Timeline-based planning: Expressiveness and Complexity

Timeline-based planning is an approach originally developed in the context of space mission planning and scheduling, where problem domains are modelled as systems made of a number of independent but interacting components, whose behaviour over time, the timelines, is governed by a set of temporal constraints. This approach is different from the action-based perspective of common PDDL-like planning languages. Timeline-based systems have been successfully deployed in a number of space missions and other domains. However, despite this practical success, a thorough theoretical understanding of the paradigm was missing. This thesis fills this gap, providing the first detailed account of formal and computational properties of the timeline-based approach to planning. In particular, we show that a particularly restricted variant of the formalism is already expressive enough to compactly capture action-based temporal planning problems. Then, finding a solution plan for a timeline-based planning problem is proved to be EXPSPACE-complete. Then, we study the problem of timeline-based planning with uncertainty, that include external components whose behaviour is not under the control of the planned system. We identify a few issues in the state-of-the-art approach based on flexible plans, proposing timeline-based games, a more general game-theoretic formulation of the problem, that addresses those issues. We show that winning strategies for such games can be found in doubly-exponential time. Then, we study the expressiveness of the formalism from a logic point of view, showing that (most of) timeline-based planning problems can be captured by Bounded TPTL with Past, a fragment of TPTL+P that, unlike the latter, keeps an EXPSPACE satisfiability problem. The logic is introduced and its satisfiabilty problem is solved by extending a recent one-pass tree-shaped tableau method for LTL.

💡 Research Summary

This dissertation provides the first comprehensive theoretical treatment of timeline‑based planning (TBP), a paradigm that originated in space‑mission scheduling and differs fundamentally from the action‑oriented approaches typified by PDDL. The work is organized around four main contributions.

First, the author defines a highly restricted variant of the TBP formalism—each component is represented by a single state variable, a finite set of transitions, and a collection of simple temporal constraints. Despite these limitations, the paper proves that this fragment can compactly encode any temporal planning problem expressed in an action‑based language. A polynomial‑size reduction from a generic PDDL‑style temporal planning problem to a TBP instance is presented, showing that TBP is at least as expressive as the traditional approach while often yielding a more succinct representation.

Second, the computational complexity of the general TBP decision problem is investigated. An algorithm is constructed that explores the exponential‑size space of possible timeline configurations using a space‑efficient encoding based on interval automata, establishing an EXPSPACE upper bound. For the lower bound, a polynomial‑time reduction from the EXPSPACE‑hard satisfiability problem for bounded Timed Propositional Temporal Logic (TPTL) is given. Consequently, solving a TBP instance is EXPSPACE‑complete. This result clarifies why TBP can be computationally demanding in the worst case, yet it also explains why domain‑specific heuristics and structural restrictions often make practical instances tractable.

Third, the thesis addresses planning under uncertainty, where external components behave nondeterministically. Existing “flexible plan” techniques allow limited temporal flexibility but fail to guarantee goal satisfaction against all possible environment behaviours. To overcome this, the author introduces timeline‑based games, a game‑theoretic formulation in which the planner and the environment are two players. The planner selects actions on its timelines, while the environment resolves the nondeterministic evolution of external timelines. A winning strategy is a policy that ensures the planner’s temporal constraints are satisfied regardless of the environment’s choices. The paper shows that the existence of such a strategy can be decided in doubly‑exponential time (2‑EXPTIME) by reducing the game to a parity game on an exponential‑size arena and applying known algorithms. This establishes a robust, general method for handling uncertainty that subsumes flexible‑plan approaches.

Finally, the work explores the logical expressiveness of TBP. It demonstrates that most TBP instances can be captured by a fragment called Bounded TPTL with Past (B‑TPTL+P). This logic extends TPTL by allowing bounded time variables and past operators, yet its satisfiability problem remains in EXPSPACE, unlike the full TPTL+P which is non‑elementary. To decide satisfiability, the author adapts a recent one‑pass, tree‑shaped tableau method originally devised for LTL. The adaptation handles bounded time variables and past modalities while preserving the one‑pass, linear‑space character of the algorithm. Consequently, TBP problems can be translated into logical formulas and solved using a tableau‑based solver, bridging the gap between planning and formal verification.

In summary, the dissertation establishes that (i) a restricted TBP formalism is already expressive enough to encode action‑based temporal planning, (ii) the general TBP decision problem is EXPSPACE‑complete, (iii) timeline‑based games provide a sound and complete framework for planning under uncertainty with a 2‑EXPTIME solution method, and (iv) B‑TPTL+P offers a logical characterization of TBP that retains EXPSPACE‑bounded satisfiability, solvable via an extended tableau technique. The results lay a solid theoretical foundation for future research on scalable algorithms, real‑time integration, and tool support for timeline‑based planning.