Weighted automata and weighted logics with discounting

Theoretical Computer Science, 2004

We consider the optimal-reachability problem for a timed automaton with respect to a linear cost function which results in a weighted timed automaton. Our solution to this optimization problem consists of reducing it to computing (parametric) shortest paths in a ÿnite weighted directed graph. We call this graph a parametric sub-region graph. It reÿnes the region graph, a standard tool for the analysis of timed automata, by adding the information which is relevant to solving the optimal-reachability problem. We present an algorithm to solve the optimal-reachability problem for weighted timed automata that takes time exponential in O(n (| (A)| + |wmax|)), where n is the number of clocks, | (A)| is the size of the clock constraints and |wmax| is the size of the largest weight. We show that this algorithm can be improved, if we restrict to weighted timed automata with a single clock. In case we consider a single starting state for the optimal-reachability problem, our approach yields an algorithm that takes exponential time only in the length of clock constraints.

Information Sciences, 2010

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. and weighted tree automata over strong bimonoids begun in respectively . As usual, there are three different definitions of the behavior of such automata: the run semantics , the initial algebra semantics , and the transition semantics . In the run semantics, the weight of a word w is computed by summing up the weights of all successful runs of A on w where the weight of a run is the product of the weights of the involved transitions. For the other two semantics, we associate with each input symbol an n × n-matrix of transition weights assuming that A has n states. In both semantics, we multiply the vector of initial weights with the matrices corresponding to the symbols of w and with the final weight vector, but the sequence of these multiplications differs in the two semantics. In the case of semirings, these three semantics coincide for every weighted finite automaton. In contrast to this, we show by an easy example that over the unit interval with bounded sum these three semantics in general differ.

2012

This report contains the programme and the abstracts of lectures delivered at the workshop

Lecture Notes in Computer Science, 2017

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is Pspace-complete.

Theoretical Computer Science, 2004

We consider an optimal-reachability problem for a timed automaton with respect to a linear cost function which results in a weighted timed automaton. Our solution to this optimization problem consists of reducing it to a (parametric) shortest-path problem in a nite directed graph. The directed graph we construct is a re nement of the region automaton due to Alur and Dill. We present an exponential time algorithm to solve the shortest-path problem for weighted timed automata starting from a single state, and a doubly-exponential time algorithm to solve this problem starting from a zone of the state space.

Lecture Notes in Computer Science, 2011

We consider weighted finite transition systems (wts) with weights from commutative naturally ordered semirings. Such semirings comprise distributive lattices as well as the natural numbers with ordinary addition and multiplication, and the max-plus-semiring. For these systems we explore the concepts of covering and cascade product. We show a cascade decomposition result for such wts using special partitions of the state set of the system. This extends a classical result of automata theory to the weighted setting.

International Journal of Algebra and Computation, 2013

This paper addresses the problem of the validity of weighted automata in which the presence of ε-circuits results in infinite summations. Earlier works either rule out such automata or characterize the semirings in which these infinite sums are all well-defined. By means of a topological approach, we take here a definition of validity that is strong enough to insure that in any kind of semirings, any closure algorithm will succeed on every valid weighted automaton and turn it into an equivalent proper automaton. This definition is stable with respect to natural transformations of automata. The classical closure algorithms, in particular algorithms based on the computation of the star of the matrix of ε-transitions, cannot be used to decide validity. This decision problem remains open for general topological semirings. We present a closure algorithm that yields a decision procedure for the validity of automata in the case where the weights are taken in ℚ or ℝ, two cases that had neve...

Information Processing Letters, 2008

We investigate weighted automata with discounting and their behaviours over semirings and finitely generated graded monoids. We characterize the discounted behaviours of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene-Schützenberger. Here we show that the very special case of Schützenberger's result for free monoids over singleton alphabets suffices to deduce our generalization.

Lecture Notes in Computer Science

We study the problems of existence and construction of infinite schedules for finite weighted automata and one-clock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on transitions and locations, corresponding to the production and consumption of some resource (e.g. energy). We ask the question whether there exists an infinite path for which the accumulated weight for any finite prefix satisfies certain constraints (e.g. remains between 0 and some given upper-bound). We also consider a game version of the above, where certain transitions may be uncontrollable.