The Formal Semantics of Processes in PDDL

One reason why autonomous processes have not been officially incorporated into PDDL is that there is no agreed-upon semantics for them. I propose a semantics with interpretations grounded on a branching time structure. A model for a PDDL domain is then simply an interpretation that makes all its axioms, action definitions, and process definitions true. In particular, a process definition is true if and only if over every interval in which its condition is true, the process is active and all its effects occur at the right times. The Semantics Problem for PDDL In (Fox & Long 2001a), Maria Fox and Derek Long proposed extensions to PDDL for describing processes, that is, activities that could go on independent of what the executor of plans did. They also described a semantics for processes in terms of hybrid automata (Henzinger 1996). Their proposal was not adopted by the rules committee for the third International Planning Competition, mainly because most of the members preferred to focus on the narrower extension to “durative actions,” actions that took a nonzero amount of time. Another potential reason is that the semantics of processes were so complex that no one really understood them. A semantics for all of PDDL is hard to lay out, for a couple of reasons. One is that the requirement-flag system turns it into a family of languages with quite different behavior. For instance, the :open-world flag transforms the language from one in which not is handled via “negation as failure” into god-knows-what. To sidestep this complexity, I will make a variety of simplifying omissions and deliberate oversights: I will ignore the possibility of action :expansions. There will be no formalization of the fact that propositions persist in truth value until some event changes them. :vars fields in action and process definitions will be left out. I will assume effects are simple conjunctions of literals. There are no secondary preconditions (“:whens”) or universally quantified effects. I won’t discuss durative actions. They can be defined in terms of processes, as discussed by (Fox & Long 2001b) and (McDermott 2003). Fox and Long (Fox & Long 2001b) allow for “events,” which occur autonomously, like processes, but are instantaneous, like primitive actions. I won’t discuss these, although they present no particular problem. Except for the presence of autonomous processes, we’ll stay in the “classical milieu,” and in particular assume that the planner knows everything about the initial situation and the consequences of events and processes. There’s no nondeterminism, and no reason to use sensors to query the world. The Structure of Time It’s fairly traditional (McDermott 1982; 1985) to think of an action term as denoting a set of intervals; intuitively, these are the intervals over which the action occurs. To make this precise, we have to specify what we mean by “interval.” A date is a pair 〈r, i〉, where r is a nonnegative real number and i is a natural number. A situation is a mapping from propositions to truth values. A date range function is a function from real numbers to natural numbers. A situation continuum is a pair 〈C, h〉, where h is a date range function and C is a timeline, that is, a mapping from {〈r, i〉 | r ∈ nonnegative reals and 0 ≤ i < h(r)} to situations. If d1 = 〈r1, i1〉 and d2 = 〈r2, i2〉, then d1 < d2 if r1 < r2 or r1 = r2 and i1 < i2. A date 〈r, d〉 is in 〈C, h〉 if r ≥ 0 and 0 ≤ d < h(r). A situation s is in 〈C, h〉 if there exists a date d in 〈C, h〉 such that C(d) = s. The intuitive meaning of these definitions is that at any point r in time there can be a series of zero or more actions taken by the target agent (i.e., the hypothetical agent that executes plans). The range function says how many actions are actually taken at point r in that continuum. Each action takes an infinitesimal amount of time, so that “just after” time r there can be an arbitrary number h(r) of actions that precede all time points with times r′ > r. (This picture is reminiscent of nonstandard analysis (Robinson 1979).) Using this framework, a purely classical plan might be specified by giving an action A(i) for all dates 〈0, i〉 where i < h(0), and h(0) is the length of the plan. The plan is feasible if A(0) is feasible in the initial situation, and A(i + 1) is feasible in C(0, i). The plan achieves a goal G if G is true in C(0, h(0)). It may sound odd to imagine the entire