## The Computational Complexity of Probabilistic Planning (1998)

### Cached

### Download Links

- [www.cs.duke.edu]
- [www.informatik.uni-trier.de]
- [www.cs.cmu.edu]
- [jair.org]
- [www.jair.org]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of Artificial Intelligence Research |

Citations: | 80 - 5 self |

### BibTeX

@ARTICLE{Littman98thecomputational,

author = {Michael L. Littman and Judy Goldsmith and Martin Mundhenk},

title = {The Computational Complexity of Probabilistic Planning},

journal = {Journal of Artificial Intelligence Research},

year = {1998},

volume = {9},

pages = {1--36}

}

### Years of Citing Articles

### OpenURL

### Abstract

We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural definitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP PP, co-NP PP , and PSPACE. In the process of proving that certain planning problems are complete for NP PP , we introduce a new basic NP PP -complete problem, E-Majsat, which generalizes the standard Boolean satisfiability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-Majsat could be important for the creation of efficient algorithms for a wide variety of problems.

### Citations

2389 | Computational complexity - Papadimitriou - 1994 |

878 | Planning and acting in partially observable stochastic domains - Kaelbling, Littman, et al. - 1998 |

645 |
Planning for conjunctive goals
- Chapman
- 1985
(Show Context)
Citation Context ...he computational complexity of determining whether a valid plan exists, of determining whether a plan exists of a given cost, and of finding the valid plans themselves under a variety of assumptions (=-=Chapman, 1987-=-; Bylander, 1994; Erol, Nau, & Subrahmanian, 1995; Backstrom, 1995; Backstrom & Nebel, 1995). These results provide lower bounds (hardness results) for analogous probabilistic planning problems since ... |

399 | Systematic nonlinear planning - McAllester, Rosenblitt - 1991 |

317 | The complexity of Markov decision processes - Papadimitriou, Tsitsiklis - 1987 |

301 | The computational complexity of propositional STRIPS planning - Bylander - 1994 |

288 | Improving elevator performance using reinforcement learning - Crites, Barto - 1996 |

284 | Acting optimally in partially observable stochastic domains - Cassandra, Kaelbling, et al. - 1994 |

264 | An algorithm for probabilistic planning - Kushmerick, Hanks, et al. - 1995 |

239 |
TD-Gammon, a self-teaching backgammon program, achieves master level play
- Tesauro
- 1994
(Show Context)
Citation Context ...mulation (random sampling) to find approximately optimal solutions to probabilistic planning problems. Some empirical successes have been obtained with the related approach of reinforcement learning (=-=Tesauro, 1994-=-; Crites & Barto, 1996), but, once again, the worst-case complexity of probabilistic planning is not known to be any lower for approximation by simulation. 6. Conclusions In this paper, we explored th... |

231 | On the hardness of approximate reasoning
- Roth
- 1996
(Show Context)
Citation Context ...ng class #P has already been recognized by the artificial-intelligence community as an important complexity class in computations involving probabilistic quantities, such as belief-network inference (=-=Roth, 1996-=-). 2 Complexity of Probabilistic Planning planning domain is one that is described by a two-stage temporal Bayes network (Boutilier et al., 1998) or similar notation. Second, the problem of determinin... |

226 | Exploiting structure in policy construction - Boutilier, Dearden, et al. - 1995 |

224 |
Computational Complexity of Probabilistic Turing Machines
- Gill
- 1977
(Show Context)
Citation Context ...L functions g and h. While #L functions have nonnegative integer values by definition, GapL functions may have negative integer values (for example, if g always returns zero). Probabilistic logspace (=-=Gill, 1977-=-), PL, is the class of sets A for which there exists a nondeterministic logarithmically space-bounded machine N such that x 2 A if and only if the number of accepting paths of N on x is greater than i... |

203 | Probabilistic planning with information gathering and contingent execution - Draper, Hanks, et al. - 1994 |

193 | PP is as hard as the polynomial-time hierarchy - TODA - 1991 |

186 |
A survey of algorithmic methods for partially observed Markov decision processes
- Lovejoy
- 1991
(Show Context)
Citation Context ...istic plans perform nearly as well as much larger and more complicated deterministic plans; this notion is often exploited in the field of randomized algorithms. Work by Platzman (1981) (described by =-=Lovejoy, 1991-=-) shows how the idea of randomized plans can come in handy for planning in partially observable domains. 3. Looping Plans Looping plans can be applied to infinite-horizon control. The complexity of pl... |

