Generic local computation

Many problems of artificial intelligence, or more generally, many problems of information processing, have a generic solution based on local computation on join trees or acyclic hypertrees. There are several variants of this method all based on the algebraic structure of valuation algebras. A strong requirement underlying this approach is that the elements of a problem decomposition form a join tree. Although it is always possible to construct covering join trees, if the requirement is originally not satisfied, it is not always possible or not efficient to extend the elements of the decomposition to the covering join tree. Therefore in this paper different variants of an axiomatic framework of valuation algebras are introduced which prove sufficient for local computation without the need of an extension of the factors of a decomposition. This framework covers the axiomatic system proposed by Shenoy and Shafer (1990) [1]. A particular emphasis is laid on the important special cases of idempotent algebras and algebras with some notion of division. It is shown that all well-known architectures for local computation like the Shenoy–Shafer architecture, Lauritzen–Spiegelhalter and HUGIN architectures may be adapted to this new framework. Further a new architecture for idempotent algebras is presented. As examples, in addition to the classical instances of valuation algebras, semiring-based valuation algebras, Gaussian potentials and the relational algebra are presented.     

Free semiring-representations and nondeterminism

Semirings provide a simple abstract model of the syntax for a nondeterministic programming language. Each element of a semiring is a nondeterministic program segment, and the semiring operations (+ and ?) correspond to nondeterministic or and program composition. This is analogous to using an algebraic theory for the abstract syntax of a deterministic language. In the case of algebraic theories, an algebra provides the semantics, and free algebras (which always exist) are particularly important. For a semiring, semantics is provided by a representation as a system of relations. This paper examines the question of when free representations exist. Unlike free algebras, free representations do not always exist. It is shown that a semiring has free representations generated by arbitrary sets of variables iff it has a free representation generated by a single variable. Examples of semirings are given that do not have free representations. However, for an important class of semirings, free representations are always available. This class consists of semirings which arise when nondeterminism is freely added to a deterministic programming language.     

Idempotent and tropical mathematics; complexity of algorithms and interval analysis

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics, constructions and algorithms are usually simpler than their traditional analogs. We examine in particular algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other “good” operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent semirings are examined. In this case, basic algorithms, like their interval versions, are polynomial. This situation is very different from the traditional linear algebra case, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.     

Algebraic structures of soft sets associated with new operations

Recently new operations have been defined for soft sets. In this paper, we study some important properties associated with these new operations. A collection of all soft sets with respect to new operations give rise to four idempotent monoids. Then with the help of these monoids we can study semiring (hemiring) structures of soft sets. Some of these semirings (hemirings) are actually lattices. Finally, we show that soft sets with a fixed set of parameters are MV algebras and BCK algebras.     

Normal design algebra

We generalise the designs of the Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, for example, since they form a Kleene and omega algebra and a test semiring. We apply our framework to investigate symmetric linear recursion and its relation to tail-recursion. This substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, pre-image, convergence and Noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.     

值域为坡代数的加权图和模糊图的范畴关系

坡代数是元素的乘法小于等于每个因子的加法幂等的半环。以坡代数为取值域,证明了加权图范畴可以作为余反射子范畴嵌入到模糊图范畴中去;如果坡代数的乘法有单位元,则这种嵌入还具有反射性。     

Idempotent Interval Analysis and Optimization Problems

Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of "Idempotent Mathematics" with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined. This revised version was published online in August 2006 with corrections to the Cover Date.     

An algebra of product families

Experience from recent years has shown that it is often advantageous not to build a single product but rather a family of similar products that share at least one common functionality while having well-identified variabilities. Such product families are built from elementary features that reach from hardware parts to software artefacts such as requirements, architectural elements or patterns, components, middleware, or code. We use the well established mathematical structure of idempotent semirings as the basis for a product family algebra that allows a formal treatment of the above notions. A particular application of the algebra concerns the multi-view reconciliation problem that arises when complex systems are modelled. We use algebraic integration constraints linking features in one view to features in the same or a different view and show in several examples the suitability of this approach for a wide class of integration constraint formulations. Our approach is illustrated with a Haskell prototype implementation of one particular model of product family algebra.     

An algebra of hybrid systems

Hybrid systems are heterogeneous systems characterised by the interaction of discrete and continuous dynamics. We present a trajectory-based algebraic model for describing hybrid systems; the trajectories used are closely related to streams. The algebra is based on left quantales and left semirings and provides a new application for these algebraic structures. We show that hybrid automata, which are probably the standard tool for describing hybrid systems, can conveniently be embedded into our algebra. Moreover we point out some important advantages of the algebraic approach. In particular, we show how to handle Zeno effects, which are excluded by most other authors. The development of the theory is illustrated by a running example and a larger case study.     

