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1053 Learning Recursive Rule Sets
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In the above discussion, we ignored the possibility that new literals added to the rule body could refer to the target predicate itself (ie, the predicate occurring in the rule head) However, if we include the target predicate in the input list of Predicates, then FOIL will consider it as well when generating candidate literals This will allow it to form recursive rules-rules that use the same predicate in the body and the head of the rule For instance, recall the following rule set that provides a recursive definition of the Ancestor relation IF IF
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Parent (x, y) Parent (x, z ) A Ancestor(z, y )
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Ancestor(x, y) Ancestor@, y)
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Given an appropriate set of training examples, these two rules can be learned following a trace similar to the one above for GrandDaughter Note the second rule is among the rules that are potentially within reach of FOIL'S search, provided Ancestor is included in the list Predicates that determines which predicates may be considered when generating new literals Of course whether this particular rule would be learned or not depends on whether these particular literals outscore competing candidates during FOIL'S greedy search for increasingly specific rules Cameron-Jones and Quinlan (1993) discuss several examples in which FOIL has successfully discovered recursive rule sets They also discuss important subtleties that arise, such as how to avoid learning rule sets that produce infinite recursion
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1054 Summary of FOIL
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To summarize, FOIL extends the sequential covering algorithm of CN2 to handle the case of learning first-order rules similar to Horn clauses To learn each rule FOIL performs a general-to-specific search, at each step adding a single new literal to the rule preconditions The new literal may refer to variables already mentioned in the rule preconditions or postconditions, and may introduce new variables as well At each step, it uses the Foil-Gain function of Equation (101) to select among the candidate new literals If new literals are allowed to refer to the target predicate, then FOIL can, in principle, learn sets of recursive rules While this introduces the complexity of avoiding rule sets that result in infinite recursion, FOIL has been demonstrated to successfully learn recursive rule sets in several cases
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In the case of noise-free training data, FOIL may continue adding new literals to the rule until it covers no negative examples To handle noisy data, the search is continued until some tradeoff occurs between rule accuracy, coverage, and complexity FOIL uses a minimum description length approach to halt the growth of rules, in which new literals are added only when their description length is shorter than the description length of the training data they explain The details of this strategy are given in Quinlan (1990) In addition, FOIL post-prunes each rule it learns, using the same rule post-pruning strategy used for decision trees ( 3)
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A second, quite different approach to inductive logic programming is based on the simple observation that induction is just the inverse of deduction! In general, machine learning involves building theories that explain the observed data Given some data D and some partial background knowledge B, learning can be described as generating a hypothesis h that, together with B, explains D Put more precisely, assume as usual that the training data D is a set of training examples, each of the form (xi, f (xi)) Here xi denotes the ith training instance and f (xi) denotes its target value Then learning is the problem of discovering a hypothesis h, such that the classification f (xi) of each training instance xi follows deductively from the hypothesis h, the description of xi, and any other background knowledge B known to the system
(102) (V(xi, f (xi)) E D) (B Ah A xi) f (xi) The expression X F Y is read "Y follows deductively from X," or alternatively "X entails Y" Expression (102) describes the constraint that must be satisfied by the learned hypothesis h; namely, for every training instance xi, the target classification f (xi) must follow deductively from B, h, and xi As an example, consider the case where the target concept to be learned is "pairs of people (u, v) such that the child of u is v," represented by the predicate Child(u, v) Assume we are given a single positive example Child(Bob, Sharon), where the instance is described by the literals Male(Bob), Female(Sharon), and Father(Sharon, Bob) Furthermore, suppose we have the general background knowledge Parent (u, v) t Father (u, v) We can describe this situation in the terms of Equation (102) as follows: xi : Male(Bob), Female(Sharon), Father(Sharon, Bob)
f (xi) :
Child(Bob, Sharon)
In this case, two of the many hypotheses that satisfy the constraint (B Ah A xi) tf (xi) are hl : Child(u, v) t Father(v, u)
h2 : Child(u, v)
Parent (v, u)
Note that the target literal Child(Bob, Sharon) is entailed by hl AX^ with no need for the background information B In the case of hypothesis h2, however, the situation is different The target Child(Bob, Sharon) follows from B ~ h AX^, but 2 not from h2 AX^ alone This example illustrates the role of background knowledge in expanding the set of acceptable hypotheses for a given set of training data It also illustrates how new predicates (eg, Parent) can be introduced into hypotheses (eg, h2), even when the predicate is not present in the original description of the instance xi This process of augmenting the set of predicates, based on background knowledge, is often referred to as constructive induction The significance of Equation (102) is that it casts the learning problem in the framework of deductive inference and formal logic In the case of propositional and first-order logics, there exist well-understood algorithms for automated deduction Interestingly, it is possible to develop inverses of these procedures in order to automate the process of inductive generalization The insight that induction might be performed by inverting deduction appears to have been first observed by the nineteenth century economist W S Jevons, who wrote:
Induction is, in fact, the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation, so that the question of relative importance cannot arise Who thinks of asking whether addition or subtraction is the more important process in arithmetic But at the same time much difference in difficulty may exist between a direct and inverse operation; it must be allowed that inductive investigations are of a far higher degree of difficulty and complexity than any questions of deduction (Jevons 1874)
In the remainder of this chapter we will explore this view of induction as the inverse of deduction The general issue we will be interested in here is designing inverse entailment operators An inverse entailment operator, O(B, D ) takes the training data D = { ( x i ,f (xi))}and background knowledge B as input and produces as output a hypothesis h satisfying Equation (102)
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