講座內(nèi)容：(Max,+) automata are weighted automata with weights(multiplicities) in the $(mathbb{R} cup {-infty},max,+)$ semiring. They have a strong expressive power in terms of timed Petri nets: every 1-safe timed Petri net can be represented by a special (max,+) automaton, called heap model(or heap automaton). We have proposed recently a direct and compositional transformation of (max,+) automata into timed Petri nets. (Max,+) automata and corresponding formal power series can be studied both algebraically and co-algebraically. Co-algebraic approach in the category of Sets, where formal power series form a final co-algebra(with weighted automaton structure defined in terms of left quotiens) is limited to deterministic weighted automata. On the other hand, the co-algebraic approach in the category of Vec of vector spaces can be used for general nondeterministic weighted automata.

The synchronous product of (max,+) automata will be presented that clearly separates the quantitative (timing) and logical (support language) aspects. Interestingly, the state explosion problem is not an issue for the quantitative aspect as the state set of the synchronous product is the union and not the cartesian product of local carrier sets. However, resulting nondeterministic automata are difficult to use in performance evaluation and supervisory control. Therefore, we have studied determinization of (max, +) automata(which is not always possible in terms of a finite state automaton) and proposed new sufficient conditions for termination of the determinization procedure based on the normalization.

We will also discuss an alternative approach based on the concept of fairness well known in the concurrency theory.