Merging the local and global approaches to probabilistic satisfiability

AbstractThe probabilistic satisfiability problem is to verify the consistency of a set of probability values or intervals for logical propositions. The (tight) probabilistic entailment problem is to find best bounds on the probability of an additional proposition. The local approach to these problems applies rules on small sets of logical sentences and probabilities to tighten given probability intervals. The global approach uses linear programming to find best bounds. We show that merging these approaches is profitable to both: local solutions can be used to find global solutions more quickly through stabilized column generation, and global solutions can be used to confirm or refute the optimality of the local solutions found. As a result, best bounds are found, together with their step-by-step justification

Parameter Learning of Logic Programs for Symbolic-Statistical Modeling

We propose a logical/mathematical framework for statistical parameter learning of parameterized logic programs, i.e. definite clause programs containing probabilistic facts with a parameterized distribution. It extends the traditional least Herbrand model semantics in logic programming to distribution semantics, possible world semantics with a probability distribution which is unconditionally applicable to arbitrary logic programs including ones for HMMs, PCFGs and Bayesian networks. We also propose a new EM algorithm, the graphical EM algorithm, that runs for a class of parameterized logic programs representing sequential decision processes where each decision is exclusive and independent. It runs on a new data structure called support graphs describing the logical relationship between observations and their explanations, and learns parameters by computing inside and outside probability generalized for logic programs. The complexity analysis shows that when combined with OLDT search for all explanations for observations, the graphical EM algorithm, despite its generality, has the same time complexity as existing EM algorithms, i.e. the Baum-Welch algorithm for HMMs, the Inside-Outside algorithm for PCFGs, and the one for singly connected Bayesian networks that have been developed independently in each research field. Learning experiments with PCFGs using two corpora of moderate size indicate that the graphical EM algorithm can significantly outperform the Inside-Outside algorithm

Exploiting Uncertainty for Querying Inconsistent Description Logics Knowledge Bases

The necessity to manage inconsistency in Description Logics Knowledge Bases~(KBs) has come to the fore with the increasing importance gained by the Semantic Web, where information comes from different sources that constantly change their content and may contain contradictory descriptions when considered either alone or together. Classical reasoning algorithms do not handle inconsistent KBs, forcing the debugging of the KB in order to remove the inconsistency. In this paper, we exploit an existing probabilistic semantics called DISPONTE to overcome this problem and allow queries also in case of inconsistent KBs. We implemented our approach in the reasoners TRILL and BUNDLE and empirically tested the validity of our proposal. Moreover, we formally compare the presented approach to that of the repair semantics, one of the most established semantics when considering DL reasoning tasks