Tropical Semiring Research Articles

Equational Theories of Tropical Semirings

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.

Context-Free Recognition with Weighted Automata

We introduce the definition of language recognition with weighted automata, a generalization of the classical definition of recognition with unweighted acceptors. We show that, with our definition of recognition, weighted automata can be used to recognize a class of languages that strictly includes regular languages. The class of languages accepted depends on the weight set which has the algebraic structure of a semiring. We give a generic linear time algorithm for recognition with weighted automata and describe examples with various weight sets illustrating the recognition of several classes of context-free languages. We prove, in particular, that the class of languages equivalent to the language of palindromes can be recognized by weighted automata over the (+,ċ)-semiring, and that the class of languages equivalent to the Dyck language of first order D 1 '*can be recognized by weighted automata over the real tropical semiring. We also prove that weighted automata over the real tropical semiring can be used to recognize regular expressions.

Representations and complete semiring morphisms

We investigate the connection between representations and complete semiring morphisms. In some semirings, e.g., for power series over the Boolean semiring, both concepts coincide. But there exist also semirings, e.g., the semiring of power series over the tropical semiring, where the concept of a complete semiring morphism is more general than that of a representation.

A complete system of identities for one-letter rational expressions with multiplicities in the tropical semiring

We construct a complete system of rational identities for the equational theory of the algebra of one-letter rational expressions with multiplicities in the tropical semiring.

THE EQUALITY PROBLEM FOR RATIONAL SERIES WITH MULTIPLICITIES IN THE TROPICAL SEMIRING IS UNDECIDABLE

We show that the equality problem for rational series with multiplicities in the tropical semiring is undecidable.

Some consequences of a Fatou property of the tropical semiring

We show that the equatorial semiring ▪ is a Fatou extension of the tropical semiring ▪. 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 therefore a question of I. Simon.

Finite tree automata with cost functions

Cost functions for tree automata are mappings from transitions to (tuples of) polynomials over some semiring. We consider four semirings, namely, N the semiring of nonnegative integers, A the “arctical semiring”. T the tropical semiring and F the semiring of finite subsets of nonnegative integers. We show: for semirings N and A it is decidable in polynomial time whether or not the costs of accepting computations is bounded; for F it is decidable in polynomial time whether or not the cardinalities of occurring cost sets are bounded. In all three cases, we derive explicit upper bounds. For semiring T , we prove decidability of boundedness as well, but obtain a polynomial-time algorithm only in case that the degrees of occurring polynomials are at most 1. For N and A , we extend our results to multidimensional cost functions.

On semigroups of matrices over the tropical semiring

The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. Our method is used to recover results of Hashiguchi, Leung and the author in a unified combinatorial framework