175 |
Probabilistic automata
- Rabin
- 1963
(Show Context)
Citation Context ...(1996) (infinite-horizon). This last result is derived by noting the isomorphism of the infinitehorizon problem to the emptiness problem for probabilistic finite-state automata, which is undecidable (=-=Rabin, 1963-=-). 5.2 Buridan The buridan planner (Kushmerick et al., 1995) finds partially ordered plans for propositional domains in the PSO representation. There are two identifiable differences between the probl... |

156 | The complexity of stochastic games
- Condon
- 1992
(Show Context)
Citation Context ...ts. 2. It is also possible to formulate the objective as one of maximizing expected total discounted reward (Boutilier et al., 1995), but the two formulations are essentially polynomially equivalent (=-=Condon, 1992-=-). The only difficulty is that compactly represented domains may require discount factors exponentially close to one for this equivalence to hold. This is discussed further in Section 5. 3. We assume ... |

152 | Decision theoretic planning: Structural assumptions and computational leverage - Boutilier, Dean, et al. - 1999 |

150 | Complexity results for SAS+ planning
- Bäckström, Nebel
- 1995
(Show Context)
Citation Context ...exists, of determining whether a plan exists of a given cost, and of finding the valid plans themselves under a variety of assumptions (Chapman, 1987; Bylander, 1994; Erol, Nau, & Subrahmanian, 1995; =-=Backstrom, 1995-=-; Backstrom & Nebel, 1995). These results provide lower bounds (hardness results) for analogous probabilistic planning problems since deterministic planning is a special case. In deterministic plannin... |

133 | Complexity, decidability and undecidability results for domain-independent planning. Arti Intelligence to appear. A more detailed version is available as - Erol, Nau, et al. - 1992 |

115 | Computing optimal policies for partially observable decision processes using compact representations - Boutilier, Poole - 1996 |

110 | Anytime synthetic projection: Maximizing the probability of goal satisfaction - Drummond, Bresina - 1990 |

83 | Probabilistic propositional planning: representations and complexity
- Littman
- 1997
(Show Context)
Citation Context ... This problem is closely related to that of solving a Markov decision process (Puterman, 1994) when it is expressed in a compact representation. In previous work (Goldsmith, Lusena, & Mundhenk, 1996; =-=Littman, 1997-=-a), we examined the complexity of determining whether an effective plan exists for completely observable domains; the problem is EXP-complete in its general form and PSPACE-complete when limited to po... |

74 | On some central problems in computational complexity - Simon - 1975 |

70 | MAXPLAN: A new approach to probabilistic planning - Majercik, Littman - 1998 |

67 | Abstraction and approximate decision-theoretic planning - Dearden, Boutilier - 1997 |

62 | Parallel computation for well-endowed rings and space bounded probabilistic machines - Borodin, Cook, et al. - 1983 |

53 | Complexity classes defined by counting quantifiers
- Torán
- 1991
(Show Context)
Citation Context ...us output. For any class C ` PSPACE, it is the case that NP C ` PSPACE, and therefore NP PSPACE = PSPACE. The primary oracle-defined class we consider is NP PP . It equals the " NP m " closu=-=re of PP (Tor'an, 1991-=-), which can be seen as the closure of PP under polynomial-time disjunctive reducibility with an exponential number of queries (each of the queries computable in polynomial time from its index in the ... |

50 | A very hard log-space counting class - Àlvarez, Jenner - 1993 |

37 | Approximating value trees in structured dynamic programming - Boutilier, Dearden - 1996 |

32 | Epsilon-safe planning - Goldman, Boddy - 1994 |

32 |
Finite-Memory Control of Partially Observable Systems
- Hansen
- 1998
(Show Context)
Citation Context ... paper and searching for optimal size-bounded plans instead of optimal unrestricted plans; in fact, recent planners for both propositional domains (Majercik & Littman, 1998a, 1998b) and flat domains (=-=Hansen, 1998-=-) are motivated by these results. 28 Complexity of Probabilistic Planning Domain Type Horizon Type Size-Bounded Plan Unrestricted Plan flat polynomial NP-complete PSPACE-complete propositional polynom... |

