Hierarchy and statistical heuristic search

43

Bo Zha ng Tsinghua University, Beijing, P.R. China

Ling Zha ng Anqing Teachers" College, Anhui, P.R. China

In this paper we present a framework for a quotient structure model of hierarchy. As an applica- tion of this model, a new statistical heuristic search technique is introduced. Its computational complex- ity is discussed theoretically and some experimental results are showed as well.

l . I n l r o d u c l i o n

second level called INROOM plan is the level of planning robot movements within the rooms. Generally, two (or more) level planning is ex- pected to be less complex than the original one.

In this paper, we present a framework for a quotient structure view of hierarchy. Based on this model, a new statistical heuristic search technique is discussed.

The human being is good at describing and observing things from different granularities and moving back and forth between them freely, i.e., solving problems hierarchically. This is one of the main characteristics of human intelligence [1]. In order to improve the capacity of computer prob- lem solving, it is appealing to look for some for- malism of hierarchy.

In human problem solving, it is generally recog- nized that the complexity of dealing with all the details of a problem at once would be computa- tionally intractable when the problem is rather complicated. Conversely, if the original problem is divided into several abstraction levels, i.e. differ- ent granularities, by ignoring some details tempor- arily, then the problem might be solved by going from coarse level down to fine one gradually. This hierarchical solving methodology, hierarchy for short, has been used in a wide range of domains, such as search, planning, design, debugging and reasoning.

Take findpath problem of an indoor robot as an example, if the robot environment is rather complicated, it would be difficult to find a colli- sion-free path at once by considering all the de- tails of obstacles. The essence of hierarchical tech- nique is to break the problem into manageable abstraction levels by ignoring some details. For example, the path planning might have two levels. The first one referred to as INTERROOM plan is the planning of navigation from room to room without considering any details within rooms. The

North-Holland Future Generation Computer Systems 6 (1990) 43-47

2. lhe hierarchical problem sohing model

A problem at hand may be represented by a triple (X, F, f ) , where X is the problem domain, F is the domain structure, f ( . ) are some domain properties. Take findpath problem above as an example. X consists of all paths and obstacles within indoor robot world. F is the geometrical structure of robot environment and f ( - ) is the spatial properties, such as connectivity, distance and dimension etc. So f is a property function from X to Y, where Y is a set of real number, n-dimensional space or topological space etc. f is either a single-valued or multiple-valued function.

Problem solving is to analyze X, F and f . In general, domain X is quite large and complicated, thus it would be difficult to deal with all the details of X at once. Based on hierarchy X is first reduced to IX]. Instead of the original problem (X, F, f ) a new more simple one ([X], [F], [ f ] ) is investigated. Then under the guidance of in- vestigating [X], some properties of domain X are easily obtained.

In hierarchy it seems that a collection of objects in X is decomposed into different classes based on some requirements, then consider those equiv- alence classes as new elements which compose a new domain [X] with coarser grain than X. In the

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44 B. Zhang, L. Zhang / Hierarchy and statistical heuristic search

example above, the objects in the same room of indoor robot world X compose an equivalence class called room which is an element of INTER- ROOM level.

In mathematics, this means that we construct a quotient set [X] from X.

Now we have the following quotient structure model of hierarchy.

Assume (X, F, f ) is a problem at hand. Given an equivalence relation R on X, obtain a quotient set [X] under R. Considering [X] as a new do- main, new [F] and I f ] on [X] are induced from F and f on X. We have a new problem ([X], [F], I f ] ) and call introducing hierarchical technique R into X.

How to construct ([X], [F], [ f ] ) from (X, F, f ) ? In [3] we made a detailed discussion. Here we only give some main points.

As viewed from quotient structure, a hierarchy corresponds to an equivalence relation R or a partition {A~} = {[x] I x ~ X}, where [x] = { Y I x R y }, x R y denotes x is equivalent to y.

Given X, let R be all equivalence relations on X. For R1 and R 2 ~ R, If Vx, y ~ X and xR~y =, x R 2 y , then R~ is said to be finer than R 2 denoted by R 2 < R].

Proposition !. R defined as above is a complete partial-ordered lattice under relation ' < ' [3].

Quotient structure [F]

[F] represents the relationship between ele- ments [x] ~ [X]. Obviously, [F] must mirror some characteristics of F to a certain extent, so it is necessary to induce [F] from F.

