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Proceedings of the Ninth National Conference on Arti- cial Intelligence, AAAI Press, Menlo Park, CA, 1991 Search Reduction in Hierarchical Problem Solving Craig A. Knoblock School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 cak@cs.cmu.edu Abstract ysis and few empirical demonstrations of the search re- It has long been recognized that hierarchical prob- ductions. Both Newell et al. 1962] and Minsky 1963] lem solving can be used to reduce search. Yet, present analyses that show that identifying interme- there has been little analysis of the problem- diate states can reduce the depth of the search, but solving method and few experimental results. these analyses assume that the intermediate states are This paper provides the rst comprehensive an- given. Korf 1987] provides an analysis of abstraction alytical and empirical demonstrations of the e ec- planning with macros, but his analysis assumes you tiveness of hierarchical problem solving. First, the are given a hierarchy of macro spaces, so that once a paper shows analytically that hierarchical prob- problem is solved in the macro space, the problem is lem solving can reduce the size of the search space solved. abstrips Sacerdoti, 1974] provides the best from exponential to linear in the solution length empirical demonstration to date, but these results are and identi es a su cient set of assumptions for in a single problem-solving domain on a small set of such reductions in search. Second, it presents em- problems. pirical results both in a domain that meets all of This paper describes hierarchical problem solving, these assumptions as well as in domains in which shows that this method can reduce the size of the these assumptions do not strictly hold. Third, the search space from exponential to linear in the solu- paper explores the conditions under which hierar- tion length, presents the assumptions that make this chical problem solving will be e ective in practice. reduction possible, and then describes experimental re- sults in three di erent problem-solving domains. The rst set of experiments provide an empirical demon- Introduction stration of the exponential-to-linear search reduction Identifying intermediate states in a search space can in a domain that fully satis es the stated assumptions be used to decompose a problem and signi cantly re- and then explores the conditions under which hierar- duce search Newell et al., 1962, Minsky, 1963]. One chical problem solving will be e ective when the as- approach to nding intermediate states is to use hi- sumptions do not strictly hold. These experiments use erarchical problem solving Newell and Simon, 1972, the Tower of Hanoi puzzle because the highly regular Sacerdoti, 1974], where a problem is rst solved in structure of the problem space makes it easy to show an abstract problem space and the intermediate states that it satis es the assumptions. The second set of in the abstract plan are used as intermediate goals to experiments provide results in both a robot-planning guide the search at successively more detailed abstrac- and a machine-shop scheduling domain, which show tion levels. that even when the assumptions of the analysis do not While hierarchical problem solving has been used in hold, the problem-solving method can still provide sig- a number of problem solvers, there has been little anal- ni cant reductions in search. The author was supported by an Air Force Laboratory Graduate Fellowship through the Human Resources Labo- Hierarchical Problem Solving ratory at Brooks AFB. This research was sponsored by the A problem solver is given a problem space, de ned by Avionics Laboratory, Wright Research and Development the legal operators and states, and a problem, de ned Center, Aeronautical Systems Division (AFSC), U.S. Air by an initial state and goal, and it searches for a se- Force, Wright-Patterson AFB, OH 45433-6543 under Con- quence of operators that can be applied to the initial tract F33615-90-C-1465, Arpa Order No. 7597. The views and conclusions contained in this document are those of state to achieve the goal. A hierarchical problem solver the author and should not be interpreted as representing employs a hierarchy of abstract problem spaces, called the o cial policies, either expressed or implied, of the U.S. abstraction spaces, to focus this search process. In- Government. stead of attempting to solve a problem in the origi- MoveL-1-3 Solution in the most abstract space. MoveM-1-2 MoveL-1-3 MoveM-2-3 Solution in