On the Minimisation of Transition-Based Rabin Automata and the Chromatic Memory Requirements of Muller Conditions

TL;DR

This paper establishes the NP-completeness of minimizing transition-based Rabin automata and links this to the chromatic memory requirements in Muller condition games, showing that minimal chromatic memory equals minimal Rabin automaton size.

Contribution

It proves the NP-completeness of Rabin automata minimization and relates chromatic memory size in Muller games to Rabin automaton minimization, disproving a prior conjecture.

Findings

Minimization of transition-based Rabin automata is NP-complete.

The minimal chromatic memory in Muller games equals the size of a minimal Rabin automaton.

Chromatic memories are not always optimal, countering previous assumptions.

Abstract

In this paper, we relate the problem of determining the chromatic memory requirements of Muller conditions with the minimisation of transition-based Rabin automata. Our first contribution is a proof of the NP-completeness of the minimisation of transition-based Rabin automata. Our second contribution concerns the memory requirements of games over graphs using Muller conditions. A memory structure is a finite state machine that implements a strategy and is updated after reading the edges of the game; the special case of chromatic memories being those structures whose update function only consider the colours of the edges. We prove that the minimal amount of chromatic memory required in games using a given Muller condition is exactly the size of a minimal Rabin automaton recognising this condition. Combining these two results, we deduce that finding the chromatic memory requirements of a Muller condition is NP-complete. This characterisation also allows us to prove that chromatic memories cannot be optimal in general, disproving a conjecture by Kopczy\'nski.