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            <title xml:lang="en">Safra-like constructions for regular cost functions over finite words</title>
            <title xml:lang="fr">Constructions à la Safra pour les fonctions régulières de coût sur les mots finis</title>
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                <title xml:lang="fr">Constructions à la Safra pour les fonctions régulières de coût sur les mots finis</title>
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                <term xml:lang="en">Automata</term>
                <term xml:lang="en">Determinisation</term>
                <term xml:lang="en">cost functions</term>
                <term xml:lang="en">regular cost functions</term>
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              <p>Regular cost functions have been introduced recently as an extension of the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. Cost functions are functions ranging over N ∪ {∞}, and considered mod-ulo an equivalence which preserves the existence of bounds over each subset of the domain. A central result in the theory of regular cost functions over words is the duality theorem stating that the two dual forms of automata, namely Band S-automata, are equivalent. The only known proof of this result relies on the translation into an equivalent algebraic formalism.In this paper, we describe direct translations from hierarchical B-automata to history-deterministic S-automata and from hierarchical S-automata to history deterministic B-automata. We furthermore describe how to optimize the number of counters of the output automaton, and how to make them hierarchical. This construction is reminiscent of the famous construction of Safra for the determinization of automata over infinite words.</p>
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              <p>Les fonctions régulières de coût ont été introduites il y a peu comme une extension de la notion de langage régulier par des capacités de comptage, tout en préservant de nombreuses propriétés importantes comme celles de clôture ou de décidabilité. Les fonctions de coût sont des fonctions à valeur dans N ∪ {∞} considérée modulo une relation d'équivalence qui préserve l'existence d'une borne sur tout sous-ensemble du domaine. Un résultat central de cette théorie sur les mots finis est le résultat de dualité qui énonce que deux formes duales d'automates (les B- et S-automates) sont duales. La seule preuve connue de ce résultat repose sur une traduction dans un formalisme algébrique. Dans ce papier, nous décrivons une traduction directe depuis les B-automates hiérarchiques vers les S-automates déterministes en histoire, et depuis les automates S-hiérarchiques vers les B-automates déterministes en histoire. Cette construction est fortement inspirée de celle de Safra.</p>
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