Expressive Completeness ofDuration and Mean Value Calculi

This paper compares the expressive power of rst-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of two formalisms for the speciication of real-time systems, the propositional versions of duration and mean value calculi. Our results show that the propositional mean value calculus is expressively complete for monadic rst-order logic of order. A new semantics for the chop operator used in these real-time formalisms is also proposed, and the expressive completeness results achieved in the paper indicate that the new deenition might be more natural than the original one. We provide a characterization of the expressive power of the propositional duration calculus and investigate the connections between the propo-sitional duration calculus and star-free regular expressions. Finally, we show that there exists at least an exponential gap between the succinctness of the propositional duration (mean value) calculus and that of monadic rst-order logic of order.