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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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