The logic of events

An event space is a set of instantaneous events that vary both in time and specificity. The concept of an event space provides a foundation for a logical – i.e., modular and open – approach to causal reasoning. In this article, we propose intuitively transparent axioms for event spaces. These axioms are constructive in the intuitionistic sense, and hence they can be used directly for causal reasoning in any computational logical framework that accommodates type theory. We also put the axioms in classical form and show that in this form they are adequate for the representation in terms of event trees established by Shafer [40] using stronger axioms. This revised version was published online in June 2006 with corrections to the Cover Date.     

A generalized negative imaginary lemma and Riccati-based static state-feedback negative imaginary synthesis

In this paper, we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition. Then, an algebraic Riccati equation method is given to determine if a system is negative imaginary. Also, a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property. The controller synthesis procedure is based on the proposed negative imaginary lemma. Using this procedure, the closed-loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty, such as in the case of unmodeled spill-over dynamics in a lightly damped flexible structure. A numerical example is presented to illustrate the usefulness of the results.     

Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making

Intuitionistic fuzzy sets are useful for modeling uncertain data of realistic problems. In this paper, we generalize and expand the utility of complex intuitionistic fuzzy sets using the space of quaternion numbers. The proposed representation can capture composite features and convey multi-dimensional fuzzy information via the functions of real membership, imaginary membership, real non-membership, and imaginary non-membership. We analyze the order relations and logic operations of the complex intuitionistic fuzzy set theory and introduce new operations based on quaternion numbers. We also present two quaternion distance measures in algebraic and polar forms and analyze their properties. We apply the quaternion representations and measures to decision-making models. The proposed model is experimentally validated in medical diagnosis, which is an emerging application for tackling patient’s symptoms and attributes of diseases.     

Term rewriting and its application to recognizing handwritten Hindu numerals

In this paper the theoretical basis is presented and the implementation of a term rewriting system based on algebraic specifications is described. The input to this system is represented by an algebraic specification language, which forms not only the set of axioms but also the sorts, variables, operators and terms of a specific simulated theory or application. Rewriting and matching mechanisms provide the formal methodology for evaluating terms and proving assertions in an algebraic theory. Specifications are evaluated by interpreting terms by means of rewrite rules. The rules are described by the axioms of the specifications where the finite termination and congruence properties are assumed. A term rewriting system to recognize handwritten Hindu numerals is introduced as a case study. Besides rewriting, a robust algorithm is proposed to segment the numeral's image into strokes based on feature points and to identify cavity features. A syntactic representation (term) of the input image is matched and rewritten against a set of rules. Experimental results proved that the proposed system is tolerant to recognize a variety of numeral shapes with 96% successful recognition rate.     

Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach

 This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. For this purpose the well-known fuzzy inference mechanisms have to be extended or replaced by more general ones. The methods proposed so far in the literature for rule interpolation are mainly conceived for the application to fuzzy control and miss certain logical characteristics of an inference. First, a set of axioms is proposed in this paper. With this, a definition is given for the notion of interpolation, extrapolation, linear interpolation and linear extrapolation of fuzzy rules. The axioms include all the conditions that have been of interest in the previous attempts and others which either have logical characteristics or try to capture the linearity of the interpolation. A new method for linear interpolation and extrapolation of compact fuzzy quantities of the real line is suggested and analyzed in the spirit of the given definition. The method is extended to non-linear interpolation and extrapolation as well.     

从BPMN模型导出组合服务的代数规约

针对应用规约自动测试BPEL表示组合服务时需要解决BPEL服务的规约生成问题,提出了一种从BPMN模型导出BPEL规范定义的组合Web服务的由代数规约语言CASOCC-WS表示的代数规约方法。首先,定义从BPMN模型转换成基调的规则和从BPMN结构转换成正则表达式的规则,设计由正则表达式导出构成公理的项的算法;然后,提出根据所得的项人工书写公理的启发式规则;最后,实现一个从BPMN模型导出组合服务基调的工具原型。案例研究表明,该方法可以解决BPEL服务的代数规约生成问题。     

Effective homology of bicomplexes, formalized in Coq

In this paper, we present a complete formalization in the Coq theorem prover of an important algorithm in computational algebra, namely the calculation of the effective homology of a bicomplex. As a necessary tool, we encode a hierarchy of algebraic structures in constructive type theory, including graded and infinite data structures. The experience shows how some limitations of the Coq proof assistant to deal with this kind of algebraic data can be overcome by applying a separation of concerns principle; more concretely, we propose to distinguish in the representation of an algebraic structure (such as a group or a module) a behavioural part, containing operation signatures and axioms, and a structural part determining if the algebraic data is free, of finite type and so on.     

NMG-代数中同态核的结构刻画

逻辑代数上的Bosbach态与Riečan态是经典概率论中Kolmogorov公理的两种不同方式的多值化推广,也是概率计量逻辑中语义计量化方法的代数公理化,是非经典数理逻辑领域中的重要研究分支.现已证明具有Glivenko性质的逻辑代数上的Bosbach态与Riečan态等价,并且逻辑代数的Glivenko性质是研究态算子的构造和存在性的重要工具,因而是态理论中的研究热点之一.研究了NMG-代数基于核算子的Glivenko性质,证明NMG-代数具有核基Glivenko性质的充要条件是该核算子是从此NMG-代数到其像集代数的同态,并给出NMG-代数中同态核的结构刻画.这里,NMG-代数是刻画序和三角模<（[0,1/2,T_NM]）,（[1/2,1,T_M]）>的逻辑系统NMG的语义逻辑代数.     

