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37
Complexity Results for SAS+ Planning
 COMPUTATIONAL INTELLIGENCE
, 1993
"... 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 ..."
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Cited by 182 (24 self)
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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.
Complexity, Decidability and Undecidability Results for DomainIndependent Planning
 ARTIFICIAL INTELLIGENCE
, 1995
"... In this paper, we examine how the complexity of domainindependent planning with STRIPSstyle operators depends on the nature of the planning operators. We show ..."
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Cited by 151 (27 self)
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In this paper, we examine how the complexity of domainindependent planning with STRIPSstyle operators depends on the nature of the planning operators. We show
Structure and Complexity in Planning with Unary Operators
 Journal of Artificial Intelligence Research
, 2003
"... Unary operator domains  i.e., domains in which operators have a single effect  arise naturally in many control problems. In its most general form, the problem of strips planning in unary operator domains is known to be as hard as the general strips planning problem  both are pspacecomplete. H ..."
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Cited by 51 (9 self)
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Unary operator domains  i.e., domains in which operators have a single effect  arise naturally in many control problems. In its most general form, the problem of strips planning in unary operator domains is known to be as hard as the general strips planning problem  both are pspacecomplete. However, unary operator domains induce a natural structure, called the domain's causal graph. This graph relates between the preconditions and effect of each domain operator. Causal graphs were exploited by Williams and Nayak in order to analyze plan generation for one of the controllers in NASA's DeepSpace One spacecraft. There, they utilized the fact that when this graph is acyclic, a serialization ordering over any subgoal can be obtained quickly. In this paper we conduct a comprehensive study of the relationship between the structure of a domain's causal graph and the complexity of planning in this domain. On the positive side, we show that a nontrivial polynomial time plan generation algorithm exists for domains whose causal graph induces a polytree with a constant bound on its node indegree. On the negative side, we show that even plan existence is hard when the graph is a directedpath singly connected DAG.
Statevariable planning under structural restrictions: Algorithms and complexity
 ARTIFICIAL INTELLIGENCE
, 1998
"... Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying statetransition grap ..."
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Cited by 48 (4 self)
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Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying statetransition graph. The exponential size of this graph, though, makes such restrictions costly to test. Hence, we propose an intermediate approach, using a state variable model for planning and defining restrictions on the separate statetransition graphs for each state variable. We identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions. The algorithm has been implemented an it outperforms Graphplan on a number of test instances. In addition, we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions. This complexity map considers both the optimal and nonoptimal plan generation problem.
Expressive Equivalence of Planning Formalisms
 Artificial Intelligence
, 1995
"... A concept of expressive equivalence for planning formalisms based on polynomial transformations is defined. It is argued that this definition is reasonable and useful both from a theoretical and from a practical perspective; if two languages are equivalent, then theoretical results carry over an ..."
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Cited by 34 (11 self)
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A concept of expressive equivalence for planning formalisms based on polynomial transformations is defined. It is argued that this definition is reasonable and useful both from a theoretical and from a practical perspective; if two languages are equivalent, then theoretical results carry over and, more practically, we can model an application problem in one language and then easily use a planner for the other language. In order to cope with the problem of exponentially sized solutions for planning problems an even stronger concept of expressive equivalence is introduced, using the novel ESPreduction. Four different formalisms for propositional planning are then analyzed, namely two variants of STRIPS, ground TWEAK and the SAS + formalism. Although these may seem to exhibit different degrees of expressive power, it is proven that they are, in fact, expressively equivalent under ESP reduction. This means that neither negative goals, partial initial states nor multivalue...
Parallel nonbinary planning in polynomial time
 In Reiter and Mylopoulos [ 1991
, 1991
"... This paper formally presents a class of planning problems which allows nonbinary state variables and parallel execution of actions. The class is proven to be tractable, and we provide a sound and complete polynomial time algorithm for planning within this class. This result means that we are gettin ..."
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Cited by 32 (12 self)
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This paper formally presents a class of planning problems which allows nonbinary state variables and parallel execution of actions. The class is proven to be tractable, and we provide a sound and complete polynomial time algorithm for planning within this class. This result means that we are getting closei to tackling realistic planning problems in sequential control, where a restricted problem representation is often sufficient, but where the size of the problems make tractability an important issue. 1
Equivalence and Tractability Results for SAS+ Planning
 Proceedings of the 3rd International Conference on Principles on Knowledge Representation and Reasoning (KR92
, 1992
"... We define the SAS + planning formalism, which generalizes the previously presented SAS formalism. The SAS + formalism is compared with some betterknown propositionalplanning formalisms with respect to expressiveness. Contrary to intuition, all formalisms turn out to be equally expressive in a v ..."
