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By Oman P.W.

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Let E = E + , E − consist of a set of positive and negative examples (ground unit clauses) respectively. The “top” clause, denoted by , is the maximal element in L. The “bottom” clause, denoted by e,S , is the least ,L element such that B, e,S |= e. Refinement of clause C, denoted by ρe,S (C), is the set of maximal ,L clauses D such that C D e,S . Definition 2 (S-consistency). Given S = B, E, L, , e ∈ E and e,S we say that C is S-consistent iff B, E, C is satisfiable. C Definition 3 (Head-connectness).

More details can be found in the full paper [7]. S. Muggleton, R. Otero, and A. ): ILP 2006, LNAI 4455, pp. 40–42, 2007. c Springer-Verlag Berlin Heidelberg 2007 Generalized Ordering-Search for Learning Directed Probabilistic Logical Models 41 2 Generalized Ordering-Search When learning directed models one of the main problems is to avoid cycles. For propositional Bayesian networks (BNs) this can be done by assuming a causal ordering on the set of random variables (RVs). Given such an ordering, we can learn for each RV X a probability tree [3] that identifies which of the preceding RVs are most relevant for X (these will be X’s parents in the BN), and specifies how they influence X (the CPD).

Given S = B, E, L, , e ∈ E and e,S we say that C is S-consistent iff B, E, C is satisfiable. C Definition 3 (Head-connectness). , bn is said to be head-connected if and only if each body atom bi contains at least one variable found either in h or in a body atom bj , where 1 ≤ j < i. S. Muggleton, R. Otero, and A. ): ILP 2006, LNAI 4455, pp. 37–39, 2007. c Springer-Verlag Berlin Heidelberg 2007 38 S. Muggleton and A. , bi−1 with Si removed Repeat Output: Reduced clause Res Fig. 1. Quick Generalisation (QG) algorithm Definition 4 (Minimal support set).

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