Formally Verified Compositional Algorithms for Factored Transition Systems

Artificial Intelligence (AI) planning and model checking are two disciplines that found wide practical applications. It is often the case that a problem in those two fields concerns a transition system whose behaviour can be encoded in a digraph that models the system's state space. However, due to the very large size of state spaces of realistic systems, they are compactly represented as propositionally factored transition systems. These representations have the advantage of being exponentially smaller than the state space of the represented system. Many problems in AI~planning and model checking involve questions about state spaces, which correspond to graph theoretic questions on digraphs modelling the state spaces. However, existing techniques to answer those graph theoretic questions effectively require, in the worst case, constructing the digraph that models the state space, by expanding the propositionally factored representation of the syste\ m. This is not practical, if not impossible, in many cases because of the state space size compared to the factored representation. One common approach that is used to avoid constructing the state space is the compositional approach, where only smaller abstractions of the system at hand are processed and the given problem (e.g. reachability) is solved for them. Then, a solution for the problem on the concrete system is derived from the solutions of the problem on the abstract systems. The motivation of this approach is that, in the worst case, one need only construct the state spaces of the abstractions which can be exponentially smaller than the state space of the concrete system. We study the application of the compositional approach to two fundamental problems on transition systems: upper-bounding the topological properties (e.g. the largest distance between any two states, i.e. the diameter) of the state spa\ ce, and computing reachability between states. We provide new compositional algorithms to solve both problems by exploiting different structures of the given system. In addition to the use of an existing abstraction (usually referred to as projection) based on removing state space variables, we develop two new abstractions for use within our compositional algorithms. One of the new abstractions is also based on state variables, while the other is based on assignments to state variables. We theoretically and experimentally show that our new compositional algorithms improve the state-of-the-art in solving both problems, upper-bounding state space topological parameters and reachability. We designed the algorithms as well as formally verified them with the aid of an interactive theorem prover. This is the first application that we are aware of, for such a theorem prover based methodology to the design of new algorithms in either AI~planning or model checking

Using Plan Decomposition for Continuing Plan Optimisation and Macro Generation

This thesis addresses three problems in the field of classical AI planning: decomposing a plan into meaningful subplans, continuing plan quality optimisation, and macro generation for efficient planning. The importance and difficulty of each of these problems is outlined below. (1) Decomposing a plan into meaningful subplans can facilitate a number of postplan generation tasks, including plan quality optimisation and macro generation – the two key concerns of this thesis. However, conventional plan decomposition techniques are often unable to decompose plans because they consider dependencies among steps, rather than subplans. (2) Finding high quality plans for large planning problems is hard. Planners that guarantee optimal, or bounded suboptimal, plan quality often cannot solve them In one experiment with the Genome Edit Distance domain optimal planners solved only 11.5% of problems. Anytime planners promise a way to successively produce better plans over time. However, current anytime planners tend to reach a limit where they stop finding any further improvement, and the plans produced are still very far from the best possible. In the same experiment, the LAMA anytime planner solved all problems but found plans whose average quality is 1.57 times worse than the best known. (3) Finding solutions quickly or even finding any solution for large problems within some resource constraint is also difficult. The best-performing planner in the 2014 international planning competition still failed to solve 29.3% of problems. Re-engineering a domain model by capturing and exploiting structural knowledge in the form of macros has been found very useful in speeding up planners. However, existing planner independent macro generation techniques often fail to capture some promising macro candidates because the constituent actions are not found in sequence in the totally ordered training plans. This thesis contributes to plan decomposition by developing a new plan deordering technique, named block deordering, that allows two subplans to be unordered even when their constituent steps cannot. Based on the block-deordered plan, this thesis further contributes to plan optimisation and macro generation, and their implementations in two systems, named BDPO2 and BloMa. Key to BDPO2 is a decomposition into subproblems of improving parts of the current best plan, rather than the plan as a whole. BDPO2 can be seen as an application of the large neighbourhood search strategy to planning. We use several windowing strategies to extract subplans from the block deordering of the current plan, and on-line learning for applying the most promising subplanners to the most promising subplans. We demonstrate empirically that even starting with the best plans found by other means, BDPO2 is still able to continue improving plan quality, and often produces better plans than other anytime planners when all are given enough runtime. BloMa uses an automatic planner independent technique to extract and filter “self-containe” subplans as macros from the block deordered training plans. These macros represent important longer activities useful to improve planners coverage and efficiency compared to the traditional macro generation approaches

Exploiting Symmetries by Planning for a Descriptive Quotient

We eliminate symmetry from a problem before searching for a plan. The planning problem with symmetries is decomposed into a set of isomorphic subproblems. One plan is computed for a small planning problem posed by a descriptive quotient, a description of any such subproblem. A concrete plan is synthesized by concatenating instantiations of that one plan for each subproble