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Cited by 22 (6 self)
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We define the SAS + planning formalism, which generalizes the previously presented SAS formalism. The SAS + formalism is compared with some betterknown propositionalplanning formalisms with respect to expressiveness. Contrary to intuition, all formalisms turn out to be equally expressive in a very strong sense. We further present the SAS + PUS planning problem which generalizes the previously presented, tractable SASPUS problem. We prove that also the SAS +  PUS problem is tractable by devising a provably correct polynomial time algorithm for this problem. 1 Introduction Much effort has gone into finding more and more general formalisms, mainly logicbased, for plans and actions and also into finding reasoning methods for these. Although such formalisms may be important for modelling problems and comparing different approaches we most probably have to identify subproblems and devise tailored algorithms for these in order to overcome the computational difficulties involved. ...
Analyzing search topology without running any search: On the connection between causal graphs and h
 JAIR
, 2011
"... The ignoring delete lists relaxation is of paramount importance for both satisficing and optimal planning. In earlier work, it was observed that the optimal relaxation heuristic h + has amazing qualities in many classical planning benchmarks, in particular pertaining to the complete absence of local ..."
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Cited by 18 (3 self)
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The ignoring delete lists relaxation is of paramount importance for both satisficing and optimal planning. In earlier work, it was observed that the optimal relaxation heuristic h + has amazing qualities in many classical planning benchmarks, in particular pertaining to the complete absence of local minima. The proofs of this are handmade, raising the question whether such proofs can be lead automatically by domain analysis techniques. In contrast to earlier disappointing results – the analysis method has exponential runtime and succeeds only in two extremely simple benchmark domains – we herein answer this question in the affirmative. We establish connections between causal graph structure and h + topology. This results in loworder polynomial time analysis methods, implemented in a tool we call TorchLight. Of the 12 domains where the absence of local minima has been proved, TorchLight gives strong success guarantees in 8 domains. Empirically, its analysis exhibits strong performance in a further 2 of these domains, plus in 4 more domains where local minima may exist but are rare. In this way, TorchLight can distinguish “easy ” domains from “hard ” ones. By summarizing structural reasons for analysis failure, TorchLight also provides diagnostic output indicating domain aspects that may cause local minima. 1.
New islands of tractability of costoptimal planning
 JAIR
, 2008
"... We study the complexity of costoptimal classical planning over propositional state variables and unaryeffect actions. We discover novel problem fragments for which such optimization is tractable, and identify certain conditions that differentiate between tractable and intractable problems. These r ..."
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Cited by 17 (3 self)
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We study the complexity of costoptimal classical planning over propositional state variables and unaryeffect actions. We discover novel problem fragments for which such optimization is tractable, and identify certain conditions that differentiate between tractable and intractable problems. These results are based on exploiting both structural and syntactic characteristics of planning problems. Specifically, following Brafman and Domshlak (2003), we relate the complexity of planning and the topology of the causal graph. The main results correspond to tractability of costoptimal planning for propositional problems with polytree causal graphs that either have O(1)bounded indegree, or are induced by actions having at most one prevail condition each. Almost all our tractability results are based on a constructive proof technique that connects between certain tools from planning and tractable constraint optimization, and we believe this technique is of interest on its own due to a clear evidence for its robustness.
Tractable Planning for an Assembly Line
, 1995
"... The industry wants formal methods for dealing with combinatorial dynamical systems that are provably correct and fast. One example of such problems is error recovery in industrial processes. We have used a provably correct, polynomialtime planning algorithm to plan for a miniature assembly line ..."
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Cited by 11 (7 self)
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The industry wants formal methods for dealing with combinatorial dynamical systems that are provably correct and fast. One example of such problems is error recovery in industrial processes. We have used a provably correct, polynomialtime planning algorithm to plan for a miniature assembly line, which assembles toy cars. Although somewhat limited, this process has many similarities with real industrial processes. By exploring the structure of this assembly line we have extended a previously presented algorithm making the class of problems that can be handled in polynomial time larger.