Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National ConferenceonArtificial Intelligence, pages 686--691, Anaheim, CA, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The point about this approach is that interactions and conflicts can be detected early on at high levels of abstraction and, therefore, resolved before further details need to be added. This has the potential to greatly reduce planning effort without compromising solution quality (see, e.g. 1] [13]) 3.2 Toward Hierarchical Case Based Reasoning The first case based reasoners to employ multiple case reuse did so in a fashion that was analogous to early goaldirected planners. Redmond [25] describes a system that solves diagnosis problems by reusing many single cases to solve individual ....

C. Knoblock, "Search Reduction in Hierarchical Problem Solving," Proc. Ninth Nat'l Conf. Artificial Intelligence, pp. 686-691, 1991.

....can find any sequence of actions whose execution can achieve a set of goals. HTN planners only find sequences that perform abstract tasks and a domain expert can intuitively define hierarchies of abstract tasks to make the planner rapidly generate all sequences of interest. Previous research [10, 9] has shown that, under certain restrictions, hierarchical refinement search reduces the search space by an exponential factor. Subsequent research has shown that these restrictions can be dropped by reasoning during refinement about the conditions embodied by abstract actions [3, 2] These ....

....data shows the exponential growth of computation with respect to the depth at which ASPEN finds solutions. Other complexity analyses have shown that under certain restrictions different forms of hierarchical problem solving can reduce the size of the search space by an exponential factor [10, 9]. Basically, these restrictions are that an algorithm never needs to backtrack from lower levels to higher levels in the problem. In other words, subproblems introduced in different branches of the hierarchy do not interact. We do not make this assumption for our problems. However, the speedup ....

C. Knoblock. Search reduction in hierarchical problem solving. In Proc. AAAI, pages 686-- 691, 1991.

....can find any sequence of actions whose execution can achieve a set of goals. HTN planners only find sequences that perform abstract tasks and a domain expert can intuitively define hierarchies of abstract tasks to make the planner rapidly generate all sequences of interest. Previous research [10, 9] has shown that, under certain restrictions, hierarchical refinement search reduces the search space by an exponential factor. Subsequent research has shown that these restrictions can be dropped by reasoning during refinement about the conditions embodied by abstract actions [3, 2] These ....

....data shows the exponential growth of computation with respect to the depth at which ASPEN finds solutions. Other complexity analyses have shown that under certain restrictions different forms of hierarchical problem solving can reduce the size of the search space by an exponential factor [10, 9]. Basically, these restrictions are that an algorithm never needs to backtrack from lower levels to higher levels in the problem. In other words, subproblems introduced in different branches of the hierarchy do not interact. We do not make this assumption for our problems. However, the speedup ....

C. Knoblock. Search reduction in hierarchical problem solving. In Proc. AAAI, pages 686-- 691, 1991.

....using the final state of one subproblem as the initial state for the next subproblem. The intermediate states of the plan at this new level then serve as goals for the subproblems at the next level, and the process is repeated until the plan is refined into the ground space. Korf [4] and Knoblock [2] presented a complexity analysis of hierarchical problem solving, which showed that, under an ideal decomposition of a problem, a hierarchical problem solving reduces the worst case complexity of the search from exponential to linear in the solution length. Since the size of the search spaces are ....

C.A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, AAAI Press, Menlo Park, CA, 1991.

....good approximations of optimal plans. 113 1 Introduction One common approach to improving the efficiency of planning is to use a hierarchical planner based on state abstraction ignoring certain literals, either in the operator preconditions [ Sacerdoti, 1974 ] or in the whole language [ Knoblock, 1991, 1994 ] First an abstracted version of the problem instance is solved, thus not taking all details into account and resulting in a plan which is correct at this abstraction level. This plan is then used as a skeleton plan to be filled in with more detail at the next lower level a process ....

....search spaces in the general case, it is usually considered a powerful method for reducing the search effort. It has been demonstrated that the method speeds up planning considerably for certain test examples [ Knoblock, 1994; Bacchus and Yang, 1994 ] This is augmented with theoretical results [ Knoblock, 1991 ] showing that state abstraction can reduce the size of the search space from exponential to linear under certain ideal conditions. These conditions are very restrictive, however, and are not likely to be met in (m)any real applications. One of the conditions is that the hierarchy satisfies the ....