Due to the variety of F, there does not exist any general method for constructing [F] from F. In [3], we show that if (X, F ) is a topological space, quotient topology [F] can be regarded as a structure in IX]. While (X, F ) is a partial-ordered space, under certain condition, a quotient partial order [F] can be induced in [X]. If the structure F of X is given by some operations p~, P2 . . . . , P,, some operation [Pt] in [X] can be induced from Pt, and [p]], [P2] . . . . . [p ,] will define a structure [F] in [X].

Quotient property function [f]

For x ~ X, f ( x ) represents some property of element x. For a ~ [ X ] , [ f ] ( a ) reflects some

property of a. While a is a subset (equivalence class) of X, [ f ] ( a ) must mirror the global prop- erty of set a. Thus constructing [ f ] may be stated as follows. For a set A c X, given local informa- tion f ( x ) , V x ~ A, constructing [ f ] means extract- ing global information of A from local one. Man is good at extracting the global information needed.

Computational complexity in hierarchical problem solving

One of the main aims of using hierarchical technique is to reduce the computational complex- ity in problem solving. In what conditions can the aim be reached? In [3] we made a thorough study about it. One of our results is the following.

For A c X, I A I denotes the number of ele- ments on A. Assume the computat ional complex- ity for seeking a goal in A only depends on [A I, i.e. the complexity function h satisfies 1. h: R ÷ ---, R ÷ is a monotonically increasing function, where R += [0, o0) 2. h ( I h l ) < IAI , IAI ~ R +.

Definition 1. Assume )(1 is a quotient set of X, h ( . ) is a complexity function of X. Assume that besides the complexity h ( I X 1 I), an additional computation y is needed in order to estimate there are u elements in X] which might contain the goal with probabili ty p (u, y).

Let

g( X 1, y ) = fou p(u, y ) du = 1 + a ( y ) ,

where D = {1, 2 . . . . . IX1 I), a ( y ) is called signifi- cance level, g ( . ) is a goal function, (h, g) is a probabilistic model of estimating the computa- tional complexity for solving X r

Given a sequence of quotient sets (or a hierarchy having n levels) X], X z . . . . . X , = X, where X i is a quotient set of X~+ 1. Assume a t ~ X~, a~ is a set of Xt+ 1 and I a~l denotes the number of its elements. Assume for Va t ~ X/, i = 1, 2 . . . . , n, ( a t [ has same value. We have

Proposition 2. Assume [ X I - O(eN) . Let signifi- cance level a i = ( l / i ) c, c > 1, for ith level. Assume Yt - O( I ln ail) . Yt is an additional complexity cor- responding to a i. h ( I X I) > O ( N In N) . I f divid-

B. Zhang, L. Zhang / Hierarchy and statistical heuristic search 45

ing X into t ( - O(N)) hierarchical levels, the com- putational complexity for seeking a goal in X is - O ( N In N). (See [3] for details.)

This proposition indicates that under certain conditions the computational complexity for prob- lem solving in domain X( I X I - O(eN)) may re- duce to O ( N In N ) by using hierarchical tech- nique properly.

The preceding quotient structure model of hierarchy can be used in a wide range of domains. In the following section we will apply the model to heuristic search.

3. Heuristic search

Assume G is a tree having N levels (Fig. 1). Let X be all leaves of G. A i is a set of nodes in ith level of G. For Vn ~ Ai, let [n] be all leaves of subtree T(n) rooted at node n. T(n) is called an i-subtree.

Now we show that heuristic search can be regarded as hierarchical problem solving.

If G is a tree having a unique goal g at depth N, looking for the goal g by heuristic search can be considered as solving problem ( X, F, f ) , where X is all nodes at depth N. If each node at depth N is viewed as an open set, then we assign a discrete topology F to X. For each node p at depth N define a function v ( p ) such that v ( g ) < v ( p ) , where p 4= g, g is a goal node. So v (p ) is one of property functions f on X.

All nodes at i th level are a partition of X. Let

X~= { [ n ] l n ~ A , ) , i= l , 2 . . . . . N.

Obviously, X 0 < X 1< . . . < X N = X is a se- quence of quotient sets of X, where Xi_ 1 is a quotient set of X,. So a tree having N levels corresponds to a sequence of quotient sets.

S A

n I i ~ n In

[nl

T(.)

Fig. 1.

In heuristic search, given a tree (graph) G and a node evaluation function f (n ) , when a new node has been expanded, a set of values f ( n ) of its successors is computed. According to a given ex- panding rules of nodes (in A* search, a node having a lower evaluation function will be ex- panded first), the node being expanded next is d e c i d e d . . . . The expanding process is continued until a goal node is found.

