A Generalised Twinning Property for Minimisation of Cost Register Automata

Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring $\mathbb {S}$, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid $(M,\otimes )$ can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form $x :=y\otimes c$, with $c\in M$. This class is denoted by $\mathrm {CRA}_{\otimes c}(M)$.We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k=1. Given an unambiguous weighted automaton W over an infinitary group $(G,\otimes )$ realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a $\mathrm {CRA}_{\otimes c}(G)$ with at most k registers.In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from $(G,\otimes )$ : the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most $\ell$, for some natural $\ell$. Moreover, we show that if the operation $\otimes \, \mathrm {of}\, G$ is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class $\mathrm {CRA}_{\otimes c}(G)$.Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA
c
(B
^*
) is PSPACE-complete.