Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective

TL;DR

This paper generalizes fair simulation concepts for nondeterministic and probabilistic Büchi automata using coalgebraic modeling, providing soundness proofs and connections to parity games and ranking functions.

Contribution

It introduces two new coalgebraic definitions of fair simulation for nondeterministic tree automata and probabilistic word automata, with proven soundness.

Findings

The nondeterministic fair simulation is formulated via fixed-point equations and parity games.

The probabilistic fair simulation uses a ranking-function approach.

Both definitions are derived from a unified coalgebraic framework.

Abstract

Notions of simulation, among other uses, provide a computationally tractable and sound (but not necessarily complete) proof method for language inclusion. They have been comprehensively studied by Lynch and Vaandrager for nondeterministic and timed systems; for B\"{u}chi automata the notion of fair simulation has been introduced by Henzinger, Kupferman and Rajamani. We contribute to a generalization of fair simulation in two different directions: one for nondeterministic tree automata previously studied by Bomhard; and the other for probabilistic word automata with finite state spaces, both under the B\"{u}chi acceptance condition. The former nondeterministic definition is formulated in terms of systems of fixed-point equations, hence is readily translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm; the latter probabilistic definition bears a strong ranking-function flavor. These two different-looking definitions are derived from one source, namely our coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic observations, we also prove their soundness: a simulation indeed witnesses language inclusion.