31 | Expressive equivalence of planning formalisms
- Bäckström
- 1995
(Show Context)
Citation Context ...exists, of determining whether a plan exists of a given cost, and of finding the valid plans themselves under a variety of assumptions (Chapman, 1987; Bylander, 1994; Erol, Nau, & Subrahmanian, 1995; =-=Backstrom, 1995-=-; Backstrom & Nebel, 1995). These results provide lower bounds (hardness results) for analogous probabilistic planning problems since deterministic planning is a special case. In deterministic plannin... |

24 | Using Caching to Solve Larger Probabilistic Planning Problems - Majercik, Littman - 1998 |

22 | On probabilistic time and space - Jung - 1985 |

19 | The complexity of plan existence and evaluation in probabilistic domains - Goldsmith, Littman, et al. - 1997 |

18 |
Polynomial space counting problems
- Ladner
- 1989
(Show Context)
Citation Context ...esults from Gill (1977) and Simon (1975). For polynomial-space-bounded computations, PSPACE equals probabilistic PSPACE, and #PSPACE is the same as the class of polynomial-space-computable functions (=-=Ladner, 1989-=-). Note that L, NL, #L, PL and GapL are to logarithmic space what P, NP, #P, PP, and GapP are to polynomial time. Also, the notion of completeness we use in this paper relies on many-one reductions. I... |

15 | Risk-sensitive planning with probabilistic decision graphs
- Koenig, Simmons
- 1994
(Show Context)
Citation Context ... planning has addressed the problem of finding effective plans in domains in which operators have probabilistic effects (Drummond & Bresina, 1990; Mansell, 1993; Draper, Hanks, & Weld, 1994; Koenig & =-=Simmons, 1994-=-; Goldman & Boddy, 1994; Kushmerick, Hanks, & Weld, 1995; Boutilier, Dearden, & Goldszmidt, 1995; Dearden & Boutilier, 1997; Kaelbling, Littman, & Cassandra, 1998; Boutilier, Dean, & Hanks, 1998). Her... |

14 |
Complexity results for finite-horizon Markov decision processs problems
- Mundhenk, Goldsmith, et al.
- 2000
(Show Context)
Citation Context ...t to ST, the two representations are closely related; the proofs we give for ST need to be changed only slightly to work for the succinct representation (Goldsmith, Littman, & Mundhenk, 1997a, 1997b; =-=Mundhenk et al., 1997-=-b). Our results require that we restrict the succinct representation to generate transition probabilities with at most a polynomial 4 Complexity of Probabilistic Planning number of bits; the results m... |

13 | Relationship among PL - Allender, Ogihara - 1996 |

13 |
A method for planning given uncertain and incomplete information
- Mansell
- 1993
(Show Context)
Citation Context .... Introduction Recent work in artificial-intelligence planning has addressed the problem of finding effective plans in domains in which operators have probabilistic effects (Drummond & Bresina, 1990; =-=Mansell, 1993-=-; Draper, Hanks, & Weld, 1994; Koenig & Simmons, 1994; Goldman & Boddy, 1994; Kushmerick, Hanks, & Weld, 1995; Boutilier, Dearden, & Goldszmidt, 1995; Dearden & Boutilier, 1997; Kaelbling, Littman, & ... |

12 | Generating optimal policies for high-level plans with conditional branches and loops - Lin, Dean - 1995 |

11 | A feasible computational approach to infinite-horizon partially-observed Markov decision problems - Platzman - 1981 |

8 | Nonapproximability results for Markov decision processes - Lusena, Goldsmith, et al. - 1998 |

8 | Representation and evaluation of plans with loops - Smith, Williamson - 1995 |

5 | The complexity of deterministically observable finite-horizon Markov decision processes - Goldsmith, Lusena, et al. - 1996 |

5 |
Complexity classes de ned by counting quanti ers
- Toran
- 1991
(Show Context)
Citation Context ...taneous output. For any class C PSPACE, it is the case that NP C PSPACE, and therefore NP PSPACE = PSPACE. The primary oracle-de ned class we consider is NP PP . It equals the \ NP m " closure of PP (=-=Toran, 1991-=-), which can be seen as the closure of PP under polynomial-time disjunctive reducibility with an exponential number of queries (each of the queries computable in polynomial time from its index in the ... |

4 | The complexity of unobservable finite-horizon Markov decision processes - Mundhenk, Goldsmith, et al. - 1996 |

3 | A feasible computational approach to in nite-horizon partially-observed Markov decision problems - Platzman - 1981 |