On the Power of Automata Minimization in Temporal Synthesis.

Reactive synthesis is the problem of automatically constructing a reactive system from a formal specification, with the guarantee that the executions of the system align with the specification. The specification is often described in temporal logic. Some types of specifications can be converted into deterministic finite automata (DFA) as an intermediate step in synthesis, thus benefiting from the fact that DFAs can be fully minimized in polynomial time. In this work we investigate DFA-minimization algorithms in the context of temporal synthesis. In particular, we compare between the Hopcroft and Brzozowski minimization algorithms, adapting them to start from temporal-logic formulas and integrating them into an existing temporal synthesis framework. While earlier studies comparing the two algorithms for randomly-generated automata concluded that neither algorithm dominates, our results suggest that in the context of temporal-synthesis, Hopcroft's algorithm is the best choice. Analyzing the results, we observe that the reason for the poor performance of Brzozowski's algorithm is a discrepancy between theory and practice. This algorithm first constructs a DFA for the reverse language of the specification and then performs a series of operations to transform it into a minimal DFA for the specification itself. In theory, the DFA for the reverse language can be exponentially smaller, which would potentially make this algorithm more efficient than directly constructing the DFA for the original specification. In practice, however, we find that the reverse DFA is often of comparable size or even larger, which cancels the advantage that this approach could have.