Taking advantage of a very simple property to efficiently infer NFAs

Grammatical inference consists in learning a formal grammar as a finite state machine or as a set of rewrite rules. In this paper, we are concerned with inferring Nondeterministic Finite Automata (NFA) that must accept some words, and reject some other words from a given sample. This problem can naturally be modeled in SAT. The standard model being enormous, some models based on prefixes, suffixes, and hybrids were designed to generate smaller SAT instances. There is a very simple and obvious property that says: if there is an NFA of size k for a given sample, there is also an NFA of size k+1. We first strengthen this property by adding some characteristics to the NFA of size k+1. Hence, we can use this property to tighten the bounds of the size of the minimal NFA for a given sample. We then propose simplified and refined models for NFA of size k+1 that are smaller than the initial models for NFA of size k. We also propose a reduction algorithm to build an NFA of size k from a specific NFA of size k+1. Finally, we validate our proposition with some experimentation that shows the efficiency of our approach.