the next abstract space. Solution in the ground space. Figure 1: Hierarchical Problem Solving in the Tower of Hanoi nal problem space, called the ground space, a hierar- the problem is solved in the most abstract space, which chical problem solver rst searches for a solution in simply requires a one step plan that moves the largest the most abstract problem space to produce a skeletal disk (diskL) from peg1 to peg3. This creates two sub- plan. This plan is then re ned at successive levels in problems at the next level of abstraction, where the the hierarchy by inserting additional operators to pro- rst subproblem is to reach the state where the ab- duce a complete sequence of ground-level operators. stract operator can be applied, and the second sub- This problem-solving technique was rst used in gps problem is to reach the goal state. After solving these Newell and Simon, 1972] and abstrips Sacerdoti, subproblems, the problem solver repeats the process 1974] and has since been used in a number of prob- at the next level and produces a plan that solves the lem solvers, including noah Sacerdoti, 1977], molgen original problem. Ste k, 1981], nonlin Tate, 1977], and sipe Wilkins, 1984]. Analysis of the Search Reduction Hierarchical problem solvers represent abstraction spaces in various ways and employ a variety of tech- This section presents a complexity analysis of hierar- niques for re ning an abstract plan. In this paper, the chical problem solving, which shows that, under an language of each successive abstraction space is a sub- ideal decomposition of a problem, hierarchical problem set of the previous problem spaces and the operators solving reduces the worst-case complexity of the search and states in an abstraction space correspond to one or from exponential to linear in the solution length. Since more operators or states in the more detailed problem the size of the search spaces are potentially in nite, spaces. Given a hierarchy of abstraction spaces, hi- the analysis assumes the use of an admissible search erarchical problem solving proceeds as follows. First, procedure (e.g., depth- rst iterative-deepening Korf, the problem solver maps the given problem into the 1985]), which is bounded by the length of the shortest most abstract space by deleting literals from the ini- solution. tial state and goal that are not relevant to the abstract The analysis is similar to the analysis of abstraction space. Next, the problem solver nds a solution that planning with macros by Korf 1987]. Korf showed solves the abstract problem. Each of the intermediate that using a hierarchy of macros can reduce an expo- states in the abstract plan serve as goals for the sub- nential search to a linear one. However, Korf's anal- problems at the next level in the abstraction hierarchy. ysis applies to abstraction planning with macros and The problem solver then solves each of the intermediate not to hierarchical problem solving because it makes subproblems using the nal state of one subproblem as several assumptions that do not hold for the latter. the initial state for the next subproblem. The interme- The most signi cant assumption of the analysis is that diate states of the plan at this new level then serve as when the abstract problem is solved, the original prob- goals for the subproblems at the next level, and the lem is solved. Using hierarchical problem solving, once process is repeated until the plan is re ned into the a problem has been solved in the abstract space, the ground space. This approach to hierarchical problem abstract solution must still be re ned into a solution solving is formally de ned in Knoblock, 1991]. in the ground space. Consider an abstraction hierarchy for the three-disk Tower of Hanoi, where the most abstract space con- Single-Level Problem Solving tains only the largest disk, the next abstraction space contains the largest and medium-sized disk, and the For single-level problem solving, if a problem has a so- ground space contains all three disks. This hierarchy lution of length and the search space has a branching l can be used for problem solving as shown in Figure 1. factor , P in the worst-case the size of the search b then First, the initial and goal states are mapped into the space is =1 . Thus, the worst-case complexity of l i b i abstract space by dropping the smaller disks. Next, this problem is ( ). O b l Two-Level Problem Solving In general, the size of the search space with levels n Let be the ratio of the solution length in the ground k (where the ratio between the