10.4230/LIPICS.FSTTCS.2010.400
Schewe, Sven
Sven
Schewe
Beyond Hyper-Minimisation---Minimising DBAs and DPAs is NP-Complete
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2010
Article
Automata Theory
Complexity
Büchi Automata
Parity Automata
Lodaya, Kamal
Kamal
Lodaya
Mahajan, Meena
Meena
Mahajan
2010
2010-12-14
2010-12-14
2010-12-14
en
urn:nbn:de:0030-drops-28816
10.4230/LIPIcs.FSTTCS.2010
978-3-939897-23-1
1868-8969
10.4230/LIPIcs.FSTTCS.2010
LIPIcs, Volume 8, FSTTCS 2010
IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)
2013
8
34
400
411
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Lodaya, Kamal
Kamal
Lodaya
Mahajan, Meena
Meena
Mahajan
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2010
8
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
431169 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
In this paper we study the problem of minimising deterministic automata over finite and infinite words. Deterministic finite automata are the simplest devices to recognise regular languages, and deterministic \buchi, \cobuchi, and parity automata play a similar role in the recognition of $\omega$-regular languages. While it is well known that the minimisation of deterministic finite and weak automata is cheap, the complexity of minimising deterministic \buchi\ and parity automata has remained an open challenge. We establish the NP-completeness of these problems. A second contribution of this paper is the introduction of almost equivalence, an equivalence class for strictly between language equivalence for deterministic \buchi\ or \cobuchi\ automata and language equivalence for deterministic finite automata. Two finite automata are almost equivalent if they, when used as a monitor, provide a different answer only a bounded number of times in any run, and we call the minimal such automaton relatively minimal. Minimisation of DFAs, hyper-minimisation, relative minimisation, and the minimisation of deterministic \buchi\ (or \cobuchi) automata are operations of increasing reduction power, as the respective equivalence relations on automata become coarser from left to right. Besides being a natural equivalence relation for finite automata, almost equivalence is language preserving for weak automata, and can therefore also be viewed as a generalisation of language equivalence for weak automata to a more general class of automata. From the perspective of \buchi\ and \cobuchi\ automata, we gain a cheap algorithm for state-space reduction that also turns out to be beneficial for further heuristic or exhaustive state-space reductions put on top of it.
LIPIcs, Vol. 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), pages 400-411