Predicate Liftings and Functor Presentations in Coalgebraic Expression Languages

TL;DR

This paper introduces a generic coalgebraic expression language with a Kleene-type theorem, connecting expressions to finite systems across various system types using predicate liftings with a preservation property.

Contribution

It develops a unified expression language for finite coalgebras and proves a Kleene-type correspondence, extending previous work with a semantics based on predicate liftings.

Findings

Established a Kleene-type theorem for the expression language.

Unified semantics for various system types including weighted and probabilistic.

Showed the preservation property holds for Moss liftings.

Abstract

We introduce a generic expression language describing behaviours of finite coalgebras over sets; besides relational systems, this covers, e.g., weighted, probabilistic, and neighbourhood-based system types. We prove a generic Kleene-type theorem establishing a correspondence between our expressions and finite systems. Our expression language is similar to one introduced in previous work by Myers but has a semantics defined in terms of a particular form of predicate liftings as used in coalgebraic modal logic; in fact, our expressions can be regarded as a particular type of modal fixed point formulas. The predicate liftings in question are required to satisfy a natural preservation property; we show that this property holds in particular for the Moss liftings introduced by Marti and Venema in work on lax extensions.