8/7/2019 10.1.1.109.5296

1/6

Ordering Problem Subgoals

Jie Cheng and Keki B. Irani

Artificial Intelligence Laboratory

Department of Electrical Engineering and Computer Science

The University of Michigan, Ann Arbor, MI 48109-2122, USA

Abst rac t

Most past research work on problem subgoalordering are of a heuristic nature and very little attempt has been made to reveal the inherent relationship between subgoal ordering constraints and problem operator schemata. As aresult, subgoal ordering strategies which havebeen developed tend to be either overly committed, imposing ordering on subgoals subjec

tively or randomly, or overly restricted, ordering subgoals only after a violation of orderingconstraints becomes explicit during the development of a problem solution or plan. Thispaper proposes a new approach characterizedby a formal representation of subgoal orderingconstraints which makes explicit the relationship between the constraints and the problemoperator schemata. Following this approach, itbecomes straightforward to categorize varioustypes of subgoal ordering constraints, to manipulate or extend the relational representation ofthe constraints, to systematically detect important subgoal ordering constraints from problemspecifications, and to apply the detected constraints to multiple problem instances.

1 In t roduct ion

Subgoal ordering plays such an important role in planning and problem solving that a great amount of research has been dedicated to detecting subgoal ordering constraints and applying the constraints to problemspace search control [Chapman, 1987, Dawson and Sik-lossy, 1977, Ernst and Goldstein, 1982, Sacerdoti, 1974,Sacerdoti, 1975, Sacerdoti, 1977, Tate, 1975, Waldinger,1981, Warren, 1974]. However, most of the reportedapproaches are heuristic and the subgoal ordering con

straints are not well-defined. Further, very litt le attemp thas been made to reveal the inherent relationship between subgoal ordering constraints and problem operator schemata. Chapman [1987] was the first who gavea formal account to the subgoal ordering problem, buthe has not addressed the relationship between orderingconstraints and problem operator schemata. Ernst andGoldstein [1982] tried to elucidate the relationship between ordering constraints and problem operators, but

their approach requires the use of instances of problemoperator schemata and cannot guarantee the correctness of the generated ordering of subgoals in general(see [Irani and Cheng, 1987]). Consequently, subgoalordering strategies previously developed tend to be either overly committed, imposing ordering on subgoalssubjectively or randomly, or overly restricted, orderingsubgoals only after a violation of ordering constraintsbecomes explicit during the development of a problemsolution or plan.

In our research, an approach is developed to explicitlyrepresent subgoal ordering constraints. Based on thisrepresentation, procedures are then constructed to systematica lly detect the constraints. This approach makesit possible to detect ordering constraints without gettinginvolved in planning or problem solving.

Our approach proves to be advantageous in that oncea representation for a class of constraints is constructed,its properties can be studied and the representation canbe manipulated to extend its generality or to produceformu lations for new types of constraints. Another advantage of our approach over the old ones is that muchtime formerly devoted to detecting violations of constraints and ordering/reordering partial problem solutions or plans can now be saved. The constraints can bederived from problem specifications via reasoning andhenceforth, problem subgoals can be properly orderedeven before problem solving or planning. Finally, constraints among a group of problem subgoals, once derived, can be stored and applied to multiple problemsas long as they share the same problem operators andinvolve at least these subgoals. The complexi ty of theapproach is measured to be polynomial with respect tothe number of subgoals involved, with the assumptionthat in any rule schema, all that is implied by the preconditions or the postconditions is explicitly represented.

In [Irani and Cheng, 1987], our initial results in sub-

goal ordering have been reported. In this paper, wepresent an extension to our previous work. The paperis organized as follows. Fi rs t, an extended representation schema for subgoal ordering constraints is defined.The features of the represented constraints are then discussed. A method to reduce the complexity of constrain tdetection is described and then several procedures fordetecting such constraints are presented. Fina lly, an example is used to further illustrate this approach.

Cheng and Irani 931

8/7/2019 10.1.1.109.5296

2/6

2 Representing Subgoal Ord eringConstraints

In this section, basic notations used in the paper areint roduced , the in it ia l results of our research reported in[Irani and Cheng, 1987] are briefly reviewed, and thenan extended schema for representing subgoal orderingconstraints is presented.

2.1 Notat ions

To represent subgoal ordering constraints, three components of a problem model are needed, namely, a statespace, a set of problem operators and a goal specificati on . It is assumed tha t each state is represented by aconjunction of propositions. G is a conjunction of literals each of which is a subgoal. W i th this restri ction,the goal G can be equivalently represented as a set ofsubgoal literals.

In this paper, s is a symbol representing a state. T(with a suitable subscript) denotes a problem operatorwhich transforms one state into another, preckand postkare the precondition formula and postcondition formularespectively of operator Tk. Sf stands for a subset ofstates in which predicate formula / holds true. g (witha suitable subsc ript) denotes a subgoal. A problem solution is a sequence of states, (S1 S2, Sn), where slis an initial state and sn is the first state that satisfiesthe goal condi tion. We say "a subgoal g is achieved atthe m-th step of a solution (s\,S2, ..,sn) if s, Sg form < i < n and if m > 1 then s m_ i Sg. g is said to betrivially achieved in a solution ifm = 1. We say "a sub-goal gi precedes a subgoal gj in a solution (si, S2, -, Sn)"ifgi(gj) is achieved at the rni(rri2)-th step of the solutionand mi