Abstract
It is a well known fact that not all max-plus automata can be determinized, i.e. transformed into deterministic max-plus automata with the same behavior. A classical sequentialization procedure, extended in the literature to max-plus automata, succeeds in computing an equivalent deterministic max-plus automaton for important subclasses of max-plus automata. This procedure is based on the normalization of state vectors in order to detect and merge states which have similar future behavior. In this paper, a novel and weaker condition is proposed that still guarantees this property. This allows for a considerable improvement of the existing determinization procedure, because it terminates for a larger class of max-plus automata.
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Notes
q ∈ Qis said to be accessibleif there exists w ∈Σ∗and qi ∈ Qisuch that\(| q_{i} \overset {w}{\rightsquigarrow } q|\geq 1\). qis said to beco-accessibleif there exists w ∈Σ∗and qf ∈ Qfsuch that\(| q \overset {w}{\rightsquigarrow } q_{f}|\geq 1\). If all its states are accessible and co-accessible, then G is said to be trim.
If, for all w ∈Σ∗, there exists a polynomial function\(P:\mathbb {N} \rightarrow \mathbb {N}\), such that there is at most P(|w|) paths in\(Q_{i} \overset {w}{\rightsquigarrow } Q_{f}\), thenGis said to be polynomially ambiguous.
A max-plus automaton is polynomially ambiguous iff for all q ∈ Q and for all w ∈Σ∗, there is at most one path in \(q \overset {w}{\rightsquigarrow } q\) (see Kirsten 2008).
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Acknowledgements
We thank the anonymous reviewers for their insightful comments and suggestions. The formulation related to tropical polyhedra given in Lemma 6 has been indicated by one anonymous referee.
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The work of Jan Komenda is supported by RVO 67985840 and GAČ R grant 19-06175J.
The previous version of this work was presented at MSR 2017, Marseille, France (Lahaye et al. 2017)
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Lahaye, S., Lai, A., Komenda, J. et al. A contribution to the determinization of max-plus automata. Discrete Event Dyn Syst 30, 155–174 (2020). https://doi.org/10.1007/s10626-019-00309-6
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DOI: https://doi.org/10.1007/s10626-019-00309-6