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## Downward refinement and the efficiency of hierarchical problem solving (1994)

Venue: | ARTI CIAL INTELLIGENCE |

Citations: | 63 - 1 self |

### Citations

3257 |
An introduction to probability theory and its applications, vol. II, 2nd edn
- Feller
- 1971
(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 ... |

1694 |
Human problem solving
- Simon, Newell
- 1972
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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... |

873 |
Data Structures and Algorithms
- Aho, Hopcroft, et al.
- 1983
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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... |

730 | 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 ... |

672 | Where the really hard problems are
- Cheeseman, Kanefsky, et al.
- 1991
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Citation Context ... solutions are unrefinable, and it can take a great deal of work to discover that they are unrefinable. The existence of such a phase boundary agrees with recent empirical studies of Cheeseman et al. =-=[5]-=-, who found that the hard cases of many problems tend to cluster in the phase boundary between very many solutions and very few solutions. 2.5.2 A concrete solution does not exist We derived an expres... |

655 | Concrete mathematics
- Graham, Knuth, et al.
- 1989
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Citation Context ... ; where (x) k = x(x \Gamma 1) \Delta \Delta \Delta (x \Gamma k+1). We can now derive the required result through standard techniques for the manipulation of sums involving binomial coefficients (see =-=[11]-=-). 2.5 Predictions of the Model We can now examine what these expressions tell us about the complexity of hierarchical problem-solving. First, we consider the case where our initial assumptions hold. ... |

527 |
Planning in a Hierarchy of Abstraction Spaces
- Sacerdoti
- 1974
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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... |

448 |
Generating Project Networks
- Tate
- 1977
(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... |

435 | A Structure for Plans and Behavior - Sacerdoti - 1977 |

295 |
Practical Planning: Extending the Classical AI Planning Paradigm
- Wilkins
- 1988
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Citation Context ...ystem to have in order to guarantee the quality of the hierarchies it generates. 3.4.1 Abstrips In the past, many abstract planning systems have relied on the user to provide an abstraction hierarchy =-=[34, 30, 36]-=-. One of the first systems that semi-automatically generated its own abstraction hierarchies was Abstrips[25]. In addition to the domain specification Abstrips also requires a user-defined partial ord... |

240 | Steps toward artificial intelligence
- Minsky
(Show Context)
Citation Context ... 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 decomposition of the problem, we can refine it to a concrete solution with much less effort than if we had searched directly for a concrete solut... |

228 |
Planning as search: A quantitative approach
- Korf
- 1987
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Citation Context ... is one of the most effective techniques in practice, as demonstrated, e.g., in the work of Sacerdoti [25] and also Newell and Simon [23]. Analytical models that have been presented in the literature =-=[15, 18]-=- predict that an exponential speed-up is possible from this technique. However, these models ignore both of the problems presented above. In particular, they assume that every abstract solution 3 can ... |

195 |
All I know: A study in autoepistemic logic
- Levesque
- 1990
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Citation Context ...uld include type predicates, e.g., IsDoor(Door12), predicates indicating the status of the robot, e.g., Inroom(Room1), and 8 Hence, \Sigma \Phi satisfies "All I know is \Phi" in the sense of=-= Levesque [19]-=-. 9 This corresponds to the possible worlds notion of knowledge, where more knowledge corresponds to a smaller set of accessible worlds [12]. 27 Room 1 Room2 Door12 Robot Figure 4: Part of a simple ro... |

159 |
Domain-independent planning: Representation and plan generation
- Wilkins
- 1984
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97 |
Planning with Constraints
- Stefik
- 1980
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90 |
The process of creative thinking
- Newell, Shaw, et al.
- 1962
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Citation Context ... 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 decomposition of the problem, we can refine it to a concrete solution with much less effort than if we had searched directly for a concrete solut... |

80 |
A guide to the modal logics of knowledge and belief
- Halpern, Moses
- 1990
(Show Context)
Citation Context ...Phi satisfies "All I know is \Phi" in the sense of Levesque [19]. 9 This corresponds to the possible worlds notion of knowledge, where more knowledge corresponds to a smaller set of accessib=-=le worlds [12]-=-. 27 Room 1 Room2 Door12 Robot Figure 4: Part of a simple robot planning domain. predicates indicating the status of the doors, e.g., Open(Door12). If S = fInroom(Room1)g, then the set of possible wor... |