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Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI-91 [ 1991 ] , pages 686--691.

...., Abtweak [ Yang and Tenenberg, 1990 ] and Prodigy [ Knoblock, 1994 ] Observe that the idea of hierarchical abstraction can be combined with both standard operator search and HTN planning, as well as with deductive planning. Hierarchical abstraction has been analyzed by, for example, Knoblock [ Knoblock, 1991b ] Knoblock tried to characterize under which restrictions an exponential reduction of the search space is possible. Using b for the branching factor and l for the length of the shortest solution, he found that the worst case size of the search space can be reduced from O(b l ) to O(l) under ....

....used in hierarchical abstraction [ Knoblock, 1994 ] In fact, the whole method we use can be viewed as a variant of two level abstraction. However, while hierarchical abstraction is a general method which is not formally well understood it can at some occasions speed up planning considerably [ Knoblock, 1991b ] and at other occasions be disastrous [ Backstrom and Jonsson, 1995 ] our, more restricted method, is provably correct and runs in polynomial time. Before presenting the method formally, we need a number of definitions. We begin by showing how a SAS problem instance can be restricted to ....

Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI-91 [ 1991 ] , pages 686--691.

....In order to reduce the search effort, one must choose an appropriate search strategy together with appropriate, typically domain dependent, heuristics in order to reduce the search effort. Some form of hierarchical abstraction is often used to make planning more efficient. State abstraction [95, 87] first solves an abstract version of the problem instance and refines this solution at less and less abstract levels by inserting more detailed subplans. Operator sequences can be abstracted as macro operators [88] i.e. treating a whole sequence as a single operator, which are used as ordinary ....

....also recursively output prefixes of prefixes etc. Recent research on the 3S problem turned out the be closely related to the work on state abstraction in planning, leading to some new insight into this area. State abstraction is known to speed up planning exponentially under ideal circumstances [87], but these circumstances are probably rare in practice. It is also known that state abstraction can make planning slower in other cases, but it seems to be assumed to be relatively harmless slowdowns. We have shown [58] that not only can planning be exponentially slower with state abstraction, we ....

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Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI-91 [43], pages 686--691.

....for example, GW92, KTY91, Pla81, Ten87] Progress has also been made in understanding better the notion of reduction in complexity. On the one hand, experiments with systems implementing abstraction have provided a quantitative evaluation of the performances of abstraction (see, for example, [Kno91b, KME91]) On the other hand, the theories of abstraction have allowed the development of models that explains why abstraction reduces the search space (see, for instance, Kor87, Kno91b] Especially in the last few years, there has been some debate on the benefits abstraction provides. As a result, both ....

.... implementing abstraction have provided a quantitative evaluation of the performances of abstraction (see, for example, Kno91b, KME91] On the other hand, the theories of abstraction have allowed the development of models that explains why abstraction reduces the search space (see, for instance, [Kor87, Kno91b]) Especially in the last few years, there has been some debate on the benefits abstraction provides. As a result, both negative and positive results can be found in the literature (see, for instance, SP92, BJ95] and [Giu96, BGSW96] Finally, we must not forget that abstraction was born as (and ....

C.A. Knoblock. Search reduction in hierarchical problem solving. In Proc. of the 9th NationalConference on Artificial Intelligence. AAAI Press, 1991. 16

.... In many domains finding a plan at all requires a considerable amount of search and there has been work on improving the efficiency of a problem solver with machine learning techniques [Mitchell et al. 1983, Laird et al. 1986, Gratch and DeJong, 1992, Korf, 1985, Veloso, 1992, Minton, 1988, Knoblock, 1991b, Etzioni, 1990] We call this speed up learning. However these mechanisms have paid none or little attention to the quality of the solutions obtained. Here we briefly present some examples of speed up learning systems in the context of PRODIGY. PRODIGY s explanation based learning system (EBL) ....

....the problem solving space but does not guarantee that the solution obtained is the best one (see [Carbonell et al. 1992] for an example) However it may lead to produce shorter solutions since the abstractions focus the problem solver on the parts of the problem that should be solved first. Knoblock, 1991b] presents experiments in which the use of abstractions produces solutions that are about 10 shorter than those produced by PRODIGY in certain domains. Note that the measure of plan quality used in this case is the length of the plan, and it seems that these results do not extend to other ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686--691, Anaheim, CA, 1991. AAAI Press/The MIT Press.

