Synthesis of quantifier-free DNF sentences from inconsistent samples of strings with EF games and SAT

Abstract

We present a method to extract knowledge in terms of quantifier-free sentences in disjunctive normal form from noisy samples of classified strings. We show that the problem to find such a sentence is NP-complete, and our approach for solving it is based on a reduction to the Boolean satisfiability problem. Moreover, our method bounds the number of disjuncts and the maximum number of literals per clause since sentences with few clauses and few literals per clause are easier to interpret. As the logic we are considering defines exactly the class of locally threshold testable (LTT) languages, our results can be useful in grammatical inference when the goal is to find a model of an LTT language from a sample of strings. We also use results of the Ehrenfeucht–Fraïssé game over strings in order to handle consistent and inconsistent samples of strings.