Internal axioms for domain semirings

New axioms for domain operations on semirings and Kleene algebras are proposed. They generalise the relational notion of domain-the set of all states that are related to some other state-to a wide range of models. They are internal since the algebras of state spaces are induced by the domain axioms. They are simpler and conceptually more appealing than previous two-sorted external approaches in which the domain algebra is determined through typing. They lead to a simple and natural algebraic approach to modal logics based on equational reasoning. The axiomatisations have been developed in a new style of computer-enhanced mathematics by automated theorem proving, and the approach itself is suitable for automated systems analysis and verification. This is demonstrated by a fully automated proof of a modal correspondence result for Löb’s formula that has applications in termination analysis.     

半环诱导的赋值代数的解轮廓和解扩展

在由半环诱导的赋值代数中,通过半环同态与赋值的复合得到了一个新的赋值代数。证明了嵌入同态保持赋值的解轮廓,推广了赋值的解扩展定理。     

On Aperiodic and Star-Free Formal Power Series in Partially Commuting Variables

Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non-commuting, then the product of any two aperiodic series is again aperiodic. We also show that if K is idempotent and the matrix monoids over K have a Burnside property (satisfied, e.g. by the tropical semiring), then the aperiodic and the star-free series coincide. This generalizes a classical result of Schützenberger (Inf. Control 4:245–270, 1961) for aperiodic regular languages and subsumes a result of Guaiana et al. (Theor. Comput. Sci. 97:301–311, 1992) on aperiodic trace languages. This work partly supported by the DAAD-PROCOPE project Temporal and Quantitative Analysis of Distributed Systems.     

Local arc consistency for non-invertible semirings, with an application to multi-objective optimization

We extend algorithms for local arc consistency proposed in the literature in order to deal with (absorptive) semirings that may not be invertible. As a consequence, these consistency algorithms can be used as a pre-processing procedure in soft Constraint Satisfaction Problems (CSPs) defined over a larger class of semirings, such as those obtained from the Cartesian product of two (or more) semirings. One important instance of this class of semirings is adopted for multi-objective CSPs. First, we show how a semiring can be transformed into a novel one where the + operator is instantiated with the least common divisor (LCD) between the elements of the original semiring. The LCD value corresponds to the amount we can “safely move” from the binary constraint to the unary one in the arc consistency algorithm. We then propose a local arc consistency algorithm which takes advantage of this LCD operator.     

State-morphism pseudo-effect algebras

The notion of a state-morphism on pseudo-effect algebras is introduced, and pseudo-effect algebras with distinguished state-morphisms are studied under the name state-morphism pseudo-effect algebras (SMPEAs). It is shown that every SMPEA admits a representation as a (total) state-morphism algebra, and some results from the general theory of state-morphism algebras (that is, algebras endowed with a distinguished idempotent endomorphism called a state-morphism), recently developed by Botur and Dvure?enskij, can be applied. In particular, it is shown that under suitable conditions, a SMPEA can be embedded into a so-called diagonal one, realized by a direct product of the SMPEA with itself endowed with a suitable natural state-morphism.     

Kleene modules and linear languages

A Kleene algebra (K, +, ·, *, 0, 1) is an idempotent semiring with an iteration * as axiomatised by Kozen. We consider left semiring modules (A, +, 0, :) over Kleene algebras. We call such a left semiring module a Kleene module if each linear equation x = a + r : x has a least solution, where : is the product from K × A to A. The linear context-free languages can be viewed as a Kleene module A over a Kleene algebra R of binary regular word relations. Thus, the simultaneous linear fixed-point operator μ on languages can be reduced to iteration * on R and the scalar product :.     

Gröbner bases in exterior algebra

We show that the Buchberger algorithm for commutative polynomials over a field may be generalised to an algebraic structure which embeds such polymomials, the exterior polynomial algebra, and which is a natural domain for linear geometry. In particular, those finite sets of exterior polynomials which induce confluent reduction relations are characterised, and a means of algorithmically constructing them from a given set presented. A distinguished subset of such bases consists of the exterior algebra version of Gröbner bases. We characterise such bases and demonstrate how to construct them algorithmically from a given finite set of exterior polynomials.     

粗代数研究

在粗糙集的代数方法研究中,一个重要的方面是从粗糙集的偶序对((下近似集,上近似集()表示入手,通过定义偶序对的基本运算,从而构造出相应粗代数,并寻找能够抽象刻画偶序对性质的一般代数结构.其中最有影响的粗代数分别是粗双Stone代数、粗Nelson代数和近似空间代数,它们对应的一般代数结构分别是正则双Stone代数、半简单Nelson代数和预粗代数.通过建立这些粗代数中算子之间的联系,证明了:(a) 近似空间代数可转化为半简单Nelson代数和正则双Stone代数;(b) 粗Nelson代数可转化为预粗代数和正则双Stone代数;(c) 粗双Stone代数可化为预粗代数和半简单Nelson代数,从而将3个不同角度的研究统一了起来.     

DNA计算的基本代数原理(一)

在文中,通过引进M-粘连代数、M-剪切代数与M-重组代数等概念,达到了使与标志集M相关联的DNA计算可按所引进代数中给出的代数定律来推演的目的.进而研究了这些代数的一系列性质,得到了许多有价值的结果.它们对DNA计算的理论与应用研究必将带来极大的便利.