levels is ) is: k space to the solution length in the abstract space. Thus, is the solution length in the abstract space. l X l kn;1 + ;1 l X k + ;2 l X k + + l X k (5) Since each operator in the abstract space corresponds k i i i i b b b b kn kn to one or more operators in the ground space, the k i=1 i =1 i=1 i =1 branching factor of the abstract space is bounded by The rst term in the formula accounts for the search P the branching factor of the ground space, . The size of the search tree in the abstract space is =1 , which b l=k b i in the most abstract space. Each successive term ac- counts for the search in successive abstraction spaces. i Thus, after solving the rst problem, there are ;1 is ( k ). In addition, the analysis must include the use l n l=k subproblems that will have to be solved at the next O b of this abstract solution to solve the original problem. The abstract solution de nes subproblems. The l level. Each of these problems are of size , since k k size of each problem is the number of steps (solution k is the ratio of the solution lengths between adjacent abstraction levels. At the next level there are ;2 length) in the ground space required to transform an n l=k subproblems ( ;1) each of size k, and so on. In initial state into a goal state +1 , which is repre- Si Si k l=k n sented as ( +1 ). d Si Si the nal level there are subproblems each of size k. l k The nal solution will therefore be of length = . l X X X k l ( l) The maximum reduction in search can be obtained k ;1 k d S ( 0 S1 ) ( 1 S2 ) d S d S l S k b i + b i + + b i (1) by setting the number of levels to log ( ), where the n k l =1 =1 =1 base of the logarithm is the ratio between levels. Sub- stituting log ( ) for in Formula 5 above produces the i i i l n which is ( O l k b dmax ), where following formula: k dmax 0 maxl i k ;1 ( d Si Si +1 ) (2) X k b i + k X k b i + k 2 X k i b + + k logk (l);1 X k i b (6) In the ideal case, the abstract solution will divide the i=1 =1 i i =1 i =1 problem into subproblems of equal size, and the length From Formula 6, it follows that the complexity of the of the nal solution using abstraction will equal the search is: length of the solution without abstraction. In this case, the abstract solution divides the problem into sub- l ((1 + + 2 + + logk ( );1 ) ) O k (7) k k l b k : problems of length . k k The standard summation formula for a nite geometric l max = l=k = (3) series with terms, where each term increases by a n factor of , is: d k b b b k Assuming that the planner can rst solve the abstract 1 + + 2 + + = ;; 1 +1 n problem and then solve each of the problems in the k 1 k(8) k n k : ground space without backtracking across problems, k then the size of the space searched in the worst case is Using this equation to simplify Formula 7, it follows the sum of the search spaces for each of the problems. that the complexity is: X l k b i + l X k b i (4) ( ;1 1 ) = ( ;1 ) logk ( ) ; O k ;1 k (9) l b k O l k k b : k =1 i i =1 Since and are assumed to be constant for a given b k The complexity of this search is: ( k + ). The l l k problem space and abstraction hierarchy, the complex- high-order term is minimized whenp = , which oc- O b k b ity of the entire search space is ( ). O l p l k Assumptions of the Analysis k curs whenp = . Thus, when = , the complexity p k l k l is (O ), compared to the original complexity of l b l The analysis above makes the following assumptions: ( ). O b l 1. The number of abstraction levels is log of the solu- tion length. Thus, the number of abstraction levels k Multi-Level Problem Solving must increase with the size of the problems. Korf 1987] showed that a hierarchy of macro spaces can reduce the expected search time from ( ) to 2. The ratio between levels is the base of the logarithm, k. O s O (log ), where is the size of the search space. This s s section proves an analogous result { that multi-level 3. The problem is decomposed into subproblems that are hierarchical problem solving can reduce the size of the all of equal size. If all the other assumptions hold, search space for a problem of length from ( ) to l O b l the complexity of the search will be the complexity ( ). O l of the largest subproblem in the search. 