Synthesis of a DNF formula from a sample of strings using Ehrenfeucht–Fraïssé games

Abstract

In order to define a DNF version of first-order sentences over strings in which atomic sentences represent substring properties of strings, we use results of the Ehrenfeucht–Fraïssé game over strings. Then, given a sample of strings and the number of disjunctive clauses, we investigate the problem of finding a DNF formula that is consistent with the sample. We show that this problem is NP-complete, and we solve it by a translation into Boolean satisfiability. We also present an extension of this problem that is robust concerning noisy samples. We solve the generalized version by a codification into the maximum satisfiability problem. As first-order logic over strings defines exactly the class of locally threshold testable (LTT) languages, our results can be useful in the grammatical inference framework when the goal is to find a model of a LTT language from a sample of strings.