Regular expression length via arithmetic formula complexity

Abstract

We prove lower bounds on the length of regular expressions for finite languages by methods from arithmetic circuit complexity. First, we show a reduction: the length of a regular expression for a language L⊆{0,1}n is bounded from below by the minimum size of a monotone arithmetic formula computing a polynomial that has L as its set of exponent vectors. This result yields lower bounds for the language of all words with exactly k ones and n−k zeros and for the language of all Dyck words of length 2n. Second, we adapt a lower bound method for multilinear arithmetic formulas by so-called log-product polynomials to regular expressions. With this method we show almost tight lower bounds for the language of all n-bit binary numbers that are divisible by a given odd integer p and for the language of all permutations of {1,…,n}.