4. The hierarchical planner produces the shortest solu- rst iterative-deepening search, a depth- rst search, tion. The analysis holds as long as the length of the and a depth- rst search on a slightly modi ed version nal solution is linear in the length of the optimal of the problem. The experiments compare the CPU solution. time required to solve problems that range from one 5. There is only backtracking within a subproblem. This to seven disks. The graphs below measure the prob- requires that a problem can be decomposed such lem size in terms of the optimal solution length, not that there is no backtracking across abstraction lev- the number of disks, since the solution to a problem els or across subproblems within an abstraction level. with disks is twice as long as the solution to a prob- n lem with ; 1 disks. For example, the solution to the n The assumptions above are su cient to produce an six-disk problem requires 63 steps and the solution to exponential-to-linear reduction in the size of the search the seven-disk problem requires 127 steps. space. The essence of the assumptions is that the ab- Figure 2 compares prodigy with and without hi- straction divides the problem into ( ) constant size O l erarchical problem solving using depth- rst iterative- subproblems that can be solved in order. deepening to solve the problems and subproblems. As Consider the abstraction of the Tower of Hanoi de- the analytical results predict, the use of abstraction scribed in the previous section. It is ideal in the sense with an admissible search procedure produces an ex- that it meets all of the assumptions listed above. First, ponential reduction in the amount of search. The re- the number of abstraction levels is (log2( )). For an O l sults are plotted with the problem size along the x-axis n-disk problem the solution length is 2 ; 1, and the l n and the number of nodes searched along the y-axis. number of abstraction levels is , which is (log2 ( )). n O l With abstraction the search is linear in the problem Second, the ratio between the levels is the base of the size and without abstraction the search is exponential. logarithm, which is 2. Third, these subproblems are ef- In the Tower of Hanoi, the use of an admissible search fectively all of size one, since each subproblem requires produces optimal (shortest) solutions both with and inserting one additional step to move the disk added without abstraction. at that abstraction level. Fourth, using an admissible search strategy, the hierarchical problem solver pro- CPU Time (sec.) 600 Prodigy duces the shortest solution. Fifth, the only backtrack- 500 Hierarchical Prodigy ing necessary to solve the problem is within a subprob- lem. 400 Since these assumptions are su cient to reduce the 300 size of the search space from exponential to linear in 200 the length of the solution, it follows that hierarchi- cal problem solving produces such a reduction for the 100 Tower of Hanoi. While these assumptions hold in this 0 0 20 40 60 80 100 120 140 domain, they will not hold in all problem domains. Solution Size Yet, hierarchical problem solving can still provide sig- ni cant reductions in search. The next section explores Figure 2: Comparison using depth- rst iterative- the search reduction in the Tower of Hanoi in practice, deepening in the Tower of Hanoi. and the section following that explores the search re- duction in more complex domains where many of the Admissible search procedures such as breadth- rst assumptions do not strictly hold. search or depth- rst iterative-deepening are guaran- teed to produce the shortest solution1 and to do so Search Reduction: Theory vs. Practice usually requires searching most of the search space. The previous section showed analytically that hierar- However, these methods are impractical in more com- chical problem solving can produce an exponential-to- plex problems, so this section also examines the use linear reduction in the size of the search space. This of hierarchical problem solving with a nonadmissible section provides empirical con rmation of this result search procedure. Figure 3 compares the CPU time and then explores the conditions under which hierar- for problem solving with and without abstraction using chical problem solving will be e ective in practice. The depth- rst search. As the graph shows, the use of ab- experiments were run on the Tower of Hanoi both with straction produces only a modest reduction in search. and without using the abstraction hierarchy described This is because, using depth- rst search, neither con- in the preceding sections. The abstractions were auto- guration is performing much search. When the prob- matically generated by the alpine system Knoblock, lem solver makes a mistake it simply proceeds adding 1990] and then used in a hierarchical version of the steps to undo the mistakes. Thus, the number of nodes