The Expressive Power of Temporal and First-Order Metric Logics

Abstract

The First-Order Monadic Logic of Order ($\textit{FO}[<]$) is a prominent logic for the specification of properties of systems evolving in time. The celebrated result of Kamp [14] states that a temporal logic with just two modalities Until and Since has the same expressive power as $\textit{FO}[<]$ over the standard discrete time of naturals and continuous time of reals. An influential consequence of Kamp’s theorem is that this temporal logic has emerged as the canonical Linear Time Temporal Logic ($LTL )$. Neither $LTL$ nor $\textit{FO}[<]$ can express over the reals properties like P holds exactly after one unit of time. Such local metric properties are easily expressible in $\textit{FO}[<,+1]$ - the extension of $\textit{FO}[<]$ by +1 function. Hirshfeld and Rabinovich [10] proved that no temporal logic with a finite set of modalities has the same expressive power as $\textit{FO}[<,+1]$.

$\textit{FO}[<,+1]$ lacks expressive power to specify a natural global metric property “the current moment is an integer.” Surprisingly, we show that the extension of $\textit{FO}[<,+1]$ by a monadic predicate “x is an integer” is equivalent to a temporal logic with a finite set of modalities.