corresponding inductive data type, and we show how our framework can extend classical initial algebra semantics to these types. For instance, we show that our theory naturally produces partially inductive functions on lists, changes in list element types, and tree pruning functions.

math.CT↗

arXiv2024

Measuring data types

In this article, we combine Sweedler's classic theory of measuring coalgebras -- by which $k$-algebras are enriched in $k$-coalgebras for $k$ a field -- with the theory of W-types -- by which the categorical semantics of inductive data types in functional programming languages are understood. In our main theorem, we find that under some hypotheses, algebras of an endofunctor are enriched in coalgebras of the same endofunctor, and we find polynomial endofunctors provide many interesting examples of this phenomenon. We then generalize the notion of initial algebra of an endofunctor using this enrichment, thus generalizing the notion of W-type. This article is an extended version of arXiv:2303.16793, it adds expository introductions to the original theories of measuring coalgebras and W-types along with some improvements to the main theory and many explicitly worked examples.

math.CT↗