  In this paper we present an algorithm for performing runtime verification of
a bounded temporal logic over timed runs. The algorithm consists of three
elements. First, the bounded temporal formula to be verified is translated into
a monadic first-order logic over difference inequalities, which we call monadic
difference logic. Second, at each step of the timed run, the monadic difference
formula is modified by computing a quotient with the state and time of that
step. Third, the resulting formula is checked for being a tautology or being
unsatisfiable by a decision procedure for monadic difference logic.
  We further provide a simple decision procedure for monadic difference logic
based on the data structure Difference Decision Diagrams. The algorithm is
complete in a very strong sense on a subclass of temporal formulae
characterized as homogeneously monadic and it is approximate on other formulae.
The approximation comes from the fact that not all unsatisfiable or
tautological formulae are recognised at the earliest possible time of the
runtime verification.
  Contrary to existing approaches, the presented algorithms do not work by
syntactic rewriting but employ efficient decision structures which make them
applicable in real applications within for instance business software.
