We consider K-semialgebras for a commutative semiring K that are at the same time Σ-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. Examples of such algebras include the algebras of Σ-tree series over an alphabet A with coe...

In this lecture, we summarize the interprocedural-dataflow-analysis method of Esparza et al., which generalizes Newton’s method from a numerical-analysis algorithm for finding roots of real-valued functions to a method for finding fixed-points of systems of equations over semirings. As in its real-valued counterpart, each iteration of the method of Esparza et al. solves a simpler “linearized” p...

We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only non-negative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associative-...

There are presented the investigations in the framework of pseudoanalysis semirings whose operations are generated with functions of one variable. Based on the operations with generators there was developed a theory of generalized functions in analogy to the Mikusiński’s operators, which enables the construction of a generalized solution of the Burgers equation. There are considered extensions ...

To study power series generated by Lindenmayer systems we de ne L algebraic systems and series over arbitrary commutative semirings. We establish closure and xed point properties of L algebraic series. We show how the framework of L algebraic series can be used to de ne D0L, 0L, E0L, DT0L, T0L and ET0L power series. We generalize for power series the classical result stating that D0L languages ...

Two types of simulations for weighted tree automata (wta) are considered. Wta process trees and assign a weight to each of them. The weights are taken from a semiring. The two types of simulations work for wta over additively idempotent, commutative semirings and can be used to reduce the size of wta while preserving their semantics. Such reductions are an important tool in automata toolkits.