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, viewing words as vectors. This result yields lower bounds for the binomial language of all words with exactly k ones and n−k zeros and for the language of all Dyck words of length 2n. We also determine the blow-up of language operations (intersection and shuffle) of regular expressions for finite languages. 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 binary numbers with n bits that are divisible by a given odd integer p, for the language of all words of length n over a k letter alphabet with an even number of occurrences of each letter and for the language of all permutations of {1,. .. , n}.