The Planning Spectrum - One, Two, Three, Infinity

• Authors: Marco Pistore;Moshe Y. Vardi
• Venue: LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
• Year: 2003

Linear Temporal Logic (LTL) is widely used for definingconditions on the execution paths of dynamic systems. Inthe case of dynamic systems that allow for nondeterministicevolutions, one has to specify, along with an LTL formula\varphi, which are the paths that are required to satisfy theformula. Two extreme cases are the universal interpretationA.\varphi, which requires to satisfy the formula for all thepossible execution paths, and the existential interpretationE.\varphi, which requires to satisfy the formula for some executionpaths. When LTL is applied to the definition of goals inplanning problems on nondeterministic domains, these twoextreme cases are too restrictive. It is often impossible todevelop plans that achieve the goal in all the nondeterministicevolutions of a system, and it is too weak to require thatthe goal is satisfied by some executions. In this paper weexplore alternative interpretations of an LTL formula thatare between these extreme cases. We define a new languagethat permits an arbitrary combination of the A and E quantifiers,thus allowing, for instance, to require that each finiteexecution can be extended to an execution satisfying an LTLformula (AE.\varphi), or that there is some finite execution whoseextensions all satisfy an LTL formula (EA.\varphi). We show thatonly eight of these combinations of path quantifiers are relevant,corresponding to an alternation of the quantifiers oflength one (A and E), two (AE and EA), three (AEA andEAE), and infinity ((AE)\omega and (EA)\omega). We also presents a planning algorithm for the new language, that is based onan automata-theoretic approach, and studies its complexity.