Malte Helmert

Concise finite-domain representations for PDDL planning tasks

We introduce an efficient method for translating planning tasks specified in the standard PDDL formalism into a concise grounded representation that uses finite-domain state variables instead of the straight-forward propositional encoding. Translation is performed in four stages. Firstly, we transform the input task into an equivalent normal form expressed in a restricted fragment of PDDL. Secondly, we synthesize invariants of the planning task that identify groups of mutually exclusive propositions which can be represented by a single finite-domain variable. Thirdly, we perform an efficient relaxed reachability analysis using logic programming techniques to obtain a grounded representation of the input. Finally, we combine the results of the third and fourth stage to generate the final grounded finite-domain representation. The presented approach has originally been implemented as part of the Fast Downward planning system for the 4th International Planning Competition (IPC4). Since then, it has been used in a number of other contexts with considerable success, and the use of concise finite-domain representations has become a common feature of state-of-the-art planners.

Complexity results for standard benchmark domains in planning

The efficiency of AI planning systems is usually evaluated empirically. For the validity of conclusions drawn from such empirical data, the problem set used for evaluation is of critical importance. In planning, this problem set usually, or at least often, consists of tasks from the various planning domains used in the first two international planning competitions, hosted at the 1998 and 2000 AIPS conferences. It is thus surprising that comparatively little is known about the properties of these benchmark domains, with the exception of BLOCKSWORLD, which has been studied extensively by several research groups.In this contribution, we try to remedy this fact by providing a map of the computational complexity of non-optimal and optimal planning for the set of domains used in the competitions. We identify a common transportation theme shared by the majority of the benchmarks and use this observation to define and analyze a general transportation problem that generalizes planning in several classical domains such as LOGISTICS, MYSTERY and GRIPPER. We then apply the results of that analysis to the actual transportation domains from the competitions. We next examine the remaining benchmarks, which do not exhibit a strong transportation feature, namely SCHEDULE and FREECELL.Relating the results of our analysis to empirical work on the behavior of the recently very successful FF planning system, we observe that our theoretical results coincide well with data obtained from empirical investigations.

Exhibiting Knowledge in Planning Problems to Minimize State Encoding Length

In this paper we present a general-purposed algorithm for transforming a planning problem specified in Strips into a concise state description for single state or symbolic exploration.

Sound and Complete Landmarks for And/Or Graphs

Landmarks for a planning problem are subgoals that are necessarily made true at some point in the execution of any plan. Since verifying that a fact is a landmark is PSPACE-complete, earlier approaches have focused on finding landmarks for the delete relaxation Π+. Furthermore, some of these approaches have approximated this set of landmarks, although it has been shown that the complete set of causal delete-relaxation landmarks can be identified in polynomial time by a simple procedure over the relaxed planning graph. Here, we give a declarative characterisation of this set of landmarks and show that the procedure computes the landmarks described by our characterisation. Building on this, we observe that the procedure can be applied to any delete-relaxation problem and take advantage of a recent compilation of the m-relaxation of a problem into a problem with no delete effects to extract landmarks that take into account delete effects in the original problem. We demonstrate that this approach finds strictly more causal landmarks than previous approaches and discuss the relationship between increased computational effort and experimental performance, using these landmarks in a recently proposed admissible landmark-counting heuristic.

A Stochastic Local Search Approach to Vertex Cover

We introduce a novel stochastic local search algorithm for the vertex cover problem. Compared to current exhaustive search techniques, our algorithm achieves excellent performance on a suite of problems drawn from the field of biology. We also evaluate our performance on the commonly used DIMACS benchmarks for the related clique problem, finding that our approach is competitive with the current best stochastic local search algorithm for finding cliques. On three very large problem instances, our algorithm establishes new records in solution quality.

