Register Complexity of Additive Regular Functions

Additive Cost Register Automata (ACRA) map strings to integers using a finite set of registers that are updated using assignments of the form “x := y + c” at every step. The corresponding class of additive regular functions has multiple equivalent characterizations, appealing closure properties, and decidable analysis questions. In this paper, we define the register complexity of an additive regular function to be the minimum number of registers that an ACRA needs to compute it. We first characterize the register complexity by a necessary and sufficient condition regarding the largest subset of registers whose values can be made far apart from one another. We then use this condition to design a pspace algorithm to compute the register complexity of a given ACRA, and establish a matching lower bound. Our results also lead to a machineindependent characterization of the register complexity of additive regular functions.