Quantitative -calculus and CTL dened over constraint semirings 1

Model checking and temporal logics are boolean. The answer to the model checking question does a system satisfy a property? is either true or false, and properties expressed in temporal logics are dened over boolean propositions. While this classic approach is enough to specify and verify boolean temporal properties, it does not allow to reason about quantitative aspects of systems. Some quantitative extensions of temporal logics has been already proposed, especially in the context of probabilistic systems. They allow to answer questions like with which probability does a system satisfy a property? We present a generalization of two well-known temporal logics: CTL and the calculus. Both extensions are dened over c-semirings, an algebraic structure that captures quantitative aspects like quality of service or soft constraints. Basically, a csemiring consists of a domain, an additive operation and a multiplicative operation, which satisfy some properties. We present the semantics of the extended logics over transition systems, where a formula is interpreted as a mapping from the set of states to the domain of the c-semiring, and show that the usual connection between CTL and -calculus does not hold in general. In addition, we reason about the complexity of computing the logics and illustrate some applications of our framework, including boolean model checking.

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