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O(B, D) = h such that (V(xi,f (xi))E D) (B ~ h
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xi) F f (xi)
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Of course there will, in general, be many different hypotheses h that satisfy (V(X~, f (xi)) E D ) (B A h A xi) F f (xi) One common heuristic in ILP for choosing among such hypotheses is to rely on the heuristic known as the Minimum Description Length principle (see Section 66) There are several attractive features to formulating the learning task as finding a hypothesis h that solves the relation (V(xi,f (xi))E D ) (B A h A xi) F f (xi)
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This formulation subsumes the common definition of learning as finding some general concept that matches a given set of training examples (which corresponds to the special case where no background knowledge B is available) By incorporating the notion of background information B, this formulation allows a more rich definition of when a hypothesis may be said to "fit" the data Up until now, we have always determined whether a hypothesis
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(eg, neural network) fits the data based solely on the description of the hypothesis and data, independent of the task domain under study In contrast, this formulation allows the domain-specific background information B to become part of the definition of "fit" In particular, h fits the training example (xi, f (xi)) as long as f (xi) follows deductively from B A h A xi By incorporating background information B, this formulation invites learning methods that use this background information to guide the search for h, rather than merely searching the space of syntactically legal hypotheses The inverse resolution procedure described in the following section uses background knowledge in this fashion
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At the same time, research on inductive logic programing following this formulation has encountered several practical difficulties
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The requirement @ f (xi)) E D) (B A h A xi) t f (xi) does not naturally x (' ,i accommodate noisy training data The problem is that this expression does not allow for the possibility that there may be errors in the observed description of the instance xi or its target value f (xi) Such errors can produce an inconsistent set of constraints on h Unfortunately, most formal logic frameworks completely lose their ability to distinguish between truth and falsehood once they are given inconsistent sets of assertions The language of first-order logic is so expressive, and the number of hypotheses that satisfy (V(xi, f (xi)) E D) (B A h A xi) t f (xi) is SO large, that the search through the space of hypotheses is intractable in the general case Much recent work has sought restricted forms of first-order expressions, or additional second-order knowledge, to improve the tractability of the hypothesis space search Despite our intuition that background knowledge B should help constrain the search for a hypothesis, in most ILP systems (including all discussed in this chapter) the complexity of the hypothesis space search increases as background knowledge B is increased (However, see s 11 and 12 for algorithms that use background knowledge to decrease rather than increase sample complexity)
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In the following section, we examine one quite general inverse entailment operator that constructs hypotheses by inverting a deductive inference rule
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107 INVERTING RESOLUTION
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A general method for automated deduction is the resolution rule introduced by Robinson (1965) The resolution rule is a sound and complete rule for deductive inference in first-order logic Therefore, it is sensible to ask whether we can invert the resolution rule to form an inverse entailment operator The answer is yes, and it is just this operator that forms the basis of the CIGOL program introduced by Muggleton and Buntine (1988)
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It is easiest to introduce the resolution rule in propositional form, though it is readily extended to first-order representations Let L be an arbitrary propositional literal, and let P and R be arbitrary propositional clauses The resolution rule is
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