Planning is a notoriously hard combinatorial search problem. In many interesting domains, current planning algorithms fail to scale up gracefully. By combining a general, stochastic search algorithm and appropriate problem encodings based on propositional logic, we are able to solve hard planning problems many times faster than the best current planning systems. Although stochastic methods have been shown to be very e ective on a wide range of scheduling problems, this is the rst demonstration of its power on truly challenging classical planning instances. This work also provides a new perspective on representational issues in planning.

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I present several computational complexity results for propositional STRIPS planning, i.e., STRIPS planning restricted to ground formulas. Different planning problems can be defined by restricting the type of formulas, placing limits on the number of pre- and postconditions, by restricting negation in pre- and postconditions, and by requiring optimal plans. For these types of restrictions, I show when planning is tractable (polynomial) and intractable (NPhard) . In general, it is PSPACE-complete to determine if a given planning instance has any solutions. Extremely severe restrictions on both the operators and the formulas are required to guarantee polynomial time or even NP-completeness. For example, when only ground literals are permitted, determining plan existence is PSPACE-complete even if operators are limited to two preconditions and two postconditions. When definite Horn ground formulas are permitted, determining plan existence is PSPACE-complete even if operators are limited t...

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In recent work we showed that planning problems can be efficiently solved by general propositional satisfiability algorithms (Kautz and Selman 1996). A key issue in this approach is the development of practical reductions of planning to SAT. We introduce a series of different SAT encodings for STRIPS-style planning, which are sound and complete representations of the original STRIPS specification, and relate our encodings to the Graphplan system of Blum and Furst (1995). We analyze the size complexity of the various encodings, both in terms of number of variables and total length of the resulting formulas. This paper complements the empirical evaluation of several of the encodings reported in Kautz and Selman (1996). We also introduce a novel encoding based on the theory of causal planning, that exploits the notionof "lifting" from the theorem-proving community. This new encoding strictly dominates the others in terms of asymptotic complexity. Finally, we consider further reductions i...

Computational efficiency is a central concern in the design of knowledge representation systems. In order to obtain efficient systems, it has been suggested that one should limit the form of the statements in the knowledge base or use an incomplete inference mechanism. The former approach is often too restrictive for practical applications, whereas the latter leads to uncertainty about exactly what can and cannot be inferred from the knowledge base. We present a third alternative, in which knowledge given in a general representation language is translated (compiled) into a tractable form — allowing for efficient subsequent query answering. We show how propositional logical theories can be compiled into Horn theories that approximate the original information. The approximations bound the original theory from below and above in terms of logical strength. The procedures are extended to other tractable languages (for example, binary clauses) and to the first-order case. Finally, we demonstrate the generality of our approach by compiling concept descriptions in a general framebased language into a tractable form.

We have previously reported a number of tractable planning problems defined in the SAS+ formalism. This report complements these results by providing a complete map over the complexity of SAS+ planning under all combinations of the previously considered restrictions. We analyze the complexity both of finding a minimal plan and of finding any plan. In contrast to other complexity surveys of planning we study not only the complexity of the decision problems but also of the generation problems. We prove that the SAS+-PUS problem is the maximal tractable problem under the restrictions we have considered if we want to generate minimal plans. If we are satisfied with any plan, then we can generalize further to the SAS+-US problem, which we prove to be the maximal tractable problem in this case.

In this paper, we examine how the complexity of domain-independent planning with STRIPS-style operators depends on the nature of the planning operators. We show

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.

In this paper, we show that in the best-known version of the blocks world (and several related versions), planning is difficult, in the sense that finding an optimal plan is NP-hard. However, the NP-hardness is not due to deleted-condition interactions, but instead due to a situation which we call a deadlock. For problems that do not contain deadlocks, there is a simple hill-climbing strategy that can easily find an optimal plan, regardless of whether or not the problem contains any deleted-condition interactions. The above result is rather surprising, since one of the primary roles of the blocks world in the planning literature has been to provide examples of deleted-condition interactions such as creative destruction and Sussman's anomaly. However, we can explain why deadlocks are hard to handle in terms of a domain-independent goal interaction which we call an enabling-condition interaction, in which an action invoked to achieve one goal has a side-effect of making it easier to achi...