Recognizable Sets with Multiplicities in the Tropical Semiring
نویسنده
چکیده
The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the nite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of nite automata. We review here these results, their applications and point out some open problems.
منابع مشابه
Some consequences of a Fatou property of the tropical semiring
1 Abstract We show that the equatorial semiring Z min = (Zf+1g; min; +) is a Fatou extension of the tropical semiring M = (Nf+1g; min; +). This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring. We also deduce from it the decidability of the limitedness problem for the equatorial semiring, solving th...
متن کاملThe Equality Problem for Rational Series with Multiplicities in the Tropical Semiring is Undecidable
1 0 Introduction The tropical semiring is the semiring denoted by M which has support Nf+1g and operations ab = minfa; bg and ab = a+b. It was rst introduced in the context of cost minimization in Operations Research. However it appeared that M plays in fact a central role in several decision problems concerning rational languages (see 15] for a survey of the tropical semiring theory and of its...
متن کاملSimulation vs. Equivalence
For several semirings S, two weighted finite automata with multiplicities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called “proper”. It is known that the Boolean semiring, the semiring of natural numbers, the ring of integers, all finite commutative positively ordered semirings and all fields are proper. The semiring S is Noetherian...
متن کاملTropical Algebraic Sets, Ideals and an Algebraic Nullstellensatz
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring – a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra....
متن کاملCut sets as recognizable tree languages
A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzi ed recognizable tree language. In this sense, su cient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime lter of the carrier set, is a recognizable tree language. Moreov...
متن کامل