Computing perfect heuristics in polynomial time: on bisimulation and merge-and-shrink abstraction in optimal planning

A* with admissible heuristics is a very successful approach to optimal planning. But how to derive such heuristics automatically? Merge-and-shrink abstraction (M&S) is a general approach to heuristic design whose key advantage is its capability to make very fine-grained choices in defining abstractions. However, little is known about how to actually make these choices. We address this via the well-known notion of bisimulation. When aggregating only bisimilar states, M&S yields a perfect heuristic. Alas, bisimulations are exponentially large even in trivial domains. We show how to apply label reduction - not distinguishing between certain groups of operators - without incurring any information loss, while potentially reducing bisimulation size exponentially. In several benchmark domains, the resulting algorithm computes perfect heuristics in polynomial time. Empirically, we show that approximating variants of this algorithm improve the state of the art in M&S heuristics. In particular, a simple hybrid of two such variants is competitive with the leading heuristic LM-cut.

Merge-and-Shrink Abstraction: A Method for Generating Lower Bounds in Factored State Spaces

Many areas of computer science require answering questions about reachability in compactly described discrete transition systems. Answering such questions effectively requires techniques to be able to do so without building the entire system. In particular, heuristic search uses lower-bounding (“admissible”) heuristic functions to prune parts of the system known to not contain an optimal solution. A prominent technique for deriving such bounds is to consider abstract transition systems that aggregate groups of states into one. The key question is how to design and represent such abstractions. The most successful answer to this question are pattern databases, which aggregate states if and only if they agree on a subset of the state variables. Merge-and-shrink abstraction is a new paradigm that, as we show, allows to compactly represent a more general class of abstractions, strictly dominating pattern databases in theory. We identify the maximal class of transition systems, which we call factored transition systems, to which merge-and-shrink applies naturally, and we show that the well-known notion of bisimilarity can be adapted to this framework in a way that still guarantees perfect heuristic functions, while potentially reducing abstraction size exponentially. Applying these ideas to planning, one of the foundational subareas of artificial intelligence, we show that in some benchmarks this size reduction leads to the computation of perfect heuristic functions in polynomial time and that more approximate merge-and-shrink strategies yield heuristic functions competitive with the state of the art.

Planning with h + in theory and practice

Many heuristic estimators for classical planning are based on the socalled delete relaxation, which ignores negative effects of planning operators. Ideally, such heuristics would compute the actual goal distance in the delete relaxation, i.e., the cost of an optimal relaxed plan, denoted by h+. However, current delete relaxation heuristics only provide (often inadmissible) estimates to h+ because computing the correct value is an NP-hard problem. In this work, we consider the approach of planning with the actual h+ heuristic from a theoretical and computational perspective. In particular, we provide domain-dependent complexity results that classify some standard benchmark domains into ones where h+ can be computed efficiently and ones where computing h+ is NP-hard. Moreover, we study domain-dependent implementations of h+ which show that the h+ heuristic provides very informative heuristic estimates compared to other state-of-the-art heuristics.

Pattern database heuristics for fully observable nondeterministic planning

When planning in an uncertain environment, one is often interested in finding a contingent plan that prescribes appropriate actions for all possible states that may be encountered during the execution of the plan. We consider the problem of finding strong cyclic plans for fully observable nondeterministic (FOND) planning problems. The algorithm we choose is LAO*, an informed explicit state search algorithm. We investigate the use of pattern database (PDB) heuristics to guide LAO* towards goal states. To obtain a fully domain-independent planning system, we use an automatic pattern selection procedure that performs local search in the space of pattern collections. The evaluation of our system on the FOND benchmarks of the Uncertainty Part of the International Planning Competition 2008 shows that our approach is competitive with symbolic regression search in terms of problem coverage, speed, and plan quality.

The Causal Graph Revisited for Directed Model Checking

Directed model checking is a well-established technique to tackle the state explosion problem when the aim is to find error states in large systems. In this approach, the state space traversal is guided through a function that estimates the distance to nearest error states. States with lower estimates are preferably expanded during the search. Obviously, the challenge is to develop distance functions that are efficiently computable on the one hand and as informative as possible on the other hand. In this paper, we introduce the causal graph structure to the context of directed model checking. Based on causal graph analysis, we first adapt a distance estimation function from AI planning to directed model checking. Furthermore, we investigate an abstraction that is guaranteed to preserve error states. The experimental evaluation shows the practical potential of these techniques.