A Semantics for ADL as Progression in the Situation Calculus

Lin and Reiter were the first to propose a purely declarative semantics of STRIPS by relating the update of a STRIPS database to a form of progression in the situation calculus. In this paper we show that a corresponding result can be obtained also for ADL. We do so using a variant of the situation calculus recently proposed by Lakemeyer and Levesque. Compared to Lin and Reiter this leads to a simpler technical treatment, including a new notion of progression. Introduction Lin and Reiter (Lin & Reiter 1997) were the first to propose a purely declarative semantics of STRIPS (Fikes & Nilsson 1971) by relating the update of a STRIPS database to a form of progression of a corresponding situation-calculus theory. More precisely, they show that when translating STRIPS planning problems into basic action theories of Reiter’s situation calculus (Reiter 2001), then the STRIPS mechanism of adding and deleting literals after an actionA is performed is correct in the sense that the conclusions about the future that can be drawn using the updated theory are the same as those drawn from the theory before the update. Given that today’s planning languages like PDDL (Fox & Long 2003) go well beyond STRIPS, it seems natural to ask whether Lin and Reiter’s results can be extended along these lines. One advantage would be to have a uniform framework for specifying the semantics of planning languages. Perhaps more importantly, as we will argue in more detail at the end of the paper, this would also provide a foundation to bring together the planning and action language paradigms, which have largely developed independently after the invention of STRIPS. In this paper we propose a first step in this direction by considering the ADL fragment of PDDL (Pednault 1989; 1994). In contrast to Lin and Reiter, we use a new variant of the situation calculus called ES recently proposed by Lakemeyer and Levesque (Lakemeyer & Levesque 2004). This has at least two advantages: for one, there is no need to switch the language when translating formulas of the planning language into the new situation calculus because there are no situation terms to worry about (in ES, situations occur only in the semantics); for another, semantic definitions like progression become simpler as it is no longer necessary to consider arbitrary first-order structures but only certain ones over a fixed universe of discourse. As Lakemeyer and Levesque recently showed (Lakemeyer & Levesque 2005), these simplifications do not lead to a loss of expressiveness. In fact, they show that second-order ES captures precisely the non-epistemic fragment of the situation calculus and the action language Golog (Levesque et al. 1997).1 The main technical contributions of this paper are the following: we show how to translate an ADL problem description into a basic action theory of ES; we develop a notion of progression, which is similar to that of Lin and Reiter but also simpler given the semantics underlying ES; finally, we establish that updating an ADL database (called a state) after performing an action is correct in the sense that the resulting state corresponds precisely to progressing the corresponding basic action theory. The result is obtained for both closed and open-world states. With the exception of Lin and Reiter (Lin & Reiter 1997), the approaches to giving semantics to planning languages have all been meta-theoretic. When Pednault introduced ADL (1989; 1994), he provided a semantics that defined operators as mappings between first-order structures that are defined by additions and deletions of tuples to the relations and functions of that structures. He presented a method of deriving a situation calculus axiomatization from ADL operator schema, but did not show the semantic correspondence between the two. Despite the fact that PDDL was built upon ADL, it was not until PDDL2.1 that a formal semantics was provided. The focus in (Fox & Long 2003) was more on formalizing the meaning of the newly introduced temporal extensions and concurrent actions; nonetheless, the predicatelogic subset of Fox and Long’s semantics represents a generalization of Lifschitz’ state transition semantics for STRIPS (1986). However, they compile conditional effects into the preconditions of the operators, propositionalize quantifiers and only consider the case of complete state descriptions. An exhaustive study of the expressiveness and compilability of different subsets of the propositional version of ADL is given in (Nebel 2000). The paper proceeds as follows. We first introduce ES and show how basic action theories are formulated in this logic. Next, we define ADL problem descriptions and provide a 1The correspondence with the full situation calculus is close but not exact. formal semantics by mapping them into basic action theories. We then define progression and establish the correctness of updating an ADL state with respect to progression. Before concluding, we give an outlook on applying the results to combine planning and the action language Golog.