Tight bounds for cut-operations on deterministic finite automata

Abstract

We investigate the state complexity of the cut and iterated cut operation for deterministic finite automata (DFAs), answering an open question stated in [M. Berglund, et al.: Cuts in regular expressions. In Proc. DLT, LNCS 7907, 2011]. These operations can be seen as an alternative to ordinary concatenation and Kleene star modelling leftmost maximal string matching. We show that the cut operation has a matching upper and lower bound of (n − 1) · m + n states on DFAs accepting the cut of two individual languages that are accepted by n- and m-state DFAs, respectively. In the unary case we obtain max(2n−1,m+n−2) states as a tight bound. For accepting the iterated cut of a language accepted by an n-state DFA we find a matching bound of 1 + (n + 1) · F(1, n + 2,−n + 2; n + 1 | −1) states on DFAs, where F refers to the generalized hypergeometric function. This bound is in the order of magnitude Θ((n − 1)!). Finally, the bound drops to 2n − 1 for unary DFAs accepting the iterated cut of an n-state DFA and thus is similar to the bound for the cut operation on unary DFAs.