Component Properties of Forgetting and Progression in the Situation Calculus

Abstract

In many tasks related to reasoning about consequences of a logical theory, it is desirable to decompose the theory into a number of weakly-related or independent components. However, a theory may represent knowledge that is subject to change due to execution of actions that have effects on some properties mentioned in the theory. Having once computed a decomposition of a theory, one would like to know whether a decomposition has to be computed again in the theory obtained from taking into account changes resulting from execution of an action. In the paper, we address this problem in the scope of the situation calculus, where a change of an initial theory is related to the notion of progression. We undertake a study of the decomposability and inseparability properties known from the literature. We contribute by studying these properties wrt progression and the related notion of forgetting. We provide negative examples and identify cases when these properties are preserved under progression of initial theories and under forgetting in local-effect basic action theories of the situation calculus.

Progression of Decomposed Local-Effect Action Theories

“…In our paper, we adopt the following notion that was introduced in [38] and applied to the study of modularity in [17].…”

“…Therefore, potentially it is more computationally complex than syntactic partitioning, which splits a theory into syntactic subsets of axioms. However, the research on algorithmic properties of decomposability (see e.g., [39,9,17,35,38]) shows that deciding whether a theory is ∆-decomposable turns out to be not harder than deciding the entailment in the underlying logic. Studying the complexity of decomposability in different logics is an ongoing research topic.…”

“…Studying the complexity of decomposability in different logics is an ongoing research topic. An algorithm for computing decomposition components can be obtained, e.g., from a procedure of computing uniform interpolants (if the logic enjoys efficient uniform interpolation, see Proposition 2 in [38] and [35]), or by applying the technique of eliminating non-∆-symbols from the axioms of a theory. The technique is described in [17] for the logics EL and DL-Lite, which is further studied in [41] and can be extended to more expressive Description Logics.…”

“…Modularity of theories has been established as an important research topic in knowledge representation. It includes both theoretical and practical aspects of modularity of theories formulated in different logical languages (L), ranging from weak (but practical) description logics (DLs) such as EL and DL-Lite to more expressive logics [13,15,17,18,38,47], to cite a few. Surprisingly, this research topic is little explored in the context of reasoning about actions.…”

“…This paper considers the decomposability and inseparability properties of logical theories. These properties are well known in research on modularization in the area of knowledge representation [17,38,18,32], but have not been studied previously in the scope of the situation calculus. Both properties are concerned with subdividing theories into components to facilitate reasoning.…”