139 research outputs found
Weighted Automata and Monadic Second Order Logic
Let S be a commutative semiring. M. Droste and P. Gastin have introduced in
2005 weighted monadic second order logic WMSOL with weights in S. They use a
syntactic fragment RMSOL of WMSOL to characterize word functions (power series)
recognizable by weighted automata, where the semantics of quantifiers is used
both as arithmetical operations and, in the boolean case, as quantification.
Already in 2001, B. Courcelle, J.Makowsky and U. Rotics have introduced a
formalism for graph parameters definable in Monadic Second order Logic, here
called MSOLEVAL with values in a ring R. Their framework can be easily adapted
to semirings S. This formalism clearly separates the logical part from the
arithmetical part and also applies to word functions.
In this paper we give two proofs that RMSOL and MSOLEVAL with values in S
have the same expressive power over words. One proof shows directly that
MSOLEVAL captures the functions recognizable by weighted automata. The other
proof shows how to translate the formalisms from one into the other.Comment: In Proceedings GandALF 2013, arXiv:1307.416
Characterizing matrices with -simple image eigenspace in max-min semiring
A matrix is said to have -simple image eigenspace if any eigenvector
belonging to the interval is the unique solution of the system in
. The main result of this paper is a combinatorial characterization of such
matrices in the linear algebra over max-min (fuzzy) semiring.
The characterized property is related to and motivated by the general
development of tropical linear algebra and interval analysis, as well as the
notions of simple image set and weak robustness (or weak stability) that have
been studied in max-min and max-plus algebras.Comment: 23 page
Representing quantum structures as near semirings
In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Łukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Łukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Łukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras
Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande
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