Synthesis of a DNF Formula From a Sample of Strings

We define a propositional substring logic (PS) in which atomic sentences represent substring properties of strings. We also investigate the following variation of the boolean function synthesis (BFS) problem: given a sample of classified strings, find a PS formula in disjunctive normal form with the minimum number of clauses and consistent with the sample. We call this problem PS formula synthesis (PSFS). The advantages of using PS is that it is as expressive as first-order logic (FO) over strings with the successor relation, and PS formulas are more succinct than FO formulas. We show that PSFS is NP-complete, and we propose an algorithm to solve PSFS via a reduction to the BFS problem.