## Downward Refinement and the Efficiency of Hierarchical Problem Solving (1993)

Venue: | Artificial Intelligence |

Citations: | 56 - 1 self |

### BibTeX

@ARTICLE{Bacchus93downwardrefinement,

author = {Fahiem Bacchus and Qiang Yang},

title = {Downward Refinement and the Efficiency of Hierarchical Problem Solving},

journal = {Artificial Intelligence},

year = {1993},

volume = {71},

pages = {43--100}

}

### OpenURL

### Abstract

Analysis and experiments have shown that hierarchical problem-solving is most effective when the hierarchy satisfies the downward refinement property (DRP), whereby every abstract solution can be refined to a concrete-level solution without backtracking across abstraction levels. However, the DRP is a strong requirement that is not often met in practice. In this paper we examine the case when the DRP fails, and provide an analytical model of search complexity parameterized by the probability of an abstract solution being refinable. Our model provides a more accurate picture of the effectiveness of hierarchical problem-solving. We then formalize the DRP in Abstrips-style hierarchies, providing a syntactic test that can be applied to determine if a hierarchy satisfies the DRP. Finally, we describe an algorithm called Highpoint that we have developed. This algorithm builds on the Alpine algorithm of Knoblock in that it automatically generates abstraction hierarchies. However, it uses th...

### Citations

2053 |
An introduction to probability theory and its applications. Vol. II. Second edition
- Feller
- 1971
(Show Context)
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962 |
Human problem solving
- Newell, Simon
- 1972
(Show Context)
Citation Context ...n using the intermediate states of the abstract solution as intermediate goals to decompose the search for the non-abstract solution. This technique has been used in a number of problem-solvers in AI =-=[23, 24, 25, 26, 29, 30, 33, 32, 10]-=-. It has long been known that the identification of intermediate states that decompose a problem can significantly reduce search [22, 21]. Hence, once we have an abstract solution providing a decompos... |

707 |
Data structures and algorithms
- Aho, Hopcroft, et al.
- 1983
(Show Context)
Citation Context ...g graph, using both the partial order relation and refinement probabilities as guides to compute the order. The procedure is a simple modification of a standard topological sort algorithm (see, e.g., =-=[1]-=-). The standard topological sort works by first placing all nodes with no in-edges in a queue. Nodes are removed from this queue, and when a node is removed from the queue all of its out-edges are rem... |

577 | Where the Really Hard Problems Are
- Cheeseman, Kenefsky, et al.
- 1991
(Show Context)
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533 | Concrete mathematics
- Graham, Knuth, et al.
- 1989
(Show Context)
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457 | Branching processes
- Athreya, Ney
- 1972
(Show Context)
Citation Context ... be a bad node or (b) the root could be good but somewhere among its descendants all the good paths terminate before reaching level 0. Case (b) can be analyzed using the theory of branching processes =-=[7, 2]-=-. A branching process is a probabilistic model with wide ranging applications. In its basic form we start with a single parent who produces some number of offspring. There is a fixed probability that ... |

441 |
Planning in a hierarchy of abstraction spaces
- Sacerdoti
- 1974
(Show Context)
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379 |
Generating project networks
- Tate
- 1977
(Show Context)
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331 | A Structurefor Plans and Behavior - Sacerdoti - 1977 |

255 |
Practical Planning: Extending the Classical AI Planning Paradigm
- Wilkins
- 1988
(Show Context)
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202 |
Planning as search: A quantitative approach
- Korf
- 1987
(Show Context)
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187 |
All I Know: A study in autoepistemic logic
- Levesque
- 1990
(Show Context)
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185 | Steps toward artificial intelligence
- Minsky
- 1961
(Show Context)
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135 | Domain-independent Planning: Representation and Plan Generation
- Wilkins
- 1984
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A guide to the modal logic of knowledge and belief
- Halpern, Moses
- 1985
(Show Context)
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- Knoblock
- 1991
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70 |
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- Stefik
- 1981
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69 |
Theorem proving with abstraction
- Plaisted
- 1981
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62 | Search Reduction in Hierarchical Problem Solving - Knoblock - 1991 |

61 |
Characterizing abstraction hierarchies for planning
- Knoblock, Tenenberg, et al.
- 1991
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- 1990
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50 |
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- 1988
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49 |
The process of creative thinking
- Newell, Shaw, et al.
- 1957
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- 1989
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- 1991
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- 1992
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- 1992
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Citation Context ...Information Technology Research Centre of Ontario. The authors' e-mail addresses are fbacchus@logos.waterloo.edu and qyang@logos.waterloo.edu. 1 Parts of Section 2 have previously appeared at AAAI-92 =-=[4]-=-, and parts of Section 3 have previously appeared at IJCAI-91 [3] 1 2 1 Introduction In this paper we examine the benefits of hierarchical problem-solving under a new, and more accurate, model of sear... |

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- Smith, Peot
- 1992
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- 1991
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Citation Context ...es multiple levels. Also worth mentioning, although not as directly relevant as the works mentioned above, is the work of Karp and Pearl [13] and its subsequent generalization by McDiarmid and Provan =-=[20]-=-. These works address the general problem of searching a tree with branches of positive costs0. They provide average case search complexity results for three different search algorithms all of which a... |

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- Christensen
- 1990
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Citation Context ...an abstraction hierarchy: a hierarchy is "good" according to Abstrips, if for every low level literal l, there is a short plan to achieve it from a state where all high level literals are tr=-=ue. Pablo [6]-=-, a successor of Abstrips, can also be viewed in this way. At a first glance, our syntactic necessary connectivity condition is similar to Abstrips since they both depend on finding short plans to ach... |

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- 1973
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Citation Context ...sed on the results of our theoretical analysis, and is completely automatic. 3.4.2 Alpine Another successor of Abstrips is Knoblock's Alpine system [14], which is related to the earlier system Lawaly =-=[27]-=-. Alpine automatically generates an abstraction hierarchy that has the ordered monotonicity property (OM). The OM property is designed to separate a problem into parts that can be solved first and the... |

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Searching for an optimal oath in a tree with random costs
- Karp, Pearl
- 1983
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Citation Context ...ical use as most work using hierarchical problem-solving utilizes multiple levels. Also worth mentioning, although not as directly relevant as the works mentioned above, is the work of Karp and Pearl =-=[13]-=- and its subsequent generalization by McDiarmid and Provan [20]. These works address the general problem of searching a tree with branches of positive costs0. They provide average case search complexi... |

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2 | Formalizing plan justi cations - Fink, Yang - 1992 |

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2 | Ste k. Planning with constraints - Mark - 1981 |

1 | Imperfect abstraction
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- 1992
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Citation Context ... as we do for this case, i.e., the case where p = 1. 7 Our analysis is a significant extension of his as it provides a model parameterized by the refinement probability p. Even more recently Williams =-=[35]-=- examined the case of imperfect abstraction. Since he is not primarily interested in Abstrips-style hierarchies as we are, he did not assume that the upwards solution property holds. In his analysis h... |