Learning definite Horn formulas from closure queries

A definite Horn theory is a set of n-dimensional Boolean vectors whose characteristic function is expressible as a definite Horn formula, that is, as conjunction of definite Horn clauses. The class of definite Horn theories is known to be learnable under different query learning settings, such as learning from membership and equivalence queries or learning from entailment. We propose yet a different type of query: the closure query. Closure queries are a natural extension of membership queries and also a variant, appropriate in the context of definite Horn formulas, of the so-called correction queries. We present an algorithm that learns conjunctions of definite Horn clauses in polynomial time, using closure and equivalence queries, and show how it relates to the canonical Guigues–Duquenne basis for implicational systems. We also show how the different query models mentioned relate to each other by either showing full-fledged reductions by means of query simulation (where possible), or by showing their connections in the context of particular algorithms that use them for learning definite Horn formulas.Peer ReviewedPostprint (author's final draft

Structure identification in relational data

This paper presents several investigations into the prospects for identifying meaningful structures in empirical data, namely, structures permitting effective organization of the data to meet requirements of future queries. We propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability. Using this framework, we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a CNF theory with desirable topology and, if the answer is positive, identifying the desired decomposition. Finally, we address the problem of expressing a given relation as a Horn theory and, if this is impossible, finding the best k-Horn approximation to the given relation. We show that both problems can be solved in time polynomial in the length of the data

Canonical Horn representations and query learning

We describe an alternative construction of an existing canonical representation for definite Horn theories, the emph{Guigues-Duquenne} basis (or GD basis), which minimizes a natural notion of implicational size. We extend the canonical representation to general Horn, by providing a reduction from definite to general Horn CNF. We show how this representation relates to two topics in query learning theory: first, we show that a well-known algorithm by Angluin, Frazier and Pitt that learns Horn CNF always outputs the GD basis independently of the counterexamples it receives; second, we build strong polynomial certificates for Horn CNF directly from the GD basis.Postprint (published version

Inductive Logic Programming in Databases: from Datalog to DL+log

In this paper we address an issue that has been brought to the attention of the database community with the advent of the Semantic Web, i.e. the issue of how ontologies (and semantics conveyed by them) can help solving typical database problems, through a better understanding of KR aspects related to databases. In particular, we investigate this issue from the ILP perspective by considering two database problems, (i) the definition of views and (ii) the definition of constraints, for a database whose schema is represented also by means of an ontology. Both can be reformulated as ILP problems and can benefit from the expressive and deductive power of the KR framework DL+log. We illustrate the application scenarios by means of examples. Keywords: Inductive Logic Programming, Relational Databases, Ontologies, Description Logics, Hybrid Knowledge Representation and Reasoning Systems. Note: To appear in Theory and Practice of Logic Programming (TPLP).Comment: 30 pages, 3 figures, 2 tables

Bayesian Logic Programs

Bayesian networks provide an elegant formalism for representing and reasoning about uncertainty using probability theory. Theyare a probabilistic extension of propositional logic and, hence, inherit some of the limitations of propositional logic, such as the difficulties to represent objects and relations. We introduce a generalization of Bayesian networks, called Bayesian logic programs, to overcome these limitations. In order to represent objects and relations it combines Bayesian networks with definite clause logic by establishing a one-to-one mapping between ground atoms and random variables. We show that Bayesian logic programs combine the advantages of both definite clause logic and Bayesian networks. This includes the separation of quantitative and qualitative aspects of the model. Furthermore, Bayesian logic programs generalize both Bayesian networks as well as logic programs. So, many ideas developedComment: 52 page

Intensional Query Processing in Deductive Database Systems.

This dissertation addresses the problem of deriving a set of non-ground first-order logic formulas (intensional answers), as an answer set to a given query, rather than a set of facts (extensional answers), in deductive database (DDB) systems based on non-recursive Horn clauses. A strategy in previous work in this area is to use resolution to derive intensional answers. It leaves however, several important problems. Some of them are: no specific resolution strategy is given; no specific methodologies to formalize the meaningful intensional answers are given; no solution is given to handle large facts in extensional databases (EDB); and no strategy is given to avoid deriving meaningless intensional answers. As a solution, a three-stage formalization process (pre-resolution, resolution, and post-resolution) for the derivation of meaningful intensional answers is proposed which can solve all of the problems mentioned above. A specific resolution strategy called SLD-RC resolution is proposed, which can derive a set of meaningful intensional answers. The notions of relevant literals and relevant clauses are introduced to avoid deriving meaningless intensional answers. The soundness and the completeness of SLD-RC resolution for intensional query processing are proved. An algorithm for the three-stage formalization process is presented and the correctness of the algorithm is proved. Furthermore, it is shown that there are two relationships between intensional answers and extensional answers. In a syntactic relationship, intensional answers are sufficient conditions to derive extensional answers. In a semantic relationship, intensional answers are sufficient and necessary conditions to derive extensional answers. Based on these relationships, the notions of the global and local completeness of an intensional database (IDB) are defined. It is proved that all incomplete IDBs can be transformed into globally complete IDBs, in which all extensional answers can be generated by evaluating intensional answers against an EDB. We claim that the intensional query processing provide a new methodology for query processing in DDBs and thus, extending the categories of queries, will greatly increase our insight into the nature of DDBs