....Une telle propri#t# peut #galement s #noncer pour la composition d algorithmes [Zilberstein, 1996] Cependant l application # des algorithmes symboliques n est pas abord#e. ffl Une notion proche, l abstraction, permet de d# nir une correction partielle dans le cas d un algorithme symbolique [Knoblock, 1991] . La connection avec un algorithme de perception est #galement possible [Washington, 1995] ffl Si l environnement d utilisation est connu # l avance, remplacer le raisonnement embarqu# par un travail de ing#ni#rie pr#paratoire plus important sur l agent, ce qui n#cessite une vision a priori de ....

Craig A. Knoblock. Search Reduction in Hierarchical Problem Solving. In Proceedings of the Ninth National Conference on Artiøcial Intelligence, Anaheim, CA, 1991.

....in which the planning agent is used changes. 2 Our mechanism also involves primitive functions of the form: if a combination f of internal variables V ever happens to be higher than some threshold M max , then directly generate a goal g. There is no other goal (either more abstract [Knoblock 91] or final [Fikes Nilsson 71] from which the goal g is regressed, either by using a causality theorem [Pednault 91] or a modal truth criterion [Chapman 87] Instead, a goal is generated as a response to the current situation of the agent. The same combination f of variables V also rates goals ....

Craig A. Knoblock. Search Reduction in Hierarchical Problem Solving. In Proceedings of the 9th National Conference on Artificial Intelligence, Anaheim, CA, 1991.

....planning techniques can be applied to solve route finding problems. Regions lend themselves to create a hierarchy of space representations. Reasoning methods and useful properties of this hierarchy can be borrowed from work on abstraction theories (Giunchiglia, Villafiorita, Walsh 1997; Knoblock 1989; 1991). Most work on hierarchical route finding assumes a given hierarchy of space representations. The automatical creation of these hierarchy has been tackled in very few works (Knoblock 1994; Maio Rizzi 1993) Causal theories were introduced in (McCain Turner 1997) for the propositional case, and ....

Knoblock, C. 1991. Search reduction in hierarchical problem solving. In AAAI-91. AAAI Press.

....aucun affinement de ce plan n affectera les litt eraux d ej a etablis dans celui ci. Il est clair que la DRP est la propri et e la plus int eressante qu une hi erarchie puisse satisfaire, pouvant conduire sous certaines conditions id eales a une r eduction exponentielle du cout de la recherche [12] N eanmoins, il s agit d un crit ere tr es fort et donc, qui ne peut etre garanti que dans des cas tr es particuliers. L OMP est plus r ealiste et plus int eressante pour des probl emes g en eraux qui se posent en pratique. Lorsque cette propri et e est v erifi ee, beaucoup de sources de ....

....montr es dans le cadre du syst eme I XTET mais, dans la mesure o u ils reposent essentiellement sur Domaines facteur d G n G =p G nombre nombre taille de gain attributs taches plan Hilares 1.4 2 1.5 3 3 10 Iares 2.7 3 1.5 5 11 19 Columbus prob. 1 1.7 4 1.7 16 11 10 prob. 2 2.3 4 1. 7 16 11 12 Ariane 3.9 6 6 6 0 56 Finition 8.8 6 4 18 12 18 pi ece Table 2: Gains globaux li es a la hi erarchisation hi erarchie ordonn ee hi erarchie hors ligne: dynamique: Noeuds 305 59 Retour arri eres 14 0 Temps CPU 14.1 s 2.8 s Table 3: Gains li es a l approche de moindre engagement l approche ....

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C.A. Knoblock. Search Reduction in Hierarchical Problem Solving. In Proceedings AAAI-91, pages 686--691, 1991.

....exists an operator that transforms state v to state w. 10 and predicate relaxation [ Christensen, 1990 ] Of these paper III discusses state abstraction. In state abstraction, certain literals are ignored, either in the operator preconditions [ Sacerdoti, 1974 ] or in the whole language [ Knoblock, 1991, 1994 ] First an abstracted version of the problem instance is solved, thus not taking all details into account, which results in a plan that is correct at this abstraction level. This plan is then used as a skeleton plan to be filled in with more detail at the next lower level a process ....

Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI-91 [ 1991 ] , pages 686--691.

....performance improvement by exploiting locality in temporal projection. The notions of subgoals and abstract events play important roles in our localized reasoning algorithm. Our approach of subgoal and abstraction is different from previous work of using subgoal [Kor87] and abstraction [Chr90] [Kno91] [YT90] in reducing the search effort in planning. In this paper, subgoals and abstract events are derived according to the structure in event ordering and search space inherent from the individual problem instances. Given an instance of the temporal projection problem, the goal can be decomposed ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI-91, pages 686--691. AAAI, 1991.

....Durfee Dept. of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109 monty caen.engin.umich.edu, durfee caen.engin.umich.edu Abstract Knoblock and Korf have determined that abstraction can reduce search at a single agent from exponential to linear complexity [ Knoblock, 1991; Korf, 1987 ] We extend their results by showing how concurrent problem solving among multiple agents using abstraction can further reduce search to logarithmic complexity. We empirically validate our formal analysis by showing that it correctly predicts performance for the Towers of Hanoi ....

....as a distributed search [Durfee and Montgomery, 1991] Therefore, when attempting to improve performance in a distributed problem solving system, it is natural to see how recent results in single agent search can be applied to such multi agent systems. Recent results by Knoblock and Korf [Knoblock, 1991; Korf, 1987] show that abstraction can reduce search from exponential to linear complexity by dividing a large problem into a number of smaller problems. We extend these results in Section 2 by showing that if these smaller problems are distributed to different agents to be solved in parallel, ....

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Knoblock, Craig A. 1991. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence.

....abstract operators with collections of more detailed (less abstract) operators. Eventually, a complete plan is built which contains only primitive operators. Studies have shown that such hierarchical approaches can significantly reduce search while developing high quality solution plans ( 1] [11]) 6 3.2 Introducing Hierarchical Case Based Reasoning Hierarchical planning has motivated the development of hierarchical case based reasoning, which also solves problems within a hierarchy of abstraction spaces, by storing, retrieving, and adapting abstract cases as well as concrete design ....

C. Knoblock, "Search Reduction in Hierarchical Problem Solving" Proceedings of the 9 th National Conference on Artificial Intelligence, Anaheim, USA, 1991, pp. 686-691.

....planning systems, such as prodigy, tweak, and snlp. This analysis is an approximation based on several simplifying assumptions about properties of planning domains. It is similar to the analysis of planning with macro operators in [Korf, 1987] and the analysis of hierarchical problem solving in [Knoblock, 1991a] Even though most real domains do not satisfy the restrictive assumptions of our formal analysis, the experimental results in Section 4 demonstrate that planning algorithms usually behave as predicted by the analysis. 3.1 Assumptions of the analysis A planning algorithm determines a search ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686-691, 1991.

....would work (called i goals) expected i goal durations, and a mapping of the i goal start and end times with respect to node problem solving time. Two extensions need to be made. First, communicating goals at a single level of detail is inappropriate and potentially wasteful in more complex domains[23]; certainly the detection of the interactions of two goals ( partial global goals ) will not always be simple [33] Secondly, many different methods may exist for accomplishing a goal, each with its own effects on duration, precision, and other goal characteristics. This makes the existing PGP ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, Anaheim, July 1991.

....we have a non abstract solution: no further search is required. Nevertheless, Korf s analysis can be viewed as demonstrating that searching for an abstract solution is significantly more efficient that searching for a non abstract solution. Knoblock s analysis of hierarchical problem solving [Kno91] is the most detailed to date, and has had a significant influence on this work. However, his analysis assumes that backtracking does not occur across abstraction levels: once an abstract solution is found we need never search for another one. Hence, Knoblock s work can be viewed as demonstrating ....

....of O(b ) where abstraction yields no benefits. Our two assumptions, then, are basic assumptions required before the abstraction hierarchy yields any interesting behavior at all. When these assumptions fail the abstraction hierarchy is simply not decomposing the problem effectively. Knoblock [Kno91] also relies on these assumptions, but his assumption of independent subproblems is phrased as an assumption that backtracking only occurs within a subproblem. This is significantly stronger as it also prohibits backtracking across abstraction levels. The tree of abstract plans will have a ....

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Craig Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the AAAI National Conference, pages 686--691, 1991.

