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.

中文翻译：

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时间综合中自动机最小化的威力

反应式综合是根据形式规范自动构建反应式系统的问题，并保证系统的执行与规范一致。规范通常用时序逻辑来描述。某些类型的规范可以转换为确定性有限自动机 (DFA) 作为合成的中间步骤，从而受益于 DFA 可以在多项式时间内完全最小化的事实。在这项工作中，我们在时间合成的背景下研究了 DFA 最小化算法。特别是，我们比较了 Hopcroft 和 Brzozowski 最小化算法，使它们从时间逻辑公式开始，并将它们集成到现有的时间综合框架中。虽然早期比较随机生成自动机的两种算法的研究得出结论，这两种算法都不占主导地位，但我们的结果表明，在时间合成的背景下，Hopcroft 算法是最佳选择。分析结果，我们观察到 Brzozowski 算法性能不佳的原因是理论与实践之间的差异。该算法首先为规范的逆向语言构造一个DFA，然后执行一系列操作将其转化为规范本身的最小DFA。理论上，反向语言的 DFA 可以呈指数级减小，这可能会使该算法比直接为原始规范构建 DFA 更有效。然而在实践中，