79 | Automatically Generating Abstractions for Problem Solving
- Knoblock
- 1991
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Citation Context ... speed-up over non-hierarchical planning. Finally, we put all of these pieces together in the design of a new hierarchy generator called Highpoint. This system builds on the Alpine system of Knoblock =-=[14]-=-, which is an automatic hierarchy generator for Abstrips-hierarchies. Highpoint takes the hierarchy suggested by Alpine and improves it using information gathered during a testing phase. These improve... |

75 |
Theorem proving with abstraction
- Plaisted
- 1981
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68 | Search reduction in hierarchical problem solving - Knoblock - 1991 |

67 |
Characterizing Abstraction Hierarchies for Planning
- Knoblock, Tenenberg, et al.
- 1991
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Citation Context ...abstraction remains important to current research: it is used in modern large scale planners [33], has been adapted to non-linear planning [36], and has been the subject of recent theoretical studies =-=[31, 16]-=-. Due to its importance it is useful to examine the special case of Abstrips hierarchies in more detail. As we will demonstrate in this section it is possible to derive more information about the down... |

56 |
Abstraction in Planning
- Tenenberg
- 1988
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Citation Context ...\Pi 0 g, such that each \Pi i is an i-th level abstract solution, and \Pi i\Gamma1 is a refinement of \Pi i . That is, there is a sequence of refinements that yield the concrete level plan. Tenenberg =-=[31]-=- has shown that this property always holds in any Abstrips-style abstraction hierarchy. In general, it will be true of any hierarchy in which abstract solutions ignore details. In this case we can gen... |

54 | ABTWEAK: abstracting a nonlinear, least commitment planner
- Yang, Tenenberg
- 1990
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Citation Context ...he work presented here originated with our observation that hierarchies don't always yield more efficiency, contrary to the predictions of these models. In fact, experiments with Abstrips and AbTweak =-=[36]-=- demonstrated that abstraction only increases search efficiency in hierarchies where most abstract solutions are refinable (i.e., where we do not have to do much backtracking in the abstraction hierar... |

33 | Abstract Theorem Proving
- Giunchiglia, Walsh
- 1989
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31 |
The Downward Refinement Property
- Bacchus, Yang
- 1991
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Citation Context ...-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 search costs, and we demonstrate how a more efficient problem-solver ... |

23 | Abstraction in Problem Solving and Learning
- Unruh, Rosenbloom
- 1989
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17 | Formalizing plan justifications - Fink, Yang - 1992 |

15 | The expected value of hierarchical problem-solving
- Bacchus, Yang
- 1992
(Show Context)
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... |

14 | An analysis of ABSTRIPS
- Knoblock
- 1992
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Citation Context ...fies that all low-level precondition literals of an operator must be simultaneously achievable, whereas Abstrips only requires that each literal be achievable individually. As pointed out by Knoblock =-=[17]-=-, in domains where interactions often occur, the existence of an individual plan for each literal does not ensure the existence of a plan for the simultaneous achievement of all of the literals. Secon... |

14 | An expected-cost analysis of backtracking and non-backtracking algorithms
- McDiarmid, Provan
- 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... |

13 | A critical look at Knoblock's hierarchy mechanism. Unpublished manuscript
- Smith, Peot
- 1992
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Citation Context ...gion. This observation is confirmed by our experimental results presented in Section 5. This problem of excessive backtracking in OM hierarchies has also been independently observed by Smith and Peot =-=[28]-=-. In conclusion, the DRP and the OM property address orthogonal issues in the design of good hierarchies. Our theoretical analysis in Section 2 indicates that there are three major criteria relevant t... |

13 | Steps Toward Arti cial Intelligence - Minsky - 1963 |

9 |
A hierarchical planner that generates its own abstraction hierarchies
- 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... |

7 | Justi plans and ordered hierarchies - Fink |

6 |
An Efficient Robot Planner Which Generates its Own Procedures, 3IJCA1
- Siklossy, Dreussi
- 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... |

4 | The downward re nement property - Bacchus, Yang - 1991 |

2 |
Searching for an optimal oath in a tree with random costs
- Karp, Pearl
- 1983
(Show Context)
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... |

2 | Justi ed plans and ordered hierarchies - Fink - 1992 |

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

2 | Planning as search: Aquantitative approach - Korf - 1985 |

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

1 | Imperfect abstraction
- Williams
- 1992
(Show Context)
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... |