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

We study the learnability of multiplicity automata in Angluin’s exact learning model, and we investigate its applications. Our starting point is a known theorem from automata theory relating the

Probabilistic I/O automata (PIOA) constitute a model for distributed or concurrent systems that incorporates a notion of probabilistic choice. The PIOA model provides a notion of composition, for constructing a PIOA for a composite system from a collection of PIOAs representing the components. We present a method for computing completion probability and expected completion time for PIOAs. Our method is compositional, in the sense that it can be applied to a system of PIOAs, one component at a time, without ever calculating the global state space of the system (i.e. the composite PIOA). The method is based on symbolic calculations with vectors and matrices of rational functions, and it draws upon a theory of observables, which are mappings from delayed traces to real numbers that generalize the classical "formal power series " from algebra and combinatorics. Central to the theory is a notion of representation for an observable, which generalizes the clasical notion "linear representation " for formal power series. As in the classical case, the representable observables coincide with an abstractly defined class of "rational" observables; this fact forms the foundation of our method. 1

In previous papers, we defined the probabilistic I/O automata model for specification and modeling of probabilistic concurrent systems, and we showed how certain performance measures for such systems could be computed compositionally, one component at a time, without the need for explicit construction of the full global state space. In this paper, we report on our experiences in constructing and testing a computer implemention of these compositional analysis algorithms. Our implementation, which is coded in the functional programming language Standard ML, uses exact rational arithmetic to calculate performance measures, and it is also capable of producing symbolic rational function expressions that describe the dependence of performance measures on a system parameter.

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC¹ and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC¹ and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: There is a language in AC 0 for which inversion and root extraction are GapL-complete, and there is a finite set for which inversion and root extraction are GapNC¹-complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC¹, such as GapAC^0. We initiate work in this direction by classifyi...

Formal power series, which are functions from the set of words over an alphabet A to a semiring k, are viewed coalgebraically. In summary, this amounts to supplying the set of all power series with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions of operators on power series take the shape of what we have called behavioural di#erential equations, after Brzozowski's notion of input derivative, and include many classical di#erential equations for analytic functions. The use of behavioural di#erential equations leads, amongst others, to easy definitions of and proofs about both existing and new operators on power series, as well as to the construction of finite (syntactic) nondeterministic automata, implementing them.

We investigate formal power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We establish a notion of a weighted MSO logics over pictures. The semantics of a weighted formula will be a picture series. We introduce weighted 2-dimensional on-line tessellation automata (W2OTA) and prove that for commutative semirings, the class of picture series defined by sentences of the weighted logics coincides with the family of picture series that are computable by W2OTA. Moreover, we show that the family of behaviors of W2OTA coincide precisely with the class of picture series characterized by weighted (quadrapolic) picture automata and consequently, the notion of weighted recognizability presented here is robust. However, the weighted structures can not be used to get better decidability properties than in the language case. For every commutative semiring, it is undecidable whether a given MSO formula has restricted structure or whether the semantics of a formula has empty support.

ible" collection of I/O automata can be combined into a single, larger automaton. The notion of composition depends in an essential way on a distinction made in the I/O automata model between input actions, which are stimuli applied to an automaton by its environment, output actions, which are responses made by an automaton to its environment, and internal actions, which represent internal steps in which the automaton does not interact with its environment. Output and internal actions are called locally controlled, because their occurrence is under the control of the automaton, whereas input actions are under the control of the environment, with the automaton unable to exert any influence over their occurrence. The PIOA model integrates probability and timing into the I/O automata model, while carrying over in a natural way its essential features of asynchrony and compositionality. To the original I/O automata model, two

Abstract. The theory of two-dimensional languages as a generalization of formal string languages was motivated by problems arising from image processing and pattern recognition and also concerns models of parallel computing. Here we investigate power series on pictures and assign weights to different devices, ranging from tiling systems to picture automata. We will prove that, for commutative semirings, the behaviours of weighted picture automata are precisely alphabetic projections of series defined in terms of rational operations and also coincide with the families of series characterized by weighted tiling or weighted domino systems. Thus we obtain a robust definition of recognizable picture series. The theory of two-dimensional languages is obtained when restricting to the boolean semiring. These new equivalent weighted picture devices can be used to model several interesting application-examples, e.g. the intensity of light of a picture (interpreting the alphabet as different levels of gray) or the maximal amplitude of a monochrome subpicture of a colored picture. 1

Abstract. In this contribution the Myhill-Nerode congruence relation on tree series is reviewed and a more detailed analysis of its properties is presented. It is shown that, if a tree series is deterministically recognizable over a zero-divisor free and commutative semiring, then the Myhill-Nerode congruence relation has nite index. By [Borchardt: Myhill-Nerode Theorem for Recognizable Tree Series. LNCS 2710. Springer 2003] the converse holds for commutative semi elds, but not in general. In the second part, a slightly adapted version of the Myhill-Nerode congruence relation is de ned and a characterization is obtained for all-accepting weighted tree automata over multiplicatively cancellative and commutative semirings. 1