## The Computational Complexity of Propositional STRIPS Planning (1994)

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Venue: | Artificial Intelligence |

Citations: | 306 - 3 self |

### BibTeX

@ARTICLE{Bylander94thecomputational,

author = {Tom Bylander},

title = {The Computational Complexity of Propositional STRIPS Planning},

journal = {Artificial Intelligence},

year = {1994},

volume = {69},

pages = {165--204}

}

### Years of Citing Articles

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### Abstract

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...

### Citations

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Citation Context ...reasoning task of finding a sequence of operators that achieve a goal from a given initial state. It is well-known that planning is intractable in general, and that several obstacles stand in the way =-=[9]-=-. However, there are few results that provide clear dividing lines between tractable and intractable planning. Below, I clarify several of these dividing lines by analyzing the computational complexit... |

613 | Where the really hard problems are
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Citation Context ... rather than worst-case running time, so that more expressive languages would be labeled efficient. However, while it appears that NP-complete problems are hard only for narrow ranges of the problems =-=[10, 30, 31]-=-, there is little research on problems that are PSPACE-complete. Promising directions include Musick and Russell [32], who develop a Markov model approximation for analyzing hillclimbing algorithms on... |

591 |
Principles of Artificial Intelligence
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(Show Context)
Citation Context ... extended propositional STRIPS planning if the number of relevant ground formulas is limited. 2 See Section 5 for a more complete description. Propositional STRIPS planning is equivalent to Nilsson's =-=[33]-=- simplified description of STRIPS except that propositional STRIPS planning requires that each planning instance explicitly specifies all relevant ground atomic formulas. Extended propositional STRIPS... |

429 | Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems
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Citation Context ... rather than worst-case running time, so that more expressive languages would be labeled efficient. However, while it appears that NP-complete problems are hard only for narrow ranges of the problems =-=[10, 30, 31]-=-, there is little research on problems that are PSPACE-complete. Promising directions include Musick and Russell [32], who develop a Markov model approximation for analyzing hillclimbing algorithms on... |

318 |
Relationship between nondeterministic and deterministic tape complexities
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Citation Context ...e of a state is bounded by the number of conditions. Although the length N of a solution plan might be exponential, an algorithm only needs memory for O(lg N) states to determine if a solution exists =-=[35]-=-. Namely, an algorithm Plan-Exists(S 1 ; S 2 ; N) that determines the existence of a plan from state S 1 to state S 2 of length N or less can be implemented as follows: for N = 1, check if S 1 = S 2 o... |

290 |
and easy distributions of SAT problems
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Citation Context ... rather than worst-case running time, so that more expressive languages would be labeled efficient. However, while it appears that NP-complete problems are hard only for narrow ranges of the problems =-=[10, 30, 31]-=-, there is little research on problems that are PSPACE-complete. Promising directions include Musick and Russell [32], who develop a Markov model approximation for analyzing hillclimbing algorithms on... |

210 |
Planning as search: A quantitative approach
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Citation Context ...ms of prediction in which a partial ordering of events is given, whereas the equivalent planning problem would be to find some ordering of any set of events that achieves some set of conditions. Korf =-=[25]-=- considers how various global properties of planning problems (e.g., serializable subgoals, operator decomposability, abstraction) affect the complexity of using problem space search to find plans. In... |

190 |
Reasoning about Change
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Citation Context ...inition. Of course, in the situation calculus, this is where the infamous frame problem appears [22]. This problem can be recast as preferring one model over another, as in Shoham's preference logics =-=[36]-=-. Note that it is easy to map a possible state to its model. The result of the Extend function can then be formalized as follows: Let S be a possible state, and M , its corresponding model. Let o be a... |

154 | Complexity results for SAS+ planning
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(Show Context)
Citation Context ... Note that one can always unstack all the blocks followed by making the goal stacks from the table on up. The intermediate state in this case is when all the blocks are unstacked. Backstrom and Nebel =-=[4]-=- give a generalization of this restriction and the corresponding algorithm in which state variables are used instead of propositions. The second polynomial problem is if there are a limited number of ... |

149 |
Macro-operators: A weak method for learning
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(Show Context)
Citation Context ...fect the complexity of using problem space search to find plans. In contrast, I focus exclusively on local properties of operators. However, except for Korf's own analysis of operator decomposability =-=[24]-=-, neither he nor I describe the relationship from these properties of planning problems to the properties of operators. Clearly, this is an issue that future work should address. Perhaps the most impo... |

145 | Reasoning about action I: A possible worlds approach
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Citation Context ...ring a preference ordering of all the literals so that, roughly, if two literals are true of 1 The ramification problem is that the effects of an operator in the context of a formula can be ambiguous =-=[19, 20]-=-. For example, if AsBsC is a formula in the domain theory, and if an operator deletes A, it is ambiguous whether B or C should result. 2 the previous state, and if it is inconsistent to assert both in... |

