Constructing Conditional Plans by a Theorem-Prover (1105.5465v1)

Abstract: The research on conditional planning rejects the assumptions that there is no uncertainty or incompleteness of knowledge with respect to the state and changes of the system the plans operate on. Without these assumptions the sequences of operations that achieve the goals depend on the initial state and the outcomes of nondeterministic changes in the system. This setting raises the questions of how to represent the plans and how to perform plan search. The answers are quite different from those in the simpler classical framework. In this paper, we approach conditional planning from a new viewpoint that is motivated by the use of satisfiability algorithms in classical planning. Translating conditional planning to formulae in the propositional logic is not feasible because of inherent computational limitations. Instead, we translate conditional planning to quantified Boolean formulae. We discuss three formalizations of conditional planning as quantified Boolean formulae, and present experimental results obtained with a theorem-prover.

• The paper frames conditional planning under uncertainty as a theorem-proving task using Quantified Boolean Formulae (QBF), moving beyond classical planning assumptions.
• It introduces several QBF formalizations to represent and handle the complexity of conditional plans evolving in nondeterministic environments.
• The work includes the design and empirical evaluation of a QBF theorem prover specifically developed to construct and verify these complex conditional plans.

Constructing Conditional Plans by a Theorem Prover: A Summary

This paper, "Constructing Conditional Plans by a Theorem Prover" by Jussi Rintanen, explores the domain of conditional planning under uncertainty and incomplete knowledge. The work challenges classical assumptions within automated planning, specifically those that presume a completely predictable and known environment, by introducing techniques that accommodate nondeterministic changes and multiple initial states.

The author approaches conditional planning as an automated reasoning task rather than extending traditional planning algorithms. This task is framed using quantified Boolean formulae (QBF), representing planning problems in a structured logical framework that allows for handling complex planning scenarios beyond simple linear sequences of actions. The general difficulty posed by conditional planning is formalized and analyzed using computational complexity theory, which highlights that certain conditional planning problems are unlikely to fit within the NP complexity class.

Translating Conditional Plans to QBF

The paper outlines several formalizations of translating conditional planning problems into QBF. These translations tackle the representation of plan executions, internal states, and contingent conditions in the planning process:

1. Plans with Unrestricted Transition Functions: This representation uses the formalism of finite automata to detail how plans evolve, introducing internal state transitions based on observable propositions. Such plans adapt dynamically, thus accommodating nondeterministic effects in the environment.
2. Plans as Sequences of Sets of Iterated Operators: Within this framework, plans consist of sequences where operators can be iteratively applied until conditions necessitate transitioning to a successor state.
3. Plans as Sequences of Sets of Simple Operators: This formalization represents plans as series of operators executed in a linear fashion, with each execution path leading to a potential goal state.

These QBF formalizations directly address the complexity inherent in conditional planning tasks, providing the flexibility needed to handle various environmental uncertainties. The scope of planning expands from mere sequence manipulation to encompass reasoning about quantifiers related to contingencies and outcomes.

Theorem Prover for QBF

A significant contribution of the paper is the development and implementation of a theorem prover designed to evaluate quantified Boolean formulae. This prover extends the Davis-Putnam procedure, commonly used for propositional logic satisfiability, to handle variations in quantifiers, thus supporting the verification of conditional plans. The paper provides detailed insights into the heuristics and optimizations incorporated into the theorem prover to enhance its performance.

Empirical Results and Discussion

The paper includes experimental evaluations demonstrating the application of the theorem prover to conditional planning problems, such as the classic blocks world and a series of benchmark conditions involving dynamically evolving environments. Notably, the paper illustrates scenarios where the theorem prover efficiently generates conditional plans that systematically consider all possible contingencies, emphasizing the computational feasibility of the proposed approach.

Implications and Future Directions

Rintanen's work contributes significantly to the field of AI planning by proposing a method that equips planners to operate under uncertainty and nondeterminism effectively. The theoretical grounding of the approach in quantified Boolean logic, coupled with the practical implementation of a theorem-proving system, offers a robust foundation for future research.

Future work may explore further computational optimizations and broader applications of the methodology in diverse domains involving complex interactions and planning under uncertainty. Additionally, advancing theorem-proving techniques for QBF in line with developments in satisfiability algorithms could enhance the scalability and efficiency of conditional planners.

In conclusion, the paper offers an insightful exploration of the challenges and solutions in constructing conditional plans through logical formalism and computational reasoning, paving the way for more advanced planning frameworks that can robustly handle real-world uncertainty.