Minimizing Cost Register Automata over a Field
Yahia Idriss Benalioua, Nathan Lhote, Pierre-Alain Reynier
TL;DR
This work addresses the problem of minimizing Cost Register Automata (CRA) over a field by exploiting invariants of Weighted Automata (WA), specifically the strongest Z-linear (linear) or Z-affine invariants (linear/affine hulls). It establishes a rigorous correspondence between WA invariants and the state and register requirements of linear/affine CRA, enabling precise minimization results: register minimization is decidable in 2-ExpTime and state-register minimization in NExpTime, with improvements to ExpTime in special cases such as commuting transition matrices or unary alphabets. The authors introduce two new algorithms to compute the invariants (and thus the CRA) and extend the framework to affine CRA, offering a coherent view of the tradeoffs between states and registers via the invariant dimensions and lengths. This yields practical decision procedures for WA/CRA minimization and deepens the theoretical understanding of how algebraic invariants govern automata expressiveness and resource use.
Abstract
Weighted automata (WA) are an extension of finite automata that define functions from words to values in a given semiring. An alternative deterministic model, 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, updated at each transition using the operations of the semiring. It is known that CRA with register updates defined by linear maps have the same expressiveness as WA. Previous works have studied the register minimization problem: given a function computable by a WA and an integer k, is it possible to realize it using a CRA with at most k registers? In this paper, we solve this problem for CRA over a field with linear register updates, using the notion of linear hull, an algebraic invariant of WA introduced recently by Bell and Smertnig. We then generalise the approach to solve a more challenging problem, that consists in minimizing simultaneously the number of states and that of registers. In addition, we also lift our results to the setting of CRA with affine updates. Last, while the linear hull was recently shown to be computable by Bell and Smertnig, no complexity bounds were given. To fill this gap, we provide two new algorithms to compute invariants of WA. This allows us to show that the register (resp. state-register) minimization problem can be solved in 2-ExpTime (resp. in NExpTime).