On expressive completeness of duration and mean value calculi (extended abstract)

This paper compares the expressive power of first-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of two formalisms for the specification 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 first-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 definition 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 propositional 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 first-order logic of order.