prodigy problem solver Minton et al., 1989]. searched by each con guration is roughly linear in the To evaluate empirically the use of hierarchical prob- 1 Due to the decomposition of a problem, an admissible lem solving in the Tower of Hanoi, prodigy was run search is not guaranteed to produce the optimal solutions both with and without the abstractions using a depth- for hierarchical problem solving. length of the solutions found. Problem solving with space allows the problem solver to undo its mistakes by abstraction performed better because the abstraction simply inserting additional steps. In domains that are provides some guidance on which goals to work on rst more constrained, the problem solver would be forced and thus produces shorter solutions by avoiding some to backtrack and search a fairly large portion of the unnecessary steps. search space to nd a solution. To demonstrate this claim, the next section presents results in two more complex problem-solving domains, where it would be CPU Time (sec.) 250 infeasible to use an admissible search. Prodigy Hierarchical Prodigy 200 150 Experimental Results This section describes the results of hierarchical prob- 100 lem solving in prodigy in two problem-solving do- 50 mains: an extended version of the strips robot- planning domain and a machine-shop scheduling do- 0 0 20 40 60 80 100 120 140 main. These domains were described in Minton, 1988], Solution Size where they were used to evaluate the e ectiveness of the explanation-based learning module in prodigy. Figure 3: Comparison using depth- rst search in the The abstraction hierarchies used in these experiments Tower of Hanoi. were automatically generated by alpine and are fully described in Knoblock, 1991]. The small di erence between depth- rst search with and without using abstraction is largely due to the fact Average Time (sec.) 300 Prodigy that the problems can be solved with relatively little Hierarchical Prodigy backtracking. To illustrate this point, consider a vari- 250 ant of the Tower of Hanoi problem that has the addi- 200 tional restriction that no disk can be moved twice in a 150 row Anzai and Simon, 1979]. By imposing additional 100 structure on the domain, the problem solver is forced to do more backtracking. Figure 4 compares the CPU 50 time used by the two con gurations on this variant 0 of the domain. This small amount of additional struc- 0 10 20 30 40 50 60 70 80 Solution Size ture enables the hierarchical problem solver to produce optimal solutions in linear time, while prodigy pro- Figure 5: Comparison in the robot-planning domain. duces suboptimal solutions that requires signi cantly more problem-solving time. Figures 5 and 6 compare the average CPU time on problems of increasing size both with and without us- ing hierarchical problem solving. Both problem do- CPU Time (sec.) 100 Prodigy Hierarchical Prodigy mains were tested on large sets of randomly generated problems (between 250 and 400 problems). Some of 80 60 the problems could not be solved by prodigy within 10 minutes of CPU time. These problems are included 40 in the graphs since including problems that exceed 20 the time bound underestimates the average, but pro- vides a better indication of overall performance. The 0 0 20 40 60 80 100 120 140 graphs show that on simple problems prodigy per- Solution Size forms about the same as hierarchical prodigy, but as the problems become harder the use of hierarchical Figure 4: Comparison using depth- rst search in a problem solving clearly pays o . In addition, hierarchi- variant of the Tower of Hanoi. cal problem solving produces solutions that are about 10% shorter than prodigy. The use of abstraction produces large search reduc- Unlike the Tower of Hanoi, these two problem- tions over problem solving without abstraction only solving domains do not satisfy the assumptions de- when a large portion of the search space must be ex- scribed in the analysis. There is backtracking both plored to nd a solution. In addition, the problem across subproblems and across abstraction levels, the solver can sometimes trade o solution quality for so- solutions are sometimes suboptimal, and the problems lution time by producing longer solutions rather than are not partitioned into equal size subproblems. De- searching for better ones. The Tower of Hanoi is per- spite this, the use of hierarchical problem solving in haps a bit unusual in that the structure of the search these domains still produces signi cant reductions in References Average Time (sec.) 350 Prodigy Anzai and Simon, 1979] Yuichiro Anzai and Her- Hierarchical Prodigy 300 250 bert A. Simon. The theory of learning by doing. 