From a problem solving point of view, given G, we have a sequence of quotient sets X o < X 1 < . . . < X s = X. Given a node evaluation function

f (n ) , when n is an i th level node, f ( n ) is a property function defined on X, denoted by f, (n). So heuristic search can be considered as hierarchi- cal problem solving from top level X 0 going down to X for seeking the goal g. From section 2, it is known that the key for reducing complexity is how to extract f , (n ) from v (p ) , p ~ [n].

A* search uses an additive function f , (n ) (i.e. property function).

f , ( n ) = f ( n ) = g(n ) + h ( n ) ,

n ~ ith level nodes, (1)

where g(n) is the depth of node n and h(n) is an admissible heuristic estimate h* (n), the distance from node n to goal node g, while h (n) < h * (n).

Pearl [10] made a thorough study about the relations between the precision of heuristic esti- mates f ( n ) and the average complexity of A*. He assumes a uniform m-ary tree G has a unique goal node g at depth N at an unknown location. The estimates h(n) are assumed to be random varia- bles ranging over [0, h*(n)] . E ( Z ) , the expected number of nodes expanded by A*, is called the mean complexity of A*. One of his results is the following.

If the typical distance-estimation error in- creases as a fraction power of the actual distance to the goal (e.g. 0 ( N ) = v~-), then the mean com- plexity of A* grows faster than N k for any posi- tive k regardless of the shape of the distribution functions. A necessary and sufficient condition for maintaining a polynomial search complexity is that A* be guided by heuristics with logarithmic precision (e.g. 0 ( N ) = k log N). Such heuristics are hard to come by. Especially, when

P h*i )

E ( Z ) - O ( e C n ) , C > O .

46 B. Zhang, L. Zhang / Hierarchy and statistical heuristic search

From computer problem solving point of view, taking f (n ) as a property function f ,(n), it is not a good global information. We will look for a new heuristic search method.

From statistical inference [8,9], we know that given a random variable X and a statistic a(n), when a new sample has been chosen, its a(n) is computed and statistical inference is exercised. According to a given stopping rule, observation is continued until a hypothesis H 0 is accepted or rejected.

We can consider a heuristic search as random sampling process and treat evaluation functions as random variables. By transferring the statistical inference technique to the heuristic search, we have a new search technique - statistical heuristic search, SA for short [3-7].

For applying the statistical inference method, it is necessary to extract an appropriate statistic from f ( - ) . There are many means available, one of the possible ways is the following.

Fixed n ~ G, let Tk(n ) be expanded tree in T(n) having k nodes, where T(n) is a subtree rooted at n. Let

ak(n ) = F ( f ( p ) , p ~ Tk(n)), (2)

where F is an arbitrary combination function of f (p ) , f ( p ) is an estimate of {min(v(p))}, p Tk(n), p ~ Nth depth nodes.

Iiypolhesis I . For any n ~ G, assume that {(ak(n ) - g ( n ) ) } is an independent and identically distrib- uted random variable having a finite fourth moment. Let L be the shortest path from s to g, s is the initial node, g is a goal node, when n ~ L, have #(n) = #0, n ~ L, g (n ) = #1 > #0, where #(n) is the mean of {ak(n) ) .

So given a statistical inference method S, and applying this method to a heuristic search A, we have the algorithm SA as follows. Step 1: Expanding initial node s, m successors

(subtrees) are obtained. Put these subtrees on a set U.

Step 2: Exercise the statistical inference S over U. 1. If U is empty, exit with failure (the solution path is mistakenly pruned off). 2. Expanding node n at which a(n) is minimal among each i-subtree in U (if

there exist several nodes, choose one which has the maximal depth. If there still exist several nodes, choose any one at your option), install the newly generated nodes as successors of it and exercise testing statistical hypothesis over each subtree. (1) If a node at depth N is met, exit. If

the node is a goal, success; otherwise, failure.

(2) If the hypothesis is accepted on some subtree T, remove all subtrees from U except T. Subtree type index i is in- creased by one; go to Step 2.

(3) If the hypothesis is rejected on some subtree T, remove T from U; go to Step 2.

(4) Otherwise, go to Step 2. From statistics [8,9], it is known that if statistic

{a(n)} satisfies Hypothesis I, given significance level a i = ( 1 / i ) 2 for X/, there are many statistical inference methods, for example, the Wald sequen- tial probability ratio test (SPRT) and asymptotic efficient sequential fixed-width confidence inter- vals for the mean (ASM) etc., which can make a statistical decision with an additional complexity y, - O( l ln a~l). Considering search as hierarchical problem solving, from Proposition 2, it is known that the complexity of the heuristic search con- structed by those statistical inference methods will be O ( N In N), where N is the depth at which the goal is located.

