Quantitative mu-calculus and CTL Based on Constraint Semirings

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 defined 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 [mu]-calculus. Both extensions are defined over c-semirings, an algebraic structure that captures many problems and that has been proposed as a general framework for soft constraint satisfaction problems (CSP). Basically, a c-semiring 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 [mu]-calculus does not hold in general. In addition, we reason about the feasibility of computing the logics and illustrate some applications of our framework, including boolean model checking

A modal #mu#-calculus for durational transition systems

Durational transition systems are finite transition systems where every transition is additionally equipped with a duration. We consider the problem of interpreting #mu#-formulas over durational transition systems. In case the formula contains only operations minimum, maxium, addition, and sequencing, we show that the interpretation is not only computable but (up to a linear factor) as efficiently computable as the interpretation of ordinary #mu#-formulas over finite transition systems. We extend our methods to the case where one-sided conditionals are allowed as well. (orig.)Available from TIB Hannover: RR 1843(95-08) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman