The coalgebraic $\mu$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $\mu$-calculus includes an exponential-time upper bound on satisfiability checking, which however relies on the availability of tableau rules for the next-step modalities that are sufficiently well-behaved in a formally defined sense; in particular, rule matches need to be representable by polynomial-sized codes, and the sequent duals of the rules need to absorb cut. While such rule sets have been identified for some important cases, they are not known to exist in all cases of interest, in particular ones involving either integer weights as in the graded $\mu$-calculus, or real-valued weights in combination with non-linear arithmetic. In the present work, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying one-step logic, roughly described as the nesting-free next-step fragment of the logic. The bound is realized by a generic global caching algorithm that supports on-the-fly satisfiability checking. Notably, our approach directly accommodates unguarded formulae, and thus avoids use of the guardedness transformation. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $\mu$-calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $\mu$-calculus with polynomial inequalities.
翻译:燃煤热量 $\ mu美元计算仪提供了一种通用的语义框架,用于对其分支类型超出标准关系设置的系统进行固定点逻辑,例如概率、加权或基于游戏的系统进行固定点逻辑,例如,概率、加权或基于游戏的系统。以前关于煤热热量 $\ mu美元计算仪的工作包括一个指数-时间的上限,这是对可食性检查的上限,然而,这取决于是否有表格规则,对于下一步模式而言,在正式定义的意义上已经足够完善;特别是,规则匹配需要由多元规模的代码来代表,而规则的顺序的序列性双重性则需要削减。虽然已经为一些重要案例确定了这样的规则组,但并不是在所有感兴趣的案例中都存在这样的规则组,特别是涉及等级 $\ mu美元 计算仪表的整数,或者与非正值的计算方法相配合的真值权重。在目前的工作中,我们证明在更笼统的假设下也存在相同的上层复杂性,具体地说,也就是(简单得多的) 直径直径直径的直径直径的逻辑分析。