Decision Problems for 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 a decidable equivalence problem. In this paper, we solve two decision problems for this model. First, 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 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 machine-independent characterization of the register complexity of additive regular functions. Second, we consider

two-player games over ACRAs

, where the objective of one of the players is to reach a target set while minimizing the cost. We show the corresponding decision problem to be

exptime

-complete when the costs are non-negative integers, but undecidable when the costs are integers.