Abstract
A large class of timed discrete event systems can be modeled by means of (max,+) automata, that is automata with weights in the so-called (max,+) algebra. In this contribution, specific recursive equations over (max,+) and (min,+) algebras are shown to be suitable for describing extremal behaviors of (max,+) automata. Several pertinent performance indicators can be easily derived or approximated from these representations with a low computation complexity. It is also shown how to define inputs which model exogenous influences on their dynamic evolution, and a new approach for the control of (max,+) automata is proposed.
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Notes
Beyond the scope of discrete event systems, there are important applications for image and speech processing, and more generally, weighted automata constitute a theoretical object which is extensively studied (see (Droste and Kuich 2009) for an overview).
As usual, we will often omit the multiplication sign ⊗, that is for example we write AB instead of A ⊗ B.
It must be noted that there exist automata in which |Q| < |H|, as well as others in which |Q| > |H|, and the comparison between methods complexity may then lead to contradictory conclusions.
The result is actually stated for a subclass of systems (see discussion in Section 3.4) with direct extension to any (max,+) automaton.
Note that, unlike (boolean) finite automata, nondeterministic (max,+) automata cannot always be determinized, that is transformed into equivalent deterministic (max,+) automata (see e.g. (Gaubert 1995; Lombardy and Sakarovitch 2006)). Despite the fact that it was studied by numerous researchers, this problem is still rather open, and to the best of our knowledge, strongly unambiguous (max,+) automata don’t satisfy necessarily conditions which are known in the literature to be sufficient for determinization.
At most one token can be in a place at any time.
At most a bounded number of tokens can be in a place at any time.
The case of unobservable events should be considered in future work.
Please note that the complexity of this procedure remains to be shown.
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Boukra, R., Lahaye, S. & Boimond, JL. New representations for (max,+) automata with applications to performance evaluation and control of discrete event systems. Discrete Event Dyn Syst 25, 295–322 (2015). https://doi.org/10.1007/s10626-013-0178-y
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DOI: https://doi.org/10.1007/s10626-013-0178-y