141 | Complexity, decidability and undecidability results for domainindependent planning. AIJ 76(1–2):75–88
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(Show Context)
Citation Context ...d that planning is undecidable and so clearly demonstrated the difficulty of planning in general, but did not show what features of tweak's representation are to blame for the complexity. Erol et al. =-=[16]-=- analyze tweak-like planning further, showing that planning is undecidable only if the number of constants is infinite. They also demonstrate that if the set of constants are finite, the operators are... |

136 | On the semantics of STRIPS
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(Show Context)
Citation Context ...mpoverished compared to working planners. They are intended to be tools for theoretical analysis rather than programming convenience. The results for these models apply to first-order STRIPS planning =-=[28] when ther-=-e is a limited number of relevant ground formulas. The first model of planning, called "propositional STRIPS planning," is STRIPS planning [17] in which an initial state is a finite set of g... |

128 |
Expressiveness and tractability in knowledge representation and reasoning
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(Show Context)
Citation Context ...ese dividing lines by analyzing the computational complexity of a planning problem and a variety of restricted and augmented versions. From the perspective of the expressiveness-tractability tradeoff =-=[27]-=-, I primarily consider the expressiveness of operators and formulas. 1.1 Previous Research The literature on planning is voluminous, and no attempt to properly survey the planning literature is attemp... |

125 |
Complexity Results for Planning
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(Show Context)
Citation Context ... are not encouraging for domain-independent planning. This research has been supported in part by DARPA/AFOSR contract F49620-89-C-0110. y This paper is a revised, integrated, and extended version of =-=[5]-=- and [6]. 1 Pr'ecis If the relationship between intelligence and computation is taken seriously, then intelligence cannot be explained by intractable theories because no intelligent creature has the t... |

101 |
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(Show Context)
Citation Context ...ation, the resulting partial world is taken to be the intersection of all possible worlds. Other approaches have dealt with the problem of determining the resulting state in a variety of ways. In TMM =-=[13]-=-, when a set of time tokens conflict with a new time token, all the time tokens in the set are constrained to end before or begin after the new one. This conservative approach would possibly create mo... |

84 |
Reasoning About Partially Ordered Events
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(Show Context)
Citation Context ...terature is attempted here. Instead, the reader is referred to Allen et al. [1] and Hendler et al. [23]. Despite the sizable literature, results on computational complexity are sparse. Dean and Boddy =-=[12]-=- analyze the complexity of temporal projection---given a partial ordering of events and causal rules triggered by events, determine what conditions must be true after each event. Their formalization o... |

66 | Impediments to universal preference-based default theories
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(Show Context)
Citation Context ...estion the "restricted language thesis" proposed by Levesque and Brachman [27], i.e., that the expressiveness of representations needs to be restricted so that inference is tractable. Doyle =-=and Patil [14] persuasively a-=-rgue that restricting the expressiveness of a general-purpose representation system to ensure polynomial reasoning "destroys the generality of the language and the system" and "fails to... |

41 | Complexity Results for Blocks-World Planning
- Gupta, Nau
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(Show Context)
Citation Context ... variables, and if no two operators have the same postcondition. Ratner and Warmuth [34] show that finding optimal solutions to the n \Theta n generalization of the 8-puzzle is NP-hard. Gupta and Nau =-=[21]-=- and Chenoweth [11] show that optimal blocks-world planning is NP-hard. Bacchus and Yang [2] present tractable tests for determining when an hierarchical planning problem has a property called downwar... |

41 |
AI planning: Systems and techniques
- Tate, Drummond
- 1990
(Show Context)
Citation Context ... Research The literature on planning is voluminous, and no attempt to properly survey the planning literature is attempted here. Instead, the reader is referred to Allen et al. [1] and Hendler et al. =-=[23]-=-. Despite the sizable literature, results on computational complexity are sparse. Dean and Boddy [12] analyze the complexity of temporal projection---given a partial ordering of events and causal rule... |

31 | Parallel Non-Binary Planning in Polynomial Time
- Bäckström, Klein
- 1991
(Show Context)
Citation Context ...ed. There are also some results concerning the tractability of very specialized kinds of planning. In the case where states are value assignments to finite-valued state variables, Backstrom and Klein =-=[3]-=- show that planning is tractable if each operator has one postcondition, i.e., changes the value of one variable, if the preconditions of any two operators do not require different values for non-chan... |

28 |
The downward refinement property
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- 1991
(Show Context)
Citation Context ... that finding optimal solutions to the n \Theta n generalization of the 8-puzzle is NP-hard. Gupta and Nau [21] and Chenoweth [11] show that optimal blocks-world planning is NP-hard. Bacchus and Yang =-=[2]-=- present tractable tests for determining when an hierarchical planning problem has a property called downward refinement, i.e., every abstract solution can be refined into a concrete solution. While t... |

