Making weighted containment feasible: A heuristic based on simulation and abstraction

Weighted automata map input words to real numbers and are useful in reasoning about quantitative systems and specifications. The containment problem for weighted automata asks, given two weighted automata A and B, whether for all words w, the value that A assigns to w is less than or equal to the value B assigns to w. The problem is of great practical interest, yet is known to be undecidable. Efforts to approximate weighted containment by weighted variants of the simulation pre-order still have to cope with large state spaces. One of the leading approaches for coping with large state spaces is abstraction. We introduce an abstraction-refinement paradigm for weighted automata and show that it nicely combines with weighted simulation, giving rise to a feasible approach for the containment problem. The weighted-simulation pre-order we define is based on a quantitative two-player game, and the technical challenge in the setting origins from the fact the values that the automata assign to words are unbounded. The abstraction-refinement paradigm is based on under- and over-approximation of the automata, where approximation, and hence also the refinement steps, refer not only to the languages of the automata but also to the values they assign to words.