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. ∗Corresponding author Email addresses: [email protected] (Marta Arias), [email protected] (José L. Balcázar), [email protected] (Cristina Tı̂rnăucă) Partially supported by project BASMATI (TIN2011-27479-C04-04) of Programa Nacional de Investigación, Ministerio de Ciencia e Innovación (MICINN), Spain. Partially supported by project PAC::LFO (MTM2014-55262-P) of Programa Estatal de Fomento de la Investigación Cient́ıfica y Técnica de Excelencia, Ministerio de Ciencia e Innovación (MICINN), Spain. Partially supported by grant 2014SGR 890 (MACDA) from AGAUR, Generalitat de Catalunya. Preprint submitted to a special issue of a professional journal November 6, 2015