Determinization of weighted finite automata over strong bimonoids ✩ Miroslav ´ Ciri´ c a , Manfred Droste b , Jelena Ignjatovi´ c a , Heiko Vogler c a Faculty of Sciences and Mathematics, University of Niˇ s, Viˇ segradska 33, P. O. Box 224, 18000 Niˇ s, Serbia b Institut f ¨ ur Informatik, Universit ¨ at Leipzig, D-04009 Leipzig, Germany c Faculty of Computer Science, Technische Universit¨ at Dresden, D-01062 Dresden, Germany Abstract We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, resp., the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics. Key words: Weighted automaton; Strong bimonoid; Formal power series; Determinization; Nerode automaton; Myhill automaton; Run automaton 1. Introduction Weighted finite automata are classical nondeterministic automata in which the transitions carry weights. These weights may model, e.g., the amount of resources needed for the execution of a transition, or the probability of its successful execution. The weights often form the algebraic structure of a semiring, and semiring-weighted automata have both a well elaborated theory as well as practical applications, cf. [1, 10, 13, 22, 28, 29, 35]. Recently, a number of authors investigated weighted automata with weights in more general structures, which can be viewed as semirings which might lack distributivity. Examples of such “strong bimonoids” include the real unit interval [0, 1] with t-conorm and t-norm from multivalued logic [20], the “string bimonoid” of all words over an alphabet arising in natural language processing [26], or the algebraic cost structure from algebraic path problems [23]. Important natural examples of strong bimonoids are ortho- modular lattices, which serve as a basis of quantum logics [2] where distributivity typically fails; automata based on quantum logics and with weights in orthomodular lattices were investigated in [24, 31, 32, 36, 37, 38]. Automata modeling, e.g., peak power consumption of energy and with particular strong bimonoids as weight structures were recently studied in [7, 8, 9]. Fuzzy automata and fuzzy tree automata defined by a pair of a t-conorm and a t-norm on the real unit interval were investigated in [3, 4] respectively [5], and their study for non-distributive pairs has been appraised as especially interesting. In this paper, we investigate weighted automata over arbitrary strong bimonoids and, in particular, methods and algorithms for their determinization. This continues the general study of weighted automata ✩ Research supported by Ministry of Science and Technological Development, Republic of Serbia, Grant No. 144011, and by DAAD, Grant No. D/08/02092 Email addresses: mciric@pmf.ni.ac.rs (Miroslav ´ Ciri´ c), droste@informatik.uni-leipzig.de (Manfred Droste), jejaign@yahoo.com (Jelena Ignjatovi´ c), Heiko.Vogler@tu-dresden.de (Heiko Vogler) Preprint submitted to Elsevier January 10, 2010