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

From FSL

This work has been published both in conference proceedings (CAV'06) and as a technical report. The technical report additionally contains a rewriting (using Maude) implementation of the translation of ALTL into LTL, together with a large Allen specification part of a planning application.

## CAV'06

**Allen Linear (Interval) Temporal Logic -Translation to LTL and Monitor Synthesis-**- Grigore Rosu and Saddek Bensalem
, LNCS 4144, pp 263-277, 2006**CAV'06**

*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 temporal logic (LTL). A discrete variant of ATL is defined, called Allen linear temporal logic (ALTL), whose models are -sequences of timepoints. It is shown that any ALTL formula can be linearly translated into an equivalent LTL formula, thus enabling the use of LTL techniques on ALTL requirements. This translation also implies the NP-completeness of ALTL satisfiability. Then the problem of monitoring ALTL requirements is investigated, 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 implemented and experimented with in the context of planning applications.

## Technical report

**Allen Linear (Interval) Temporal Logic -Translation to LTL and Monitor Synthesis-**- Grigore Rosu and Saddek Bensalem
UIUCDCS-R-2006-2681. January 2006.**Technical Report**

*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 temporal logic (LTL). A discrete variant of ATL is defined, called Allen linear temporal logic (ALTL), whose models are -sequences of timepoints, like in LTL. It is shown that any ALTL formula can be linearly translated into an equivalent LTL formula, thus enabling the use of LTL techniques and tools when requirements are expressed in ALTL. This translation also implies the NP-completeness of ATL satisfiability. Then the monitoring problem for ALTL is discussed, showing that it is NP-complete despite the fact that the similar problem for LTL is EXPSPACE-complete. An effective monitoring algorithm for ALTL is given, which has been implemented and experimented with in the context of planning applications.