Using Constraint Satisfaction for Learning Hypotheses in Inductive Logic Programming

Inductive logic programming (ILP) is a subfield of machine learning which uses first-order logic as a uniform representation for examples, background knowledge and hypotheses (Muggleton and De Raedt, 1994). In this paper we deal with a so called template consistency problem, which is one of essential tasks in ILP (Gottlob et al 1999). In particular, given learning examples and template T, we are looking for a substitution making T consistent with the examples. Such T is called a consistent hypothesis meaning that it entails all positive examples and no negative example. Writing variables (constants) in upper (lower) cases, we assume examples expressed as sets of ground function-free atoms, e.g. E = {arc(a,b), arc(b,c), arc(c,a)}. A hypothesis is a set of atoms where all terms are variables, e.g. H = {arc(X,Y), arc(Y,Z), arc(Z,X)}. The set represents a disjunction of atoms, negation is not allowed though a negative literal can be modeled using a special atom. The hypothesis is obtained from a template by applying substitution which is basically unification of certain variables in the template. In this paper we propose a constraint model describing which variables in the template are unified to obtain consistent hypothesis. To check the entailment we use a form of -subsumption (Plotkin 1970) which is a decidable restriction of logical entailment. Hypothesis H subsumes example E, if there exists a substitution of variables such that H E. In the above example, substitution = {X/a, Y/b, Z/c} implies that H subsumes E. The requirement that a negative example E is not subsumed by hypothesis H means that there may not exist any substitution such that H E. Constraint satisfaction techniques have been previously used in ILP, though only for subsumption checking (Maloberti, Sebag, 2004). Constraint satisfaction (CS) is basically a technology for solving combinatorial problems