Stochastic Logic Programs

One way to represent a machine learning algorithm's bias over the hypothesis and instance space is as a pair of probability distributions. This approach has been taken both within Bayesian learning schemes and the framework of U-learnability. However, it is not obvious how an Induc-tive Logic Programming (ILP) system should best be provided with a probability distribution. This paper extends the results of a previous paper by the author which introduced stochastic logic programs as a means of providing a structured deenition of such a probability distribution. Stochastic logic programs are a generalisation of stochastic grammars. A stochastic logic program consists of a set of labelled clauses p : C where p is from the interval 0; 1] and C is a range-restricted deenite clause. A stochastic logic program P has a distributional semantics, that is one which assigns a probability distribution to the atoms of each predicate in the Herbrand base of the clauses in P. These probabilities are assigned to atoms according to an SLD-resolution strategy which employs a stochastic selection rule. It is shown that the probabilities can be computed directly for fail-free logic programs and by normalisation for arbitrary logic programs. The stochastic proof strategy can be used to provide three distinct functions: 1) a method of sampling from the Herbrand base which can be used to provide selected targets or example sets for ILP experiments, 2) a measure of the information content of examples or hypotheses; this can be used to guide the search in an ILP system and 3) a simple method for conditioning a given stochastic logic program on samples of data. Functions 1) and 3) are used to measure the generality of hypotheses in the ILP system Progol4.2. This supports an implementation of a Bayesian technique for learning from positive examples only. This paper is an extension of a paper with the same title which appeared in 12]