Coalgebraic Infinite Traces and Kleisli Simulations

TL;DR

This paper extends the categorical theory of Kleisli simulations to infinite traces, providing a soundness foundation for both forward and backward simulations in various system types, aided by forward partial execution.

Contribution

It offers a theoretical foundation for Kleisli simulations' soundness with respect to infinitary traces, including conditions for backward simulation and the use of FPE in this setting.

Findings

Soundness of forward simulations with respect to infinitary traces.

Backward simulation soundness under certain conditions.

FPE enhances Kleisli simulations for infinite trace inclusion.

Abstract

Kleisli simulation is a categorical notion introduced by Hasuo to verify finite trace inclusion. They allow us to give definitions of forward and backward simulation for various types of systems. A generic categorical theory behind Kleisli simulation has been developed and it guarantees the soundness of those simulations with respect to finite trace semantics. Moreover, those simulations can be aided by forward partial execution (FPE)---a categorical transformation of systems previously introduced by the authors. In this paper, we give Kleisli simulation a theoretical foundation that assures its soundness also with respect to infinitary traces. There, following Jacobs' work, infinitary trace semantics is characterized as the "largest homomorphism." It turns out that soundness of forward simulations is rather straightforward; that of backward simulation holds too, although it requires certain additional conditions and its proof is more involved. We also show that FPE can be successfully employed in the infinitary trace setting to enhance the applicability of Kleisli simulations as witnesses of trace inclusion. Our framework is parameterized in the monad for branching as well as in the functor for linear-time behaviors; for the former we mainly use the powerset monad (for nondeterminism), the sub-Giry monad (for probability), and the lift monad (for exception).