Commutative Semiring Research Articles

Glushkov Construction For Series: The Non Commutative Case

We present an extension to multiplicities of a classical algorithm for computing a boolean automaton from a regular expression. The Glushkov construction computes an automaton with n +1 states from a regular expression with n occurences of letters. We give an extension of the Glushkov algorithm for the multiplicity case in a non commutative semiring. Next, we give four equivalent extended step by step algorithms.

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.

On memory redundancy in the BCJR algorithm for nonrecursive shift register processes

For computation of a posteriori probabilities (APPs), the well-known Bahl, Cocke, Jelinek, and Raviv (1974) algorithm is often applied. If the underlying model is a nonrecursive shift register process, it is shown that this algorithm obtains a general memory redundancy if the computation is performed over a commutative semiring with an absorbing zero element. The result is of particular interest for the BCJR algorithm carried out in the max-log domain.

On Sequences Defined by D0L Power Series

We study D0L power series over commutative semirings. We show that a sequence (cn)n≥0 of nonzero elements of a field A is the coefficient sequence of a D0L power series if and only if there exist a positive integer k and integers βi for 1 ≤ i ≤ k such that for all n ≥ 0. As a consequence we solve the equivalence problem of D0L power series over computable fields.

The Dynamic Parallel Complexity of Computational Circuits

We establish connections between parallel circuit evaluation and uniform algebraic closure properties of unary function classes. We use this connection in the development of time-efficient and processor-efficient parallel algorithms for the evaluation of algebraic circuits. Our algorithm provides a nontrivial upper bound on the parallel complexity of the circuit value problem over $\{{\Bbb R},\min,\max,+\}$ and $\{{\Bbb R}^{+},\min,\max,\times\}$. We partially answer an open question of Miller, Ramachandran, and Kaltofen by showing that circuits over a polynomial-bounded noncommutative semiring and circuits over infinite noncommutative semirings with a polynomial-bounded dimension over a commutative semiring can be evaluated in polylogarithmic time in their size and degree using a polynomial number of processors. We also present an improved parallel algorithm for Boolean circuits.

Different interpretations of triangular norms and related operations

Triangular norms (and triangular conorms and uni-norms) can be treated as semigroup, two-place function, also as commutative semiring multiplications. It can also be derived from the difference operation in a difference poset. We discuss here these different interpretations and their consequences.

Characterizations of semirings by p-injective and projective semimodules

In this paper we obtain characterizations of classes of semirings by P-injective and projective right R-semimodules. We prove that a semiring R is von Neumann regular if and only if each cyclic right R-semimodule is P-injective. Moreover, a commutative semiring R whose principal ideals are k-closed is von Neumann regular if and only if every simple R-semimodule is PP-injective. We also examine some properties of right PP-semirings, that is, semirings all of whose principal right ideals are projective. It is shown that R is a right PP-semiring if and only if the endomorphism semiring of every cyclic projective right R-semimodule is right PP.

Bideterminants, arborescences and extension of the Matrix-Tree theorem to semirings

The Matrix-Tree Theorem is a well-known combinatorial result relating the value of the minors of a certain square matrix to the sum of the weights of the arborescences (= rooted directed trees) in the associated graph. We prove an extension of this result to algebraic structures much more general than the field of real numbers, namely commutative semirings. In such structures, the first law (addition) is not assumed to be invertible, therefore the combinatorial proof given here significantly differs from earlier proofs for the standard case. In particular, it requires the use of the concept of bideterminant of a matrix, an extension of the classical concept of determinant.

An Algebraic Model for Combinatorial Problems

A new algebraic model, called the generalized satisfiability problem (GSP) model, is introduced for representing and solving combinatorial problems. The GSP model is an alternative to the common method in the literature of representing such problems as language-recognition problems. In the GSP model, a problem instance is represented by a set of variables together with a set of terms, and the computational objective is to find a certain sum of products of terms over a commutative semiring. The model is general enough to express all the standard problems about sets of clauses and generalized clauses, all nonserial optimization problems, and all $\{ 0,1\} $-linear programming problems. The model can also describe many graph problems, often in a very direct structure-preserving way. Two important properties of the model are the following;1. In the GSP model, one can naturally discuss the structure of individual problem instances. The structure of a GSP instance is displayed in a “structure tree.” The smaller the “weighted depth” or “channelwidth” of the structure tree for a GSP instance, the faster the instance can be solved by any one of several generic algorithms. 2. The GSP model extends easily so as to apply to hierarchically specified problems and enables solutions to instances of such problems to be found directly from the specification rather than from the (often exponentially) larger specified object.

Pan-operations structure

The structure of pan-addition ⊕ and pan-multiplication ⊙ of a commutative isotonic semiring ( R +, ⊕, ⊙) is analyzed. We show that if ⊕ ≠ V (supremum), then ( R +, ⊕, ⊙ ) is a g-semiring, i.e. a⊕ b = g −1( g( a) + g( b)) and d a⊙ b = g −1( g( a) · g( b)). Conclusions for pan-integrals with respect to a ⊕-measure are shown.

Seven trees in one

Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T 7 of seven-tuples of such trees. “Particularly elementary” means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X = 1 + X 2. Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all.

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.

Tensor products of spherical and equivariant immersions

In [C1, C2, C3, C4], B.-Y. Chen introduced the tensor product of two immersions of a given Riemannian manifold; he proved that the set of all immersions of the given manifold, provided with direct sum and tensor product, defines a commutative semiring. In [DDVV] we introduced I, the commutative semiring of all transversal immersions of all differentiable manifolds in Euclidean spaces, provided with the binary operations direct sum and tensor product. In this paper we further investigate which immersions define a subsemiring or a multiplicative subsemigroup ; in particular, we fix our attention on spherical immersions of differentiable manifolds, isometric and equivariant immersions of Riemannian manifolds and immersions of finite type. Denote by E the n-dimensional Euclidean space with Euclidean metric 〈 , 〉. The n-dimensional sphere with radius r is denoted by S(r). Let f : M → E be an immersion of a differentiable manifold in a Euclidean space. Then f is said to be transversal in a point p ∈M if and only if the position vector f(p) is not tangent to M at p, i.e. f(p) / ∈ f∗(TpM). If f is transversal in every point of M , then f shortly is called transversal. Consider two differentiable manifolds M and N of dimensions r resp. s and assume that f : M → E and h : N → E are two transversal immersions. Then the direct sum map f⊕h : M×N → E : (p, q) 7→ (f(p), h(q)) and the tensor product map f ⊗ h : M × N → E : (p, q) 7→ f(p) ⊗ h(q) are again two transversal immersions. We define a symmetric relation ∼ as follows : if f : M → E is an immersion and i : E ⊂ E is a linear isometric immersion, then ∗Supported by a research fellowship of the Research Council of the Katholieke Universiteit Leuven †Senior Research Assistant of the National Fund for Scientific Research (Belgium) ‡Research Fellow of the Research Council of the K.U.Leuven Received by the editors November 1993 Communicated by A. Warrinier AMS Mathematics Subject Classification : 53C40, 53B25, 58G25

DIFFERENTIATION OF K-RATIONAL EXPRESSIONS

We develop a formal differential calculus on the K-*-algebra of rational expressions with multiplicities in a commutative semiring K. We also characterize the systems [Formula: see text] of K-rational identities such that the derivative of any [Formula: see text]-consequence remains [Formula: see text]-deducible.

Orderings on semirings

Though more general definitions are sometimes used, for this paper a semiring will be defined to be a set S with two operations + (addition) and �9 (multiplication); with respect to addition, S is a commutative monoid with 0 as its identity dement�9 With respect to multiplication, S is a (generaily noncommutative) monoid with 1 as its identity element. Connecting the two algebraic structures are the distributive laws, a.(b+ c) = a.b+ a.c and (a+ b).c = a.c+ b.c for all a,b,c E S, and the requirement a.0 = 0.a = 0 for every a C S. The reader is referred to [3] and [4] for an introduction to the theory of semirings. Semirings have proven to be useful in theoretical computer science, in particular for studying automata and formal languages. Several authors have considered total order relations on semirings (e.g. [8, 9]). In this paper, we begin by looking at special classes of semirings which arise naturally in the theory of formally real (not necessarily commutative) fields and in the study of (commutative) real algebraic geometry�9 In the second section of the paper, we give characterizations of the semirings which arise in this way. Using this work as motivation, the third section of the paper suggests a new notion of ordering for commutative semirings which generalizes the concepts found in real algebraic geometry. In the most general case, these orderings turn out to have a somewhat weaker connection with total order relations than in the case of rings.

Further contributions to the study of finite fuzzy relation equations

An interesting property of the Schein rank of a fuzzy matrix stressed by K.H. Kim and F.W. Roush is formalized as definition of the basis of a set of fuzzy vectors (defined over a commutative semiring) with respect to another set of fuzzy vectors. We apply this definition and another related concept in the setting of finite fuzzy relation equations. In particular, we give a sufficient condition for the existence of the smallest solution of a max-min fuzzy equation.

Inversion of matrices over a commutative semiring

It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative semiring, then the equation AB = 1 implies BA = 1. We give two proofs: one algebraic in nature, the other more combinatorial. Both proofs use a generalization of the familiar product law for determinants over a commutative semiring.

Infinite linear systems and one counter languages

A theory of infinite linear systems and their solutions is developed. For any commutative semiring families of formal power series are introduced, which are generalizations of the one counter languages. These families are characterized in two ways: 1. (i) by solutions of infinite linear systems with a certain periodic structure; 2. (ii) by solutions of finite algebraic systems with a certain form given by matrix equations.

An algebraic characterization of some principal regulated rational cones

The aim of this paper is to deal with formal power series over a commutative semiring A. Generalizing Wechler's pushdown automata and pushdown transition matrices yields a characterization of the A-semi-algebraic power series in terms of acceptance by pushdown automata. Principal regulated rational cones generated by cone generators of a certain form are characterized by algebraic systems given in certain matrix form. This yields a characterization of some principal full semi-AFL's in terms of context-free grammars. As an application of the theory, the principal regulated rational cone of one-counter “languages” is considered.

On groupoids defined by commutators

We study matrices R, L which count the numbers of solutions of i x = j ix = j and x i = j xi = j . For slight generalizations of R, L, the relation R L = L R RL = LR characterizes associativity of a groupoid. For groupoids defined by group commutators x y x − 1 y − 1 xy{x^{ - 1}}{y^{ - 1}} the relation R L = L R RL = LR is valid. In addition one can study analogues of Green’s relations. Any I \mathcal {I} -class contains at most four H \mathcal {H} -classes in a commutator groupoid.