....in polynomial time and its solution constitutes a skeleton to be filled in by solving subproblems from the second instance. This process is referred to as interweaving and can be viewed as a restricted variant of the more general concept refinement, as used in hierarchical state abstraction [ Knoblock, 1991 ] In fact, the whole method we use can be viewed as a restricted variant of two level state abstraction. However, while state abstraction is a general method which is not formally well understood it can at some occasions speed up planning considerabley [ Knoblock, 1991 ] and at other occasions ....

....state abstraction [ Knoblock, 1991 ] In fact, the whole method we use can be viewed as a restricted variant of two level state abstraction. However, while state abstraction is a general method which is not formally well understood it can at some occasions speed up planning considerabley [ Knoblock, 1991 ] and at other occasions be disastrous [ Backstrom and Jonsson, 1995 ] our, more restricted method, is provably correct, guaranteed not to make things worse and runs in polynomial time. First we show how a SAS problem instance can be restricted to take only a subset of the variables into ....

C. A. Knoblock. Search reduction in hierarchical problem solving. In AAAI91

....in polynomial time and its solution constitutes a skeleton to be filled in by solving subproblems from the second instance. This process is referred to as interweaving and can be viewed as a restricted variant of the more general concept refinement, as used in hierarchical state 97 abstraction [ Knoblock, 1991 ] In fact, the whole method we use can be viewed as a restricted variant of two level state abstraction. However, while state abstraction is a general method which is not formally well understood it can on some occasions speed up planning considerably [ Knoblock, 1991 ] and on other occasions ....

....state 97 abstraction [ Knoblock, 1991 ] In fact, the whole method we use can be viewed as a restricted variant of two level state abstraction. However, while state abstraction is a general method which is not formally well understood it can on some occasions speed up planning considerably [ Knoblock, 1991 ] and on other occasions be disastrous [ Backstrom and Jonsson, 1995 ] our, more restricted method, is provably correct, guaranteed not to make things worse and runs in polynomial time. First we show how a SAS problem instance can be restricted to take only a subset of the variables into ....

Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI-91 [ 1991 ] , pages 686--691.

....paper [ Dean and Lin, 1995 ] 6 Related Work The related work on abstraction and decomposition is extensive. In the area planning and search assuming deterministic action models, there is the work on macro operators [ Korf, 1985 ] and hierarchies of state space operators [ Sacerdoti, 1974 ] Knoblock, 1991 ] Closely related is the work on decomposing discrete event systems modeled as (deterministic) finite state machines [ Zhong and Wonham, 1990 ] Caines and Wang, 1990 ] In the area of reinforcement learning, there is work on deterministic action models and continuous state spaces [ Moore and ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI91, pages 686--691. AAAI, 1991.

....Veloso, 1994] There has, however, been little research on the common principles underlying different types of representation changers. Researchers have developed theoretical frameworks for some special cases of representation changes, most notably generating abstraction hierarchies [Korf, 1987; Knoblock, 1991b; Knoblock et al. 1991; Bacchus and Yang, 1992; Giunchiglia and Walsh, 1992] replacing operators with macros [Korf, 1985; Korf, 1987; Mooney, 1988] and learning control rules [Cohen, 1992] A generalized model of representation changes was suggested by Korf, who formalized the concept of ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686--691, 1991.

....in the context of task planning problems. One potential research direction involves the notion of abstraction, i.e. splitting an action into several levels and solving the most abstract levels rst independently of the concerned action. These abstraction levels can be generated automatically [Knoblock, 1991] or manually [Washington and Hayes Roth, 1995] and have been demonstrated to save an exponential amount of time on some domains. It is related to anytime properties in that a planner that plans by decreasing levels of abstraction might be able to return an abstract plan when interrupted. But other ....

....is action addition. 2 Theorem 1 mainly suggests that the MTC should be regressed on the action templates themselves (i.e. action descriptions that are copied and instantiated to generate actions actually used in a plan) in order to predict some changes made by adding actions to a plan [Knoblock, 1991]. This theorem also suggests that no new unsatis ed prerequisite can result from a plan modi cation dioeerent from action addition: only existing unsatis ed need being checked which in itself leads to substantial performance improvements [Morignot, 1994] A consequence of this is that between two ....

