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The main points of this chapter include: The sequential covering algorithm learns a disjunctive set of rules by first learning a single accurate rule, then removing the positive examples covered by this rule and iterating the process over the remaining training examples It provides an efficient, greedy algorithm for learning rule sets, and an alternative to top-down decision tree learning algorithms such as ID3, which can be viewed as simultaneous, rather than sequential covering algorithms In the context of sequential covering algorithms, a variety of methods have been explored for learning a single rule These methods vary in the search strategy they use for examining the space of possible rule preconditions One popular approach, exemplified by the CN2 program, is to conduct a generalto-specific beam search, generating and testing progressively more specific rules until a sufficiently accurate rule is found Alternative approaches search from specific to general hypotheses, use an example-driven search rather than generate and test, and employ different statistical measures of rule accuracy to guide the search Sets of first-order rules (ie, rules containing variables) provide a highly expressive representation For example, the programming language PROLOG represents general programs using collections of first-order Horn clauses The problem of learning first-order Horn clauses is therefore often referred to as the problem of inductive logic programming One approach to learning sets of first-order rules is to extend the sequential covering algorithm of CN2 from propositional to first-order representations
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This approach is exemplified by the FOIL program, which can learn sets of first-order rules, including simple recursive rule sets A second approach to learning first-order rules is based on the observation that induction is the inverse of deduction In other words, the problem of induction is to find a hypothesis h that satisfies the constraint where B is general background information, X I x, are descriptions of the instances in the training data D, and f (XI) f (x,) are the target values of the training instances Following the view of induction as the inverse of deduction, some programs search for hypotheses by using operators that invert the well-known operators for deductive reasoning For example, CIGOL uses inverse resolution, an operation that is the inverse of the deductive resolution operator commonly used for mechanical theorem proving PROGOL combines an inverse entailment strategy with a general-to-specific strategy for searching the hypothesis space
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Early work on learning relational descriptions includes Winston's (1970) well-known program for learning network-style descriptions for concepts such as "arch" Banerji7s (1964, 1969) work and Michalski7s series of AQ programs (eg, Michalski 1969; Michalski et al 1986) were among the earliest to explore the use of logical representations in learning Plotkin's (1970) definition of 8-subsumption provided an early formalization of the relationship between induction and deduction Vere (1975) also explored learning logical representations, and Buchanan's (1976) META-DENDRAL program learned relational descriptions representing molecular substructures likely to fragment in a mass spectrometer This program succeeded in discovering useful rules that were subsequently published in the chemistry literature Mitchell's (1979) CANDIDATE-ELIMINATION version space algorithm was applied to these same relational descriptions of chemical structures With the popularity of the PROLOG language in the mid-1980~~ researchers began to look more carefully at learning relational descriptions represented by Horn clauses Early work on learning Horn clauses includes Shapiro's (1983) MIS and Sammut and Banerji's (1986) MARVIN Quinlan7s (1990) FOIL algorithm, discussed here, was quickly followed by a number of algorithms employing a general-to-specific search for first-order rules including MFOIL (Dieroski 1991), FOCL (Pazzani et al 1991), CLAUDIEN (De Raedt and Bruynooghe 1993), and MARKUS (Grobelnik 1992) The FOCL algorithm is described in 12 An alternative line of research on learning Horn clauses by inverse entailment was spurred by Muggleton and Buntine (1988), who built on related ideas by Sammut and Banerji (1986) and Muggleton (1987) More recent work along this line has focused on alternative search strategies and methods for constraining the hypothesis space to make learning more tractable For example, Kietz and
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Wrobel (1992) use rule schemata in their RDT program to restrict the form of expressions that may be considered, during learning, and Muggleton and Feng (1992) discuss the restriction of first-order expressions to ij-determinate literals Cohen (1994) discusses the GRENDEL program, which accepts as input an explicit description of the language for describing the clause body, thereby allowing the user to explicitly constrain the hypothesis space LavraC and DZeroski (1994) provide a very readable textbook on inductive logic programming Other useful recent monographs and edited collections include (Bergadano and Gunetti 1995; Morik et al 1993; Muggleton 1992, 1995b) The overview chapter by Wrobel(1996) also provides a good perspective on the field Bratko and Muggleton (1995) summarize a number of recent applications of ILP to problems of practical importance A series of annual workshops on ILP provides a good source of recent research papers (eg, see De Raedt 1996)
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