Decidability and Undecidability Results for Planning with Numerical State Variables

These days, propositional planning can be considered a quite well-understood problem. Good algorithms are known that can solve a wealth of very different and sometimes challenging planning tasks, and theoretical computational properties of both general STRIPS-style planning and the best-known benchmark problems have been established. However, propositional planning has a major drawback: The formalism is too weak to allow for the easy encoding of many genuinely interesting planning problems, specifically those involving numbers. A recent effort to enhance the PDDL planning language to cope with (among other additions) numerical state variables, to be used at the third international planning competition, has increased interest in these issues. In this contribution, we analyze “STRIPS with numbers” from a theoretical point of view. Specifically, we show that the introduction of numerical state variables makes the planning problem undecidable in the general case and many restrictions thereof and identify special cases for which we can provide decidability results.