Provably Shorter Regular Expressions from Finite Automata

A famous theorem due to Kleene [25] states that the regular languages admit two equivalent characterizations of entirely different nature, namely as the languages accepted by finite automata on the one hand, and as those described by regular expressions on the other hand. There are a few classical algorithms for converting finite automata into regular expressions. Those algorithms look different at first glance [6, 7, 28]. But, as Sakarovitch [32] pointed out, all of these approaches are more or less reformulations of the same underlying algorithmic idea: they can be recast as variations of the standard state elimination algorithm. The latter is found in most textbooks on automata theory, see, e.g., [36]. All of these algorithms have an upper bound of roughly 4 on the size of the resulting regular expressions, the number n being the number of states in the given finite automaton. The desire to obtain shorter regular expressions than the 4 upper bound can be traced back to the work by McNaughton and Yamada [28]. They observed that the choice of the ordering in which the states are eliminated