A Temporal Description Logic for Reasoning about Actions and Plans

A class of interval-based temporal languages for uniformly representing and reasoning about actions and plans is presented. Actions are represented by describing what is true while the action itself is occurring, and plans are constructed by temporally relating actions and world states. The temporal languages are members of the family of Description Logics, which are characterized by high expressivity combined with good computational properties. The subsumption problem for a class of temporal Description Logics is investigated and sound and complete decision procedures are given. The basic language TL-F is considered first: it is the composition of a temporal logic TL – able to express interval temporal networks – together with the non-temporal logic F – a Feature Description Logic. It is proven that subsumption in this language is an NP-complete problem. Then it is shown how to reason with the more expressive languages TLU-FU and TL-ALCF. The former adds disjunction both at the temporal and non-temporal sides of the language, the latter extends the non-temporal side with set-valued features (i.e., roles) and a propositionally complete language.

💡 Research Summary

The paper introduces a family of interval‑based temporal description logics designed to represent and reason uniformly about actions and plans. The core language, TL‑F, is obtained by composing a temporal logic TL—capable of expressing Allen’s interval algebra and metric constraints—with a non‑temporal feature description logic F (a variant of ALC with functional features). In TL‑F, actions are modeled by describing the world’s properties that hold during the interval in which the action occurs; plans are built by linking such action intervals with interval‑based constraints on world states.

The authors formalize the syntax of TL‑F, providing temporal qualifiers (at, sometime, alltime) and the usual DL constructors (conjunction, universal and existential restrictions, features, roles). Semantically, an interpretation consists of a domain of individuals together with a mapping of interval variables to concrete time intervals; the interpretation must satisfy both the temporal constraints (as a consistent interval network) and the DL constraints (as usual set‑theoretic meanings of concepts and features). Subsumption (concept inclusion) and instance recognition are defined with respect to all models of a knowledge base.

A central technical contribution is the analysis of the subsumption problem for TL‑F. By separating the temporal component (interval network consistency) from the DL component (standard DL subsumption), the authors show that TL‑F subsumption can be decided by first checking the consistency of the interval constraints (a polynomial‑time CSP) and then applying a DL subsumption test. Since the underlying DL (ALC‑F) is already NP‑complete, the combined problem is proved to be NP‑complete. The NP‑hardness proof reduces 3‑SAT to TL‑F subsumption, demonstrating that the temporal layer does not increase the worst‑case complexity.

The paper then illustrates how TL‑F can encode classic action‑planning domains such as cooking and the blocks world, showing that concurrent, overlapping, and persistent effects can be captured naturally. A sound and complete calculus for TL‑F subsumption is presented, consisting of normalization, temporal constraint propagation, and DL rule application. The calculus forms the basis for reasoning in more expressive extensions.

Two extensions are explored. TLU‑FU adds disjunction both on the temporal side (allowing alternative interval constraints) and on the non‑temporal side (allowing concept disjunction). Subsumption remains decidable; the algorithm proceeds by case‑splitting on disjuncts and recombining results. TL‑ALCF further enriches the non‑temporal part with set‑valued features (roles) and a full propositional language, yielding a logic comparable to ALC with roles. Despite the added expressivity, subsumption stays decidable, though the complexity remains at least NP‑hard. Dedicated decision procedures are sketched for both extensions.

To address the frame problem, the authors introduce homogeneity and persistence operators. Homogeneity asserts that a property holds uniformly over an interval, while persistence captures facts that remain unchanged across an action’s execution. These operators allow the representation of inertia and the explicit specification of unchanged world parts, offering a partial solution to the classic frame problem within the DL framework.

The related‑work section surveys other temporal extensions of description logics, interval‑based logics, and situation‑calculus approaches, highlighting how TL‑F uniquely combines interval semantics with DL’s subsumption‑oriented reasoning. The paper concludes by emphasizing the theoretical significance of a decidable, expressive temporal DL for plan description, retrieval, and recognition, and outlines future directions such as optimized implementations, handling of large plan libraries, and integration with probabilistic reasoning.