Craig A. Knoblock. Search Reduction in Hierarchical Problem Solving. In Proceedings of the Ninth National Conference on Artiøcial Intelligence, Anaheim, CA, 1991.

....determining independence of a clause from a predicate refinement, 4 thereby yielding an algorithm for determining irrelevance of predicate refinements. The computational savings gained by using abstractions has been demonstrated both theoretically and empirically (e.g. Bacchus and Yang, 1992; Knoblock, 1991; Ellman, 1993 ] In our case the savings achieved by abstractions will be maximized if we can identify large sets of queries for which we can create the same abstract KB, and therefore amortize the cost of creating the abstract KB over many queries. One of the key advantages of using the ....

Knoblock, Craig A. 1991. Search reduction in hierarchical problem solving. In Proceedings of AAAI-91. 686--691.

....2 and (ii) O(N 12=7 ) number of iterations when k = 3. 10 Related work The related work on abstraction and decomposition is extensive. In the area planning and search assuming deterministic action models, there is the work on macro operators [13] and hierarchies of state space operators [22] [12]. Closely related is the work on decomposing discrete event systems modeled as (deterministic) finite state machines [23] 2] In the area of reinforcement learning, there is work on deterministic action models and continuous state spaces [18] and stochastic models and discrete state spaces [11] ....

Knoblock, Craig A., Search Reduction in Hierarchical Problem Solving, Proceedings AAAI-91, Anaheim, California, AAAI, 1991, 686--691.

....solving. Our techniques provide a possible way to reduce temporal reasoning to graph reachability. Our results do not depend on any severe restrictions on the causal rules. Finally, in Section 5, we discuss the possibility of applying the concept of planning by hierarchical abstraction spaces [Knoblock, 1991] to temporal reasoning, and point out that we cannot characterize abstraction hierarchies for temporal reasoning in the same way in planning, since we need to consider the ordering constraints on events and the limited number of available events. 2 Representations and Perspectives 2.1 Temporal ....

....state space can be reduced by replacing the finite automaton determined by the problem instance with an equivalent finite automaton involving fewer states. The algorithms for minimizing finite automata can be used to reduce the size of the state space. 5 Temporal Reasoning by Abstraction Knoblock [Knoblock, 1991] discusses planning using abstraction hierarchies to reduce the size of the search space. Given an abstraction hierarchy for planning, variables are partitioned into levels. An abstract plan is searched at the highest level, and then refined level by level to incrementally achieve the goal ....

[Article contains additional citation context not shown here]

Knoblock, Craig A. 1991. Search reduction in hierarchical problem solving. In Proceedings AAAI-91. AAAI. 686--691.

....would work (called i goals) expected i goal durations, and a mapping of the i goal start and end times with respect to node problem solving time. Two extensions need to be made. First, communicating tasks at a single level of detail is inappropriate and potentially wasteful in more complex domains[Knoblock, 1991] ; certainly the detection of the interactions of two tasks will not always be simple [Robinson and Fickas, 1990, v. Martial, 1992, Durfee and Montgomery, 1991] Secondly, many different methods may exist for a task, each with its own effects on duration, precision, and other task ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, Anaheim, July 1991.

....utility of an optimal policy generated from the abstract Markov decision process and the expected utility of an optimal policy generated from the original process. The abstractions used in Dearden and Boutilier [1994] are closely related to methods described by Sacerdoti [1974] Korf [1987] Knoblock [1991], and Lin and Dean [1994] for planning and temporal reasoning in deterministic environments. There is a long history of using abstractions in both artificial intelligence and disciplines such as adaptive control that deal with Markov decision processes. Davidson and Fehling [1994] and Nicholson ....

Knoblock, Craig A., 1991, Search reduction in hierarchical problem solving, In Proceedings AAAI-91, AAAI, 686-691.

....finding good approximations of optimal plans. 1 Introduction One common approach to improving the efficiency of planning is to use a hierarchical planner based on state abstraction ignoring certain literals, either in the operator preconditions [ Sacerdoti, 1974 ] or in the whole language [ Knoblock, 1991, 1994 ] First an abstracted version of the problem instance is solved, thus not taking all details into account and resulting in a plan which is correct at this abstraction level. This plan is then used as a skeleton plan to be filled in with more detail at the next lower level a process ....

