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 \((n-1)\cdot m+n\) 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 \(1+(n+1)\cdot \mathsf {F}(\,1,n+2,-n+2;n+1\mid -1\,)\) states on DFAs, where \(\mathsf {F}\) refers to the generalized hypergeometric function. This bound is in the order of magnitude \(\varTheta ((n-1)!)\). 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.