On coalgebras with internal moves

TL;DR

This paper explores coalgebras with internal moves, framing them as coalgebras over monads, and investigates behavioral equivalences like weak bisimulation and trace semantics, providing a unified approach and new definitions.

Contribution

It formalizes coalgebras with internal moves as monadic structures and develops a uniform framework for weak bisimulation and trace semantics.

Findings

Coalgebras with internal moves are characterized as coalgebras over monads.

Necessary conditions for different saturator approaches to weak bisimulation are identified.

A new fixed point operator-based definition for trace semantics with silent moves is proposed.

Abstract

In the first part of the paper we recall the coalgebraic approach to handling the so-called invisible transitions that appear in different state-based systems semantics. We claim that these transitions are always part of the unit of a certain monad. Hence, coalgebras with internal moves are exactly coalgebras over a monadic type. The rest of the paper is devoted to supporting our claim by studying two important behavioural equivalences for state-based systems with internal moves, namely: weak bisimulation and trace semantics. We continue our research on weak bisimulations for coalgebras over order enriched monads. The key notions used in this paper and proposed by us in our previous work are the notions of an order saturation monad and a saturator. A saturator operator can be intuitively understood as a reflexive, transitive closure operator. There are two approaches towards defining saturators for coalgebras with internal moves. Here, we give necessary conditions for them to yield the same notion of weak bisimulation. Finally, we propose a definition of trace semantics for coalgebras with silent moves via a uniform fixed point operator. We compare strong and weak bisimilation together with trace semantics for coalgebras with internal steps.