Taking Advantage of a Very Simple Property to Efficiently Infer NFAs

Abstract

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.

Inference of Over-Constrained NFA of Size $$k+1$$ to Efficiently and Systematically Derive NFA of Size k for Grammar Learning

“…Of course, the notion of path can be defined and built in many ways. In [10], prefix, suffix, and hybrid approaches are proposed.…”

“…We propose two (k + 1)_NFA model extensions for which a new state and new constraints are added. The first model extension, the (k + 1)_NFA + , over-constrains the (k + 1)_NFA to be able to reduce a generated (k + 1)_NFA into a k_NFA with a reduction algorithm (see [10]). However, as shown in [10], we are not always able to reduce a (k + 1)_NFA + into a k_NFA, and the gain is really poor.…”

“…The first model extension, the (k + 1)_NFA + , over-constrains the (k + 1)_NFA to be able to reduce a generated (k + 1)_NFA into a k_NFA with a reduction algorithm (see [10]). However, as shown in [10], we are not always able to reduce a (k + 1)_NFA + into a k_NFA, and the gain is really poor. In fact, a (k + 1)_NFA + instance mainly provides information when it is unsatisfiable: in this case, we also know that there is no k_NFA.…”

“…A (k + 1)_NFA + is a (k + 1)_NFA with some extra properties that ensure that it may be reduced to a k_NFA by a reduction algorithm [10]. The extra properties of a (k + 1)_NFA + are:…”

“…The simple following example (Figure 1) shows a k_NFA + solution with k = 3, whereas there is no k_NFA with k = 2. It is possible to try to obtain a k_NFA from a (k + 1)_NFA + solution with the algorithm presented in [10]. However this algorithm rarely succeeds, and its worst-case complexity is in O(k • |S|).…”