The past five years have seen dramatic advances in planning algorithms, with an emphasis on propositional methods such as Graphplan and compilers that convert planning problems into propositional CNF formulae for solution via systematic or stochastic SAT methods. Related work on the Deep Space One spacecraft control algorithms advances our understanding of interleaved planning and execution. In this survey,we explain the latest techniques and suggest areas for future research.

A parallel satis ability testing algorithm called Parallel Modoc is presented. Par-allel Modoc is based on Modoc, which is based on propositional Model Elimination with an added capability to prune away certain branches that cannot lead to a successful subrefutation. The pruning information is encoded in a partial truth assignment called an autarky. Parallel Modoc executes multiple instances of Modoc as separate processes and allows processes to cooperate by sharing lemmas and autarkies as they are found. When a Modoc process nds a new autarky or a new lemma, it makes the informa-tion available to other Modoc processes via a \blackboard". Combining autarkies generally is not straightforward because two autarkies found by two separate pro-cesses may have con icting assignments. The paper presents an algorithm to combine two arbitrary autarkies to form a larger autarky. Experimental results show that for many of the formulas, Parallel Modoc achieves speedup greater than the number of processors. Formulas that could not be solved in an hour by Modoc were often solved by Parallel Modoc in the order of minutes, and in some cases, in seconds.

Goal-sensitive resolution methods, such as Model Elimination, have been observed to have a higher degree of search redundancy than model-search methods, Therefore, resolution methods have not been seen in high performance propositional satis ability testers. A method to reduce search redundancy in goal-sensitive resolution methods is introduced. The idea at the heart of the method is to attempt to construct a refutation and a model simultaneously and incrementally, based on sub-search outcomes. The method exploits the concept of \autarky", which can be informally described as a \self-su cient " model for some clauses, but which does not a ect the remaining clauses of the formula. Incorporating this method into Model Elimination leads to an algorithm called Modoc. Modoc is shown, both analytically and experimentally, to be faster than Model Elimination by an exponential factor. Modoc, unlike Model Elimination, is able to nd a model if it fails to nd a refutation, essentially by combining autarkies. Unlike the pruning strategies of most re nements of resolution, autarky-related pruning does not prune any successful refutation; it only prunes attempts that ultimately will be unsuccessful; consequently, it will not force the underlying Modoc search to nd an unnecessarily long refutation. To prove correctness and other properties, a game characterization of refutation search isintroduced, which demonstrates

. Random 2+p-SAT interpolates between the polynomialtime problem Random 2-SAT when p = 0 and the NP-complete problem Random 3-SAT when p = 1. At some value p = p0 0:41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods. This is reflected by a change in the typical cost scaling for a complete search method TABLEAU, seen experimentally. We show empirically the same change of of behaviour in the local search algorithm NOVELTY + , a recent variant of WSAT. Between p = 0:3 and p = 0:5 we see typical cost scaling of NOVELTY + at the 50% satisfiability point apparently change from slow polynomial growth to superpolynomial. That this behaviour is seen in two such different algorithms lends credibility to the hypothesis that there is change of typical-case complexity around p0 . Previous work linked the emergence of a backbone of fully constrained variables to the cost peak seen in Random k-SAT. Initial experiments suggest that for those...

Model elimination is a back-chaining strategy to search for and construct resolution refutations. Many formulas can be refuted more succinctly by recording certain derived clauses, called lemmas, then using them where a clause of the original formula would normally be required. However, recording too many lemmas overwhelms the proof search.

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Random-2+p-SAT interpolates between the polynomial-time problem Random-2-SAT when p = 0 and the NP-complete problem Random3 -SAT when p = 1. At some value p = p 0 0:41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods. This is reected by a change in the typical cost scaling for a complete search method Tableau, seen experimentally. We show empirically the same change of of behaviour in the local search algorithm WSat/Novelty , a recent variant of WSat. Between p = 0:3 and p = 0:5 we see typical cost scaling of WSat/Novelty at the 50% satis ability point apparently change from slow polynomial growth to superpolynomial.

The local search algorithm WSat is one of the most successful algorithms for solving the archetypal NP-complete problem of satisfiability (SAT). It is notably effective at solving Random-3-SAT instances near the so-called `satisfiability threshold', which are thought to be universally hard. However, WSat still shows a peak in search cost near the threshold and large variations in cost over different instances. Why are solutions to the threshold instances so hard to find using WSat? What features characterise threshold instances which make them difficult for WSat to solve? We make a number of significant contributions to the analysis of WSat on these high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are implicates of an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We undertake a detailed study of the behaviour of the algorithm during search and uncover some interesting patterns. These patterns lead us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for WSat are those with large backbones which are also backbone-fragile. We suggest that the decay in cost for WSat beyond the satis ability threshold, which has perplexed a number of researchers, is due to the decreasing backbone fragility. Our hypothesis makes three correct predictions. First, that a measure of the backbone robustness of an instance (the opposite to backbone fragility) is negatively correlated with the WSat ...