The preceding result shows that if a ( a ( n ) } satisfying Hypothesis I can be obtained, the SA search will have O(N In N ) complexity. Is there any such a(n) available? Does any F ( . ) in form (2) exist? Definitely, there are. For example, if v(p) is a strictly monotonic function of the dis- tance p to g and f (n ) is an admissible estimate, then taking 'mean ' as function F(-) , ak(n ) is just the property function f~ (n) we needed.

In [3], we discuss some measures which can construct a proper { a(n)} . We also compare Hy- pothesis I with the conditions which make A* search complexity polynomial and show that the former is weaker than the latter. So statistical heuristic search has better performance than A*.

We have taken 8-puzzle as an experimental model to compare SA with BF (Best-First) search. In 80 instances, SA is superior to BF with 90% and the mean decrement in the complexity is about 25-30%.

B. Zhang, L. Zhang / Hierarchy and statistical heuristic search 47

4. ('onclusio~.s

In this paper a quotient structure model of hierarchical problem solving is presented. Under the model, we give a new heuristic search method which transfers statistical inference technique to heuristic search such that search efficiency is im- proved.

In order to clarify the principle of SA, assume G is a uniform m-ary tree (Fig. 1) and the SA consists of the statistical inference method SPRT and BF heuristic search. Then the procedure of SA search is divided into two steps:

First, it identifies quickly the most promising subtree using a subtree evaluation function a(n) (i.e. ak(n ) in form (2)). Based on the statistical inference method SPRT, from nodes n n . . . . . nlm in the first level, the m search directions, i.e. the subtrees T(nat ) . . . . . T(ntm) rooted at nodes nil . . . . . him , are selected and rejected, taking the expanded nodes in each subtree as observed sam- pies. The subtrees which contain the goal with lower probability are rejected (pruned). The most promising one is selected. Then, it expands nodes within that subtree using node evaluation function

f (n ) . Assume that after the first round of the statisti-

cal inference, subtrees T(nlt ) . . . . . T(nl , ,_ l ) are pruned, only the search direction represented by subtree T(nlm ) is accepted. Similarly, the subtrees rooted at nodes n21 . . . . . n2, . in the second level are considered . . . . the process goes on hierarchi- cally until goal g is found.

Obviously, as long as the statistical decision in each level can be made in a polynomial time, through N levels (N is the depth at which the goal is located), the goal can be found in a poly- nomial time. Fortunately, under certain conditions SPRT and many other statistical inference meth-

ods can satisfy such a requirement. This is just the benefit which SA search gets from statistical in- ference.

Besides heuristic search, the quotient structure model of hierarchy has a wide range of applica- tions; it can be used for managing uncertainty or fuzzy information. Because certainty and uncer- tainty or fuzzy and clear are relative, it depends on what abstraction level we are interested in. An uncertainty information in some abstraction level may be certain when we view from some higher level, or vice versa. The same is true of fuzzy information. So the quotient structure model of hierarchy provides a good tool for managing un- certainty and fuzzy information. A more detailed discussion of the problem is presented elsewhere [3].

References

[1] J.R. Hobbs, Granularity, in Proc. IJCAI-85, 432-435. [2] M. Eisenberg, Topology (Holt, Rinehart and Winston, Inc.

New York, 1974). [3] Zhang Bo and Zhang Ling, Artificial intelligence and

mathematics, (Chinese) to appear. [4] Zhang Bo and Zhang Ling, Statistical heuristic search, J.

Computer Sci. Technol. 2 (1) (Jan. 1987) 1-11. [5] Ling Zhang and Bo Zhang, The successive SA search and

its computational complexity, in Proc. 6th ECAI (1984) 249-258.

[6] Bo Zhang and Ling Zhang, A weighted technique in heuristic search, in Proc. 9th IJCAI (1985) 1037-1039.

[7] Zhang Bo and Zhang Ling, The comparison between the statistical heuristic search and A *, J. Computer Sci. Tech- nol. 4 (2) (1989) 126-132.

[8] S. Zacks, The Theory of Statistic Inference (Wiley, New York, 1971).

[9] G.B. Wethrill, Sequential Method in Statistics (Wiley, New York, 1975).

[10] J. Pearl, Heuristics, Intelligent Search Strategies for Com- puter Problem Solving (Addison-Wesley, Reading, MA, 1984).