....search spaces in the general case, it is usually considered a powerful method for reducing the search effort. The method has been demonstrated to speed up planning considerably for certain test examples [ Knoblock, 1994; Bacchus and Yang, 1994 ] This is augmented with theoretical results [ Knoblock, 1991 ] showing that state abstraction can reduce the size of the search space from exponential to linear under certain ideal conditions. These conditions are very strong, however, and are not likely to be met in (m)any real applications. One of the conditions is that the hierarchy satisfies the ....

[Article contains additional citation context not shown here]

Craig A Knoblock. Search reduction in hierarchical problem solving. In AAAI [ 1991 ] , pages 686--691.

....in that problem space. The intermediate states of the abstract plan are then used as intermediate goals to guide the search for a more detailed plan. Knoblock showed that this method can reduce the search complexity from exponential to linear when searching for a solution with the same length [ Knoblock, 1991 ] In TBSA, the phase of organizing the hierarchical map corresponds to the step of abstraction, and the phase of reconstructing paths from the hierarchical map corresponds to the steps of problem solving and reconstitution. The major issue of TBSA and hierarchical problem solving in general, is ....

Craig A. Knoblock. Search Reduction in Hierarchical Problem Solving. In Proceedings of AAAI-91, pages 686--691, 1991.

....values of a selected subset of variables. Aggregation and abstraction are employed as tools for constructing compact and equivialent Markov decision processes or finite automata to improve computational efficiency. Different abstraction techniques have been investigated in classic STRIPS planning [Kno91b] Ten91] Sac74b] HS94] to cope with very large search spaces. 8.4 Probabilistic Inference in Bayesian Networks Localized probabilistic inference in Bayesian networks [LS88] JLO90] provides an analogy to the decomposition approach for planning under uncertainty in this thesis. In a Bayesian ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI-91, pages 686--691. AAAI, 1991.

.... about temporal intervals [All83] GS93] KG77] and in the event calculus [Eva90] Mea92] The use of subgoals and abstract events in Chaprer 4 is inspired by the similar notions of subgoals and abstraction used in search [Kor87] and in abstract and hierarchical planning [Sac74a] Chr90] Kno91a] YT90] in reducing the overall search effort. The idea of using localized reasoning to exploit locality is also studied by Lansky in GEM [Lan88] in event based planning. In GEM, locality is similarly modeled by sets of interrelated events delimited by temporal logic constraints. 5.8 Summary In ....

....(ii) O(N 12=7 ) number of iterations when k = 3. 7.5 Related work The related work on abstraction and decomposition is extensive. In the area planning and search assuming deterministic action models, there is the work on macro operators [Kor85] and hierarchies of state space operators [Sac74a] Kno91a] Closely related is the work on decomposing discrete event systems modeled as (deterministic) finite state machines [ZW90] CW90] In the area of reinforcement learning, there is work on deterministic action models and continuous state spaces [MA95] and stochastic models and discrete state ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI-91, pages 686--691. AAAI, 1991.

.... knowledge about temporal intervals [All83] GS93] KG77] and in the event calculus [Eva90] Mea92] The use of subgoals and abstract events in this paper is inspired by the similar notions of subgoals and abstraction used in search [Kor87] and in abstract and hierarchical planning [Sac74] Chr90] [Kno91] [YT90] in reducing the overall search effort. The idea of using localized reasoning to exploit locality is also studied by Lansky in GEM [Lan88] in event based planning. In GEM, locality is similarly modeled by sets of interrelated events delimited by temporal logic constraints. 10. CONCLUSION ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI91, pages 686--691. AAAI, 1991.

....the problem into easier subproblems. Nevertheless, when it works, hierarchical problem solving 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 be refined to a concrete solution without backtracking across abstraction levels, and they also ....

....the probability of a bad subtree is zero. Our expression for GoodTreeWork(n) Eq. 8, thus simplifies to GoodTreeWork(n) b k k n Gamma1 k Gamma1 . Hence, we have GoodTreeWork(n) O(b k k n Gamma1 ) p = 1: 9) This result agrees with Knoblock s previous analysis which assumed the DRP [15]. This result also shows that at p = 1 the work required to search a good tree is simply P n i=1 NodeWork(i) This is to be expected. With p = 1 we never have to backtrack, and we simply need to do the work required to refine a single abstract solution down through the levels of abstraction, ....

