NFA Reduction Algorithms by Means of Regular Inequalities

Abstract

We present different techniques for reducing the number of states and transitions in nondeterministic automata. These techniques are based on the two preorders over the set of states, related to the inclusion of left and right languages. Since their exact computation is $\mathcal{N}\mathcal{P}$-hard, we focus on polynomial approximations which enable a reduction of the NFA all the same. Our main algorithm relies on a first approximation, which can be easily implemented by means of matrix products with an $\mathcal{O}(sn^4 )$ time complexity, and optimized to an $\mathcal{O}(sn^3 )$ time complexity, where s is the average nondeterministic arity and n is the number of states. This first algorithm appears to be more efficient than the known techniques based on equivalence relations as described by Lucian Ilie and Sheng Yu. Afterwards, we briefly describe some more accurate approximations and the exact (but exponential) calculation of these preorders by means of determinization.