Perturbation: A Means for Guiding Generalization

Learning problem solving from examples suffers from three problems. F i r s t , there is a strong dependency on the order of the presented examples. Second, each example has i t s own pecul iar i t ies which must be overcome. Third, the size of the generalization space can be huge, even if the instance language is small. By adding perturbation operators to the concept tree each of these problems can be al leviated. This is demonstrated in a system which learns, through interact ion with a teacher, to solve simultaneous l inear equations. 1 . Introduction With the aid of a teacher, junior high school students can learn to solve simultaneous l inear equations. A program, given the same information, has numerous problems to overcome. One problem is focussing at tent ion: the solutions presented may contain spurious associations, hiding the essential characterist ics or features. Another problem is the extremely large space of possible rules (candidate generalizations) that match a given instance of a set of l inear equations and the given appropriate operation. For a natural representation of equations there are more than one mi l l i on rules one might in fe r . We avoid these problems by examining the effect of the same operator on a "near" example created by perturbing the given example. In th is manner we can focus attent ion on the essential features and reduce the size of the search space to several thousand poss ib i l i t i es . Once th is is done we can apply standard generalization techniques, such as described by Vere [10, 12, 13] , Michalski [ 5 ] , or Mitchel l [ 6 , 8 ] . As a side benefit th is technique also mitigates the effect of the part icular sequence of examples that the teacher presented. This research was supported by the Naval Ocean System Center under contract N00123-81-C-1165. 2. Related Work Winston [15] showed the importance of "near" misses in learning concepts about the blocks world. By using perturbations "near" examples are generated automatically. We use a re lat ional production system, somewhat l i ke Vere's [12] except that we use a bag of conditions rather than a set, to represent the program's knowledge of when to apply operators. Production systems have been successfully used to model the acquisit ion of s k i l l for poker playing [14], puzzle solving [ 1 ] , algebra problems [9 ] , arithmetic problems [2 ] , and symbolic integration [ 7 ] . Of these, Neves's [9] system learned to solve one equation in one unknown from textbook traces. The system learned both the context (preconditions) of an operator as wel l as which operator was applied, although the operator had to be known to the system. His generalization language was simpler than ours in that a constant could only be generalized to a variable. Anzai [1] gradually refined weak general problem solving methods into strong ones by acquiring strategies for the tower-of-Hanoi problem, weak methods, without some heur ist ics, would leave our program with too large a space to search. The program LEX [7] uses version spaces to describe the current hypothesis space as wel l as concept trees to direct or bias the generalizations. As it is not the main point of our work, we keep only the minimal (maximally specif ic) generalization [10] of the examples. 3. Perturbation Method Before discussing the general technique of perturbations, we w i l l i l l u s t r a t e i t s use in learning to solve simultaneous l inear equations. We adopt a re lat ional description of each equation, so equation(a) 2x-3y=-7 is stored as: {tenn(a,2*x) , term(a,-3*y),term(a,7)}. Following Mitchel l [7] and Michalski [5] we have concept trees for integers, variables, and 416 D. Kibler and B. Porter Figure 3 1 : Concept tree for integers For variables the concept tree is simpler. An algebraic variable stands for i t s e l f or can be generalized to var(X), where X is a variable in the pattern language. Simi lar ly labels are either the part icular label or a label variable. Basically we are using the typed variables of Michalski [ 5 ] . We permit generalizations by 1) deleting conditions, 2) replacing constants by variables (typed), and 3) climbing tree generalization. Disjunctive generalization is allowed by adding addit ional productions or rules. This covers a l l the generalization rules discussed by Michalski [5] except for closed interval generalization. To be more speci f ic , generalizations of equation(a) are achieved by generalizing any term according to i t s concept tree or by deleting any term. term(a,2*x) has two generalizations of a (a and label (X)), four generalizations of 2 (2, posit ive(N), nonzero (N), and integer (N)), and two generalizations of x (x and var (Y)), giving a t o ta l of 16 possible generalizations. Two equations may have four such terms as wel l as two constant terms, yielding a t o ta l of 16*16*16*16*4*4 or more than a m i l l i on possible generalizations! Note we have not counted the addit ional generalizations that come about by deleting terms. The program is set the task of learning when to apply opaque operators, i . e . operators that are hard-coded and unanalyzable by the program. The operators are: add(a,b), sub(a,b), and cross(a,b,c l ,c2) . where a and b are equation labels and c l , and c2 are integers. The add(a,b) operator replaces equation (b) by the sun of equation (a) and equation(b). Simpl i f icat ion takes place as part of the application of an operator. The operator sub(a,b) is defined s imi la r ly . The operator cross(a,b,cl,c2) replaces equation(b) by cl*equation(a) c2*equation(b) • We show how these powerful operators can be learned from simpler D. Kiblerand B. Porter 417 sub(a,b) the result ing equations are: a: 2x+3y7 b: 3 y 4 which is not simplier (sub(b,a) would be effect ive however). Since the operator is ef fect ive in example i i , the system generalizes (minimally) i t s current rule conditions with th is example yielding the new ru le : {term(a,2*x),term(a,-7), term(b,2*x),tenn(b,-5*y),term(b,-3)} => sub(a,b) The major effect is to delete the condition on the y-term of equation (a). Perturbed examples for which the operator is not effect ive are disregarded. In other 'domains th is negative information might be useful, but it is not necessary for th is domain. After generalizing with example i v f the rule becomes: {term(a,2*x) ,term(b,2*x) ,term(b,-5*y), term(b,-3)} «> sub(a,b). The effect of generalizing with example v is to allow any negative coeff ic ient for the y-term of