Allen Linear (Interval) Temporal Logic –Translation to LTL and Monitor Synthesis–

Abstract. The relationship between two well established formalisms for temporal reasoning is first investigated, namely between Allen’s interval algebra (or Allen’s temporal logic, abbreviated ATL) and linear tempo-ral logic (LTL). A discrete variant of ATL is defined, called Allen linear temporal logic (ALTL), whose models are ω-sequences of timepoints. It is shown that any ALTL formula can be linearly translated into an equiv-alent LTL formula, thus enabling the use of LTL techniques on ALTL requirements. This translation also implies the NP-completeness of ATL satisfiability. Then the problem of monitoring ALTL requirements is in-vestigated, showing that it reduces to checking satisfiability; the similar problem for unrestricted LTL is known to require exponential space. An effective monitoring algorithm for ALTL is given, which has been imple-mented and experimented with in the context of planning applications.