28 |
On the NP-Hardness of Blocks World
- Chenoweth
- 1991
(Show Context)
Citation Context ...no two operators have the same postcondition. Ratner and Warmuth [34] show that finding optimal solutions to the n \Theta n generalization of the 8-puzzle is NP-hard. Gupta and Nau [21] and Chenoweth =-=[11]-=- show that optimal blocks-world planning is NP-hard. Bacchus and Yang [2] present tractable tests for determining when an hierarchical planning problem has a property called downward refinement, i.e.,... |

23 |
Complexity Results for Serial Decomposability
- Bylander
- 1992
(Show Context)
Citation Context ...tors as a whole is a promising research approach. Korf's [24] analysis of serial decomposability and Bacchus and Yang's [2] analysis of abstraction are significant steps in this direction, though see =-=[7]-=- for negative results on the complexity of serial decomposability. More generally, I speculate that the analysis of general-purpose planning will evolve into the analysis of many different special-pur... |

22 |
The frame problem and related problems in artificial intelligence
- Hayes
- 1990
(Show Context)
Citation Context ... choose among alternative ways to resolve inconsistency, but does not specify any constraints on its definition. Of course, in the situation calculus, this is where the infamous frame problem appears =-=[22]-=-. This problem can be recast as preferring one model over another, as in Shoham's preference logics [36]. Note that it is easy to map a possible state to its model. The result of the Extend function c... |

19 |
A Representation of Action and Belief for Automatic Planning Systems
- Drummond
- 1987
(Show Context)
Citation Context ...onstrained to end before or begin after the new one. This conservative approach would possibly create more constraints than necessary to eliminate the contradiction. The plan net approach of Drummond =-=[15] requires -=-a "reconciliation set selection function" to choose among alternative ways to resolve inconsistency, but does not specify any constraints on its definition. Of course, in the situation calcu... |

15 |
An O( p jvj \Delta jej) algorithm for finding maximum matching in general graphs
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- 1980
(Show Context)
Citation Context ... edges, and because breadth-first search finds the shortest path. 2 Theorem 4.8 PLANMIN 0 2+ is polynomial. Proof: This can be reduced to maximum matching of a graph, which has a polynomial algorithm =-=[29]-=-. A matching of a graph is a set of edges such that no two edges are incident to the same vertex. Assuming that the goal is reachable, the reduction from a given PLANMIN 0 2+ instance is as follows. E... |

13 |
Complexity Results for Extended Planning
- Bylander
- 1992
(Show Context)
Citation Context ... encouraging for domain-independent planning. This research has been supported in part by DARPA/AFOSR contract F49620-89-C-0110. y This paper is a revised, integrated, and extended version of [5] and =-=[6]-=-. 1 Pr'ecis If the relationship between intelligence and computation is taken seriously, then intelligence cannot be explained by intractable theories because no intelligent creature has the time to p... |

12 | How long will it take
- Musick, Russell
- 1992
(Show Context)
Citation Context ... that NP-complete problems are hard only for narrow ranges of the problems [10, 30, 31], there is little research on problems that are PSPACE-complete. Promising directions include Musick and Russell =-=[32]-=-, who develop a Markov model approximation for analyzing hillclimbing algorithms on single postcondition problems, and Bylander [8], who shows that most PLANSAT instances under certain distributional ... |

11 |
Finding a shortest solution for the n \Theta n extension of the 15-puzzle is intractable
- Ratner, Warmuth
- 1986
(Show Context)
Citation Context ...he value of one variable, if the preconditions of any two operators do not require different values for non-changing variables, and if no two operators have the same postcondition. Ratner and Warmuth =-=[34]-=- show that finding optimal solutions to the n \Theta n generalization of the 8-puzzle is NP-hard. Gupta and Nau [21] and Chenoweth [11] show that optimal blocks-world planning is NP-hard. Bacchus and ... |

8 |
An average case analysis of planning
- Bylander
- 1993
(Show Context)
Citation Context ...re PSPACE-complete. Promising directions include Musick and Russell [32], who develop a Markov model approximation for analyzing hillclimbing algorithms on single postcondition problems, and Bylander =-=[8]-=-, who shows that most PLANSAT instances under certain distributional assumptions can be efficiently solved. An alternative approach is to restrict global properties of planning instances rather than l... |

5 |
The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 13
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Citation Context ...ference ordering of literals. For a Krom domain theory, a postcondition always implies the same set of conditions 7 The use of "Krom" to refer to this type of formula can be traced to a pape=-=r by Krom [26]-=-. 7 regardless of the current state and other postconditions. Thus, any EPLANSAT problem restricted to Krom theories is equivalent to the corresponding PLANSAT problem without any limitations on the n... |