Computer Science > Formal Languages and Automata Theory
[Submitted on 25 May 2021 (v1), last revised 5 Nov 2021 (this version, v3)]
Title:On the Minimisation of Transition-Based Rabin Automata and the Chromatic Memory Requirements of Muller Conditions
View PDFAbstract: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 Kopczyński.
Submission history
From: Antonio Casares [view email][v1] Tue, 25 May 2021 15:22:51 UTC (44 KB)
[v2] Fri, 9 Jul 2021 17:12:28 UTC (41 KB)
[v3] Fri, 5 Nov 2021 16:31:56 UTC (41 KB)
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