[Article contains additional citation context not shown here]

Craig Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the AAAI National Conference, pages 686--691, 1991.

....hard problem (see [2] and it is generally accepted that most non trivial planning problems are at least NPcomplete. In order to cope with the combinatorial explosion of the search problem, AI researchers proposed a wide variety of solutions, from search control rules [3] to hierarchical planning [8] to skeletal planning [5] More recently, we witnessed the occurrence of the stochastic planners, which trade in the completeness of the planner for the speed up of the search process (e.g. SatPlan [7] or PBR [1] are at least one order of magnitude faster than the classic planning systems) In ....

Knoblock, C. "Search Reduction in Hierarchical Problem Solving." Proceedings of the National Conference on Artificial Intelligence 686-691, 1991.

....that most nontrivial planning problems are at least NP complete. In order to cope with the combinatorial explosion of the search problem, AI researchers proposed a wide variety of solutions, from search control rules (Weld 1994; Etzioni 1993; Minton 1996) to abstraction and hierarchical planning (Knoblock 1991; Knoblock 1994) to skeletal planning (Friedland and Iamasaki 1985) More recently, we witnessed the occurrence of a new type of planning systems: the stochastic planners. The new approach trades in the completeness of the planner for the speed up of the search process. Planners like SatPlan ....

Knoblock, C. 1991. Search Reduction in Hierarchical Problem Solving. In Proceedings of the National Conference on Artificial Intelligence, Pages 686691.

No context found.

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National ConferenceonArtificial Intelligence, pages 686--691, Anaheim, CA, 1991.

No context found.

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686--691, 1991.

....linear in the size of the solution under certain assumptions. For single level planning the size of the search space is exponential in the solution length. Hierarchical planning reduces this complexity by taking a large complex problem and decomposing it into a number of smaller subproblems. See [33, 35] for a formal definition of hierarchical planning and an analysis of the search reduction. In addition, hierarchical planning can improve the performance of a learning system. For an example see [36] which describes the integration of abstraction and explanation based learning in the context of ....

....This formal definition captures the notion of plan refinement used in a number of different planners, including abstrips [53] abtweak [68] and pablo [11] 2. 4 Ordered Monotonicity Property Hierarchical planning reduces search by partitioning a problem into a number of smaller subproblems [33]. An effective partitioning of a problem requires that the subproblems can be solved without violating the conditions that were already achieved in the more abstract levels of the abstraction hierarchy. In other words, a hierarchical planner ideally finds a solution at one level and then maintains ....

[Article contains additional citation context not shown here]

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, Anaheim, CA, 1991, 686--691.

....levels of abstraction as possible as long as good properties are preserved. However, the evidence for high efficiency of planning in finer grained hierarchies is mostly empirical. Some theoretical work has been done to demonstrate high efficiency of planning in multilevel ordered hierarchies [Knoblock, 1991b] Bacchus and Yang 1992] but the results presented in these papers are based on assumptions that are too strong for most planning problems. 2.6 Data structures and basic algorithms All sets and relations discussed above must somehow be represented in the computer memory. We use the real RAM ....

....domain has n variables, it is probably better to have two levels of abstraction, each containing n=2 variables, then three levels of abstractions containing respectively 1, 1, and (n Gamma 2) variables. Some theoretical support for this intuition may be derived by the method presented in [Knoblock, 1991b] Also, it would be interesting to approach the problem of generating hierarchies with the Downward Refinement Property [Bacchus and Yang, 1991] using primary effect restricted planning. If the same primary effect restricted hierarchy is used repeatedly, it may be worthwhile to find a hierarchy ....

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686--691, 1991.

No context found.

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686-691, 1991.

No context found.

C.A. Knoblock, Search Reduction in Hierarchical Problem Solving. Proceedings of the 9 National Conference on Artificial Intelligence, 1991, 686-691, Anaheim, CA.

No context found.

Craig Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the 9th AAAI, volume 2, pages 686--691, 1991.

No context found.

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings of the Ninth National Conference on Artificial Intelligence, pages 686--691, 1991.

No context found.

Craig A. Knoblock. Search reduction in hierarchical problem solving. In Proceedings AAAI-91, pages 686--691. AAAI, 1991.

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