tions).  Given a query, a set of evidence and context atoms, relevant ground p-atoms are identified by resolving context atoms away by SLDNF resolution, and a local Bayesian network is built to calculate the probability of the query. They proved the soundness and completeness of their query evaluation procedure under the condition that programs are acyclic 88 and domains are finite.

Instead of defining a local distribution for each query, Poole (1993) defined a global distribution in his "probabilistic Horn abduction". His program consists of definite clauses and disjoint declarations of the form disjoint ([h1:p1,...,h,:pn]) which specifies aprobability distribution over the hypotheses (abducibles) { h 1 , . . , h n } . He assigned probabilities to all ground atoms with the help of the theory of logic programming, and furthermore proved that Bayesian networks are representable in his framework. Unlike previous approaches, his language contains function symbols, but the acyclicity condition imposed on the programs for his semantics to be definable seems to be a severe restriction. Also, probabilities are not defined for quantified formulas.

Bacchus et al. (1996) used a much more powerful first-order probabilistic language than clauses annotated with probabilities. Their language allows a statistically quantified term such as ‖ φ ( x ) | θ ( x ) ‖ to denote the ratio of individuals in a finite domain satisfying φ ( x ) Λ
θ( x ) to those satisfying θ( x ). Assuming that every world (interpretation for their language)
is equally likely, they define the probability of a sentence ϕ under the given knowledge base KB as the limit lim N →∞  \#worlds x ( eAKB)    where \#worlds x ( x ) is the number of
\#worlds N (KB)
possible worlds containing N individuals satisfying χ , and τ parameters used in judging approximations. Although the limit does not necessarily exist and the domain must be finite, they showed that their method can cope with difficulties arising from "direct inference" and default reasoning.

In a more linguistic vein, Muggleton (1996, and others) formulated SLPs (stochastic logic programs) procedurally, as an extension of PCFGs to probabilistic logic programs.

So, a clause C , which must be range-restricted, 67 is annotated with a probability p like p : C . The probability of a goal G is the product of such ps appearing in its refutation but with a modification such that if a subgoal g can invoke n clauses, p i : C i (1 ≤ i ≤ n ) at some refutation step, the probability of choosing k -th clause is normalized to pk/ C i =1 Pi More recently, Cussens (1999, 2001) enriched SLPs by introducing a special class of log-linear models for SLD refutations w.r.t. a given goal.  He for example considers all possible SLD refutations for the most general goal ← s(X) and defines probability P(R)
of a refutation R as P(R) = Z-lexp(Li\iv(R,i)). Here Ai is a number associated with a clause C i and v ( R , i ) is a feature, i.e. the number of occurrences of C i in R .  Z is the normalizing constant. Then, the probability assigned to s(a) is the sum of probabilities of refutation for ← s(a).