Message passing over factor graph can be considered as generalization of many well known algorithms for efficient marginalization of multivariate function. A specific instance of the algorithm is obtained by choosing an appropriate commutative semiring for the range of the function to be marginalized. Some examples are Viterbi algorithm, obtained on max-product semiring and forward-backward alg...

PAWS is a tool to analyse the behaviour of weighted automata and conditional transition systems. At its core PAWS is based on a generic implementation of algorithms for checking language equivalence in weighted automata and bisimulation in conditional transition systems. This architecture allows for the use of arbitrary user-defined semirings. New semirings can be generated during run-time and ...

In their seminal paper [1] McCulloch and Pitts have shown the strong relationship between finite automata and so-called McCullochPitts networks. Our goal is to extend this result to weighted automata. In other words, we want to integrate artificial neural networks and weighted automata. For this task, we introduce semiring artificial neural networks, that is, artificial neural networks which im...

We consider the iterates of bilinear functions over the semiring (max,+). Equivalently, our object of study can be viewed as recognizable tree series over the semiring (max,+). In this semiring, a fundamental result associates the asymptotic behaviour of the iterates of a linear function with the maximal average weight of the circuits in a graph naturally associated with the function. Here we p...

Deciding the equivalence of SQL queries is a fundamental problem in data management. As prior work has mainly focused on studying the theoretical limitations of the problem, very few implementations for checking such equivalences exist. In this paper, we present a new formalism and implementation for reasoning about the equivalences of SQL queries. Our formalism, U-semiring, extends SQL’s semir...

Let S be a complete star-omega semiring and Σ be an alphabet. For a weighted ω-pushdown automaton P with stateset {1, . . . ,n}, n ≥ 1, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair ((S≪ Σ ≫),(S≪ Σ ≫)) such that the behavior ‖P‖ of P is a component of a solution of this system. In case the basic semiring is B or N we show that there exists a mixed c...

The max-Łukasiewicz semiring is defined as the unit interval [0, 1] equipped with the arithmetics “a + b” = max(a, b) and “ab” = max(0, a + b− 1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, w...

Many “semiring-like” structures are used in Soft Constraint Satisfaction Problems (SCSPs). We review a few properties of semirings that are useful for dealing with soft constraints, highlighting the differences between alternative proposals in the literature. We then extend the semiring structure by adding the notion of division as a weak inverse operation of product. In particular, division is...