We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time $\mathcal{O}((m+n) \log n)$ on systems with $n$ states and $m$ transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was $\mathcal{O}(m n)$.
翻译:我们为构建公式提供了一种通用算法,该公式在诸如非确定性、概率或加权等等同型体系中按行为区分等值状态;过渡型的通用性是通过与煤热布拉合作,在通用煤热布拉范式中为一组配方。对于一个特定系统中的每个行为等同类,我们为该类中的每个行为等同类构建一个公式。对于以美元和美元过渡为单位的系统,算法即时计算公式、过渡系统、标记的Markov链条和许多其他类型的系统。环境逻辑是一种模式逻辑,其模式化逻辑是一般地从真菌中提取的模式;这些模式可以在后处理步骤中系统地转换成自定制模式。新的算法建立在现有的煤热对称配方精度精细算法上。在时间里运行$\macal{O}(m+n)\log n) 。对于以美元和美元过渡为单位的系统,同样的调制约束适用于其构建的公式大小。这改善了运行时间和公式模式的定制组合组合,在以往的特定变压中改进了时间和公式的缩缩数,在以前的系统上是以前的变压。