Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speech-to-text processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi’s and Elgot’s fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted first-order logic and show that aperiodic series coincide with the first-order definable ones, if the semiring is locally finite, commutative and has some aperiodicity property. 1

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Abstract. A popular and much studied class of filters for approximate string matching is based on finding common q-grams, substrings of length q, between the pattern and the text. A variation of the basic idea uses gapped q-grams and has been recently shown to provide significant improvements in practice. A major difficulty with gapped q-gram filters is the computation of the so-called threshold which defines the filter criterium. We describe the first general method for computing the threshold for q-gram filters. The method is based on a carefully chosen precise statement of the problem which is then transformed into a constrained shortest path problem. In its generic form the method leaves certain parts open but is applicable to a large variety of q-gram filters and may be extensible even to other classes of filters. We also give a full algorithm for a specific subclass. For this subclass, the algorithm has been implemented and used succesfully in an experimental comparison. 1

Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of maxplus series with respect to ambiguity.

Finite automata are classical computational devices used in a variety of large-scale applications [1]. Finite-state transducers are automata whose transitions are labeled with both an input and an output label. Some applications in text, speech and image processing require more general devices, weighted automata, to account for the variability of the input data and to rank various output hypotheses [7, 9, 8]. A weighted automaton is a finite automaton in which each transition is labeled with some weight in addition to the usual symbol. Weighted automata and transducers provide a common representation for each component of a complex system used in these applications and admit general algorithms such as composition which can be used to combine these components. The time efficiency of these systems is substantially increased when deterministic or subsequential machines are used [9] and the size of these machines can be further reduced using general minimization algorithms [9, 10]. A weighted automaton or transducer is deterministic or subsequential if it has a unique initial state and if no two transitions leaving the same state share the same input label. A general determinization algorithm for weighted automata and transducers was given by [9]. The algorithm outputs a deterministic machine equivalent to the input weighted automaton or transducer and is an extension of the classical subset construction used for unweighted finite automata. But, unlike the case of unweighted automata, not all finite-state transducers or weighted automata and transducers can be determinized using this algorithm. In fact, some machines do not admit any equivalent deterministic one. Thus, it is important to design an algorithm for testing the determinizability of finite-state transducers and weighted automata.

Abstract—A nondeterministic weighted finite automaton (WFA) maps an input word to a numerical value. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. Many of these applications require the WFAs to be deterministic, or work substantially better when the WFAs are deterministic. Unlike NFAs, which can always be determinized, not all WFAs have an equivalent deterministic weighted automaton (DWFA). In [1], Mohri describes a determinization construction for a subclass of WFA. He also describes a property of WFAs (the twins property), such that all WFAs that satisfy the twins property are determinizable and the algorithm terminates on them. Unfortunately, many natural WFAs cannot be determinized. In this paper we study approximated determinization of WFAs. We describe an algorithm that, given a WFA A and an approximation factor t ≥ 1, constructs a DWFA A ′ that t-determinizes A. Formally, for all words w ∈ Σ ∗ , the value of w in A ′ is at least its value in A and at most t times its value in A. Our construction involves two new ideas: attributing states in the subset construction by both upper and lower residues, and collapsing attributed subsets whose residues can be tightened. The larger the approximation factor is, the more attributed subsets we can collapse. Thus, t-determinization is helpful not only for WFAs that cannot be determinized, but also in cases determinization is possible but results in automata that are too big to handle. In addition, t-determinization is useful for reasoning about the competitive ratio of online algorithms. We also describe a property (the t-twins property) and use it in order to characterize t-determinizable WFAs. Finally, we describe a polynomial algorithm for deciding whether a given WFA has the t-twins property. Index Terms—Weighted automata; Determinization; I.

Abstract: We investigate weighted finite automata over strings and strong bimonoids. Such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold. We define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions. Moreover, we show that in this case the given weighted automata can be determinized. 1