Recognizable sets with multiplicities in the tropical semiring

2005 ◽  
pp. 107-120 ◽  
Author(s):  
Imre Simon
Keyword(s):  
Tropical Semiring

Languages Accepted by Weighted Restarting Automata*

2021 ◽  
Vol 180 (1-2) ◽  
pp. 151-177
Author(s):  
Qichao Wang
Keyword(s):  
General Class ◽  
Formal Languages ◽  
Input Word ◽  
Additional Requirement ◽  
Tropical Semiring

Weighted restarting automata have been introduced to study quantitative aspects of computations of restarting automata. In earlier works we studied the classes of functions and relations that are computed by weighted restarting automata. Here we use them to define classes of formal languages by restricting the weight associated to a given input word through an additional requirement. In this way, weighted restarting automata can be used as language acceptors. First, we show that by using the notion of acceptance relative to the tropical semiring, we can avoid the use of auxiliary symbols. Furthermore, a certain type of word-weighted restarting automata turns out to be equivalent to non-forgetting restarting automata, and another class of languages accepted by word-weighted restarting automata is shown to be closed under the operation of intersection. This is the first result that shows that a class of languages defined in terms of a quite general class of restarting automata is closed under intersection. Finally, we prove that the restarting automata that are allowed to use auxiliary symbols in a rewrite step, and to keep on reading after performing a rewrite step can be simulated by regular-weighted restarting automata that cannot do this.

scholarly journals TROPICAL ALGEBRAIC SETS, IDEALS AND AN ALGEBRAIC NULLSTELLENSATZ

2008 ◽  
Vol 18 (06) ◽  
pp. 1067-1098 ◽  
Author(s):  
ZUR IZHAKIAN
Keyword(s):  
Algebraic Geometry ◽  
Classical Theory ◽  
Commutative Algebra ◽  
Fundamental Theorem ◽  
Polynomial Algebra ◽  
Algebraic Sets ◽  
Tropical Semiring ◽  
Fundamental Theorem Of Algebra

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. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.

scholarly journals REACHABILITY PROBLEMS FOR PRODUCTS OF MATRICES IN SEMIRINGS

2006 ◽  
Vol 16 (03) ◽  
pp. 603-627 ◽  
Author(s):  
STÉPHANE GAUBERT ◽  
RICARDO D. KATZ
Keyword(s):  
Matrix Product ◽  
Reachability Problem ◽  
Tropical Semiring ◽  
The Matrix ◽  
Left And Right ◽  
Matrix Vector ◽  
Products Of Matrices

We consider the following matrix reachability problem: given r square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? Similarly, we define the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied on the left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any r ≥ 2 is equivalent to the specialization to r = 2. As an application of these reductions and of a theorem of Krob, we show that when r = 2, the vector and matrix reachability problems are undecidable over the max-plus semiring (ℤ∪{-∞}, max ,+). These reductions also improve known results concerning the classical zero corner problem. Finally, we show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are "positive", like the tropical semiring (ℤ∪{+∞}, min ,+).

scholarly journals THE FINITE BASIS PROBLEM FOR THE MONOID OF TWO-BY-TWO UPPER TRIANGULAR TROPICAL MATRICES

2016 ◽  
Vol 94 (1) ◽  
pp. 54-64 ◽  
Author(s):  
YUZHU CHEN ◽  
XUN HU ◽  
YANFENG LUO ◽  
OLGA SAPIR
Keyword(s):  
Finite Basis ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Tropical Semiring ◽  
Bicyclic Monoid ◽  
Nonfinitely Based

For each positive $n$, let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$. We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$) is nonfinitely based.

The tropical semiring in higher dimensions

2018 ◽  
Vol 11 (3) ◽  
pp. 477-488
Author(s):  
John Norton ◽  
Sandra Spiroff
Keyword(s):  
Higher Dimensions ◽  
Tropical Semiring

Equational Theories of Tropical Semirings

2001 ◽  
Vol 8 (21) ◽  
Author(s):  
Luca Aceto ◽  
Zoltán Ésik ◽  
Anna Ingólfsdóttir
Keyword(s):  
Equational Theory ◽  
Finite Basis ◽  
Free Algebras ◽  
Real Numbers ◽  
Binary Operations ◽  
Equational Theories ◽  
Tropical Semiring

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the <em> (max,+) semiring</em> and the <em>tropical semiring</em>. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

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