Register Minimization of Cost Register Automata over a Field

Weighted automata (WA) are an extension of finite automata that defines functions from words to values in a given semi-ring. An alternative model that is deterministic, called Cost Register Automata (CRA), was introduced by Alur et al. It enriches deterministic finite automata with a finite number of registers, which store values, are updated at each transition using the operations of the semi-ring, and are combined to produce the output. The expressiveness of a CRA depends on the number of its registers and the type of register updates allowed for each of its transitions. In particular, the class of functions computable by a CRA with register updates defined by linear (or affine) maps correspond exactly with rational functions (functions computable by a WA). A natural problem for CRA is the register minimization problem: given a function defined by a CRA, what is the minimal number of registers needed to realize this function? In this paper, we solve the register minimization problem for CRA over a field with linear (or affine) register updates, using an algebraic invariant of a WA introduced recently by Bell and Smertnig, the so-called the linear hull of the automaton. This invariant being computable, we are able to explicitly compute a CRA with linear (or affine) updates, using the minimal number of registers. Using these techniques, we are also able to solve the more general CRA minimisation problem: given a CRA and integers $k,d$, is there an equivalent linear (resp.~affine) CRA using at most $k$ states and $d$ registers?