Integration of Logic and Probability in Terminological and Inductive Reasoning

Elena Bellodi
Università di Ferrara
26 Marzo 2013


This thesis deals with Statistical Relational Learning (SRL), a research area combining principles and ideas from three important subfields of Artificial Intelligence: machine learning, knowledge representation and reasoning on uncertainty. Machine learning is the study of systems that improve their behavior over time with experience; the learning process typically involves a search through various generalizations of the examples, in order to discover regularities or classification rules. A wide variety of machine learning techniques have been developed in the past fifty years, most of which used propositional logic as a (limited) representation language. Recently, more expressive knowledge representations have been considered, to cope with a variable number of entities as well as the relationships that hold amongst them. These representations are mostly based on logic that, however, has limitations when reasoning on uncertain domains. These limitations have been lifted allowing a multitude of different formalisms combining probabilistic reasoning with logics, databases or logic programming, where probability theory provides a formal basis for reasoning on uncertainty. In this thesis we consider in particular the proposals for integrating probability in Logic Programming, since the resulting probabilistic logic programming languages present very interesting computational properties. In Probabilistic Logic Programming, the so-called “distribution semantics” has gained a wide popularity. This semantics was introduced for the PRISM language but is shared by many other languages: Independent Choice Logic, Stochastic Logic Programs, CP-logic, ProbLog and Logic Programs with Annotated Disjunctions (LPADs). A program in one of these languages defines a probability distribution over normal logic programs called worlds. This distribution is then extended to queries and the probability of a query is obtained by marginalizing the joint distribution of the query and the programs. The languages following the distribution semantics differ in the way they define the distribution over logic programs. The first part of this dissertation presents techniques for learning probabilistic logic programs under the distribution semantics.
Two problems are considered: parameter learning and structure learning, that is, the problems of inferring values for the parameters or both the structure and the parameters of the program from data. This work contributes an algorithm for parameter learning, EMBLEM, and two algorithms for structure learning (SLIPCASE and SLIPCOVER) of probabilistic logic programs (in particular LPADs). EMBLEM is based on the Expectation Maximization approach and computes the expectations directly on the Binary Decision Diagrams that are built for inference. SLIPCASE performs a beam search in the space of LPADs while SLIPCOVER performs a beam search in the space of probabilistic clauses and a greedy search in the space of LPADs, improving SLIPCASE performance. All learning approaches have been evaluated in several relational real-world domains.

The second part of the thesis concerns the field of Probabilistic Description Logics, where we consider a logical framework suitable for the Semantic Web. Description Logics (DL) are a family of formalisms for representing knowledge. Research in the field of knowledge representation and reasoning is usually focused on methods for providing high-level
descriptions of the world that can be effectively used to build intelligent applications. Description Logics have been especially effective as the representation language for formal ontologies. Ontologies model a domain with the definition of concepts and their properties and relations. Ontologies are the structural frameworks for organizing information and are used in artificial intelligence, the Semantic Web, systems engineering, software engineering, biomedical informatics, etc. They should also allow to ask questions about the concepts and instances described, through inference procedures. Recently, the issue of representing uncertain information in these domains has led to probabilistic extensions of DLs. The contribution of this dissertation is twofold: (1) a new semantics for the Description Logic SHOIN(D), based on the distribution semantics for probabilistic logic programs, which embeds probability; (2) a probabilistic reasoner for computing the probability of queries from uncertain knowledge bases following this semantics. The explanations of queries are encoded in Binary Decision Diagrams, with the same technique employed in the learning systems developed for LPADs. This approach has been evaluated on a real-world probabilistic ontology.