A Generalised Twinning Property for Minimisation of Cost Register Automata
Résumé
Weighted automata (WA) extend finite-state automata by associating weights with transitions. Unambiguous WA are such that for each word, there is at most one accepting run. Unambiguous WA are equivalent to functional WA, i.e. WA such every two runs on the same input have the same weight. Recently, cost register automata have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Given a monoid (M,⊗), we denote by CRA⊗c(M) the cost register automata whose registers take their values in M, and are updated by operations of the form x := y ⊗ c, with c ∈ M.
We introduce a twinning property and a bounded variation property parameterised by an integer k, such that the corresponding notions introduced by Choffrut are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, ⊗), we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) the function it realises satisfies the k-bounded variation property, and iii) this function can be described by a CRA⊗c(G) with at most k registers. We actually prove this result in the more general setting of finite-valued weighted automata. We show that if the operation of the group is 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 CRA⊗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.
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