200 Psychological Review, 86:124{140, 1979. 150 Knoblock, 1990] Craig A. Knoblock. Learning ab- 100 straction hierarchies for problem solving. In Proceed- 50 ings of the Eighth National Conference on Arti cial 0 Intelligence, pages 923{928, Boston, MA, 1990. 0 2 4 6 8 10 12 14 16 18 Solution Size 20 Knoblock, 1991] Craig A. Knoblock. Automatically Generating Abstractions for Problem Solving. PhD Figure 6: Comparison in the machine-shop domain. thesis, School of Computer Science, Carnegie Mellon University, 1991. Tech. Report CMU-CS-91-120. Korf, 1985] Richard E. Korf. Depth- rst iterative- search. On the harder sets of problems, the graphs deepening: An optimal admissible tree search. Ar- show that even the hierarchical problem solver begins ti cial Intelligence, 27(1):97{109, 1985. to search more. In these domains, this can be at- Korf, 1987] Richard E. Korf. Planning as search: tributed to the fact that alpine does not currently A quantitative approach. Arti cial Intelligence, nd the best abstraction hierarchies for these prob- 33(1):65{88, 1987. lems, but this is a limitation of alpine and not of the Minsky, 1963] Marvin Minsky. Steps toward arti - hierarchical problem solver. cial intelligence. In Edward A. Feigenbaum, editor, Computers and Thought, pages 406{450. McGraw- Conclusion Hill, New York, NY, 1963. While hierarchical problem solving has long been Minton et al., 1989] Steven Minton, Jaime G. Car- claimed to be an e ective technique for reducing bonell, Craig A. Knoblock, Daniel R. Kuokka, Oren search, there has been no detailed analysis and few Etzioni, and Yolanda Gil. Explanation-based learn- empirical results. This paper presented a method for ing: A problem solving perspective. Arti cial Intel- hierarchical problem solving, showed that this method ligence, 40(1-3):63{118, 1989. can produce an exponential-to-linear reduction in the Minton, 1988] Steven Minton. Learning E ective search space, and identi ed the assumptions under Search Control Knowledge: An Explanation-Based which such a reduction is possible. In addition, the pa- Approach. PhD thesis, Computer Science Depart- per provided empirical results that show that hierarchi- ment, Carnegie Mellon University, 1988. cal problem solving can reduce search in practice, even when the set of assumptions does not strictly hold. Newell and Simon, 1972] Allen Newell and Herbert There are several interesting conclusions that one A. Simon. Human Problem Solving. Prentice-Hall, can draw from the experiments. First, the degree to Englewood Cli s, NJ, 1972. which abstraction reduces search depends on the por- Newell et al., 1962] Allen Newell, J. C. Shaw, and tion of the ground-level search space that is explored Herbert A. Simon. The processes of creative think- without using hierarchical problem solving. Thus, the ing. In Contemporary Approaches to Creative Think- more backtracking in a problem, the more bene t pro- ing, pages 63{119. Atherton Press, New York, 1962. vided by the use of abstraction. Second, with a non- Sacerdoti, 1974] Earl D. Sacerdoti. Planning in a hi- admissible search procedure the use of abstraction will erarchy of abstraction spaces. Arti cial Intelligence, tend to produce shorter solutions since the abstractions 5(2):115{135, 1974. focus the problem solver on the parts of the problem Sacerdoti, 1977] Earl D. Sacerdoti. A Structure for that should be solved rst. Third, although many do- Plans and Behavior. American Elsevier, New York, mains lack the highly regular structure of the Tower NY, 1977. of Hanoi, hierarchical problem solving can still provide signi cant reductions in search. Ste k, 1981] Mark Ste k. Planning with con- straints (MOLGEN: Part 1). Arti cial Intelligence, Acknowledgements 16(2):111{140, 1981. Tate, 1977] Austin Tate. Generating project net- I am grateful to my advisor, Jaime Carbonell, for his works. In Proceedings of the Fifth International guidance and support. I would also like to thank Jane Joint Conference on Arti cial Intelligence, pages Hsu, Paul Rosenbloom, and Manuela Veloso for their 888{900, Cambridge, MA, 1977. detailed comments on the analysis, as well as Claire Wilkins, 1984] David E. Wilkins. Domain-indepen- Bono, Oren Etzioni, and Qiang Yang for their help- dent planning: Representation and plan generation. ful comments and suggestions on earlier drafts of this Arti cial Intelligence, 22(3):269{301, 1984. paper.

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