About the implementability and the power of equationally defined data abstractions

The question of implementability and expressive power of equational axiom definitions of data abstractions is faced in the paper from the point of view of computability theory.A definition of implementable algebra is given, which looks reasonable and very general. With respect to the given definition it is proved that, if the least congruence semantics is accepted, an equationally defined data algebra is implementable if and only if the least congruence on terms induced by the equational definition is decidable. Moreover, the paper shows that there are: (a) equationally defined data algebras that cannot be implemented; (b) implementable algebras that cannot be expressed in any way by equational axioms.     

非线性代数规范的直接实现技术

代数规范是支持软件规格说明和设计的一种有效的方法，代数规范的直接实现技术是该研究领域的一个主要分支，目前这方面的研究基本上局限于线性代数规范，本文介绍一个实现非线性代数规范的转换过程，从该过程可自然是导出针对不同程序设计语言的转换系统，我们已实现了一个基于Ｐａｓｃａｌ语言的转换系统。     

BDI模型信念特性研究

Agent计算的典型模型是BDI(belief,desire,intention),信念是Agent计算的重要属性．把信念拓展为知识性信念和可实现信念．Agent的知识性信念是指Agent当前具有或掌握的知识,具有知识的进化和继承特性．Agent的可实现信念是指当前不成立而在将来会成立的事情,是Agent被意识到的个性倾向和目标．用非标准世界的模态逻辑为形式化工具描述两种信念,将可能世界的可达关系作为认识的不同阶段和达到可实现信念的不同阶段,避免了“逻辑全知问题”和逻辑蕴涵的副作用问题．可实现信念满足KD公理,知识性信念满足KDT4公理．两种信念适合描述Agent的意识状态和意识模型．     

Exponential synchronization of a class of neural networks with time-varying delays.

This paper aims to present a synchronization scheme for a class of delayed neural networks, which covers the Hopfield neural networks and cellular neural networks with time-varying delays. A feedback control gain matrix is derived to achieve the exponential synchronization of the drive-response structure of neural networks by using the Lyapunov stability theory, and its exponential synchronization condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. This condition can avoid solving an algebraic Riccati equation. Both the cellular neural networks and Hopfield neural networks with time-varying delays are given as examples for illustration.     

The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.     

The rewriting logic semantics project

Rewriting logic is a flexible and expressive logical framework that unifies algebraic denotational semantics and structural operational semantics (SOS) in a novel way, avoiding their respective limitations and allowing succinct semantic definitions. The fact that a rewrite logic theory’s axioms include both equations and rewrite rules provides a useful “abstraction dial” to find the right balance between abstraction and computational observability in semantic definitions. Such semantic definitions are directly executable as interpreters in a rewriting logic language such as Maude, whose generic formal tools can be used to endow those interpreters with powerful program analysis capabilities.     

Structures for Abstract Rewriting

When rewriting is used to generate convergent and complete rewrite systems in order to answer the validity problem for some theories, all the rewriting theories rely on a same set of notions, properties, and methods. Rewriting techniques have been used mainly to answer the validity problem of equational theories, that is, to compute congruences. Recently, however, they have been extended in order to be applied to other algebraic structures such as preorders and orders. In this paper, we investigate an abstract form of rewriting, by following the paradigm of logical-system independency. To achieve this purpose, we provide a few simple conditions (or axioms) under which rewriting (and then the set of classical properties and methods) can be modeled, understood, studied, proven, and generalized. This enables us to extend rewriting techniques to other algebraic structures than congruences and preorders such as congruences closed under monotonicity and modus ponens. We introduce convergent rewrite systems that enable one to describe deduction procedures for their corresponding theory, and we propose a Knuth-Bendix–style completion procedure in this abstract framework.     

正则FI-代数的刻画及成为Boole代数的条件

正则FI-代数是仅基于蕴涵算子在一般集合上建立的逻辑代数。基于正则FI-代数的公理组以及诸多性质之间的内部联系,给出了正则FI-代数的两个公理组条件更少的刻画定理,简化了正则FI-代数的定义形式。在正则FI-代数中引入蕴涵分配性,探讨了蕴涵分配正则FI-代数的若干性质,证明了蕴涵分配正则FI-代数与Boole代数是相互等价的代数系统,给出了Boole代数的一种新的刻画,使其在形式上更接近于二值逻辑代数。     

Interpreting Abstract Interpretations in Membership Equational Logic

We present a logical framework in which abstract interpretations can be naturally specified and then verified. Our approach is based on membership equational logic which extends equational logics by membership axioms, asserting that a term has a certain sort. We represent an abstract interpretation as a membership equational logic specification, usually as an overloaded order-sorted signature with membership axioms. It turns out that, for any term, its least sort over this specification corresponds to its most concrete abstract value. Maude implements membership equational logic and provides mechanisms to calculate the least sort of a term efficiently. We first show how Maude can be used to get prototyping of abstract interpretations “for free.” Building on the meta-logic facilities of Maude, we further develop a tool that automatically checks an abstract interpretation against a set of user-defined properties. This can be used to select an appropriate abstract interpretation, to characterize the specific loss of information during abstraction, and to compare different abstractions with each other.     

From machine learning to machine reasoning

A plausible definition of “reasoning” could be “algebraically manipulating previously acquired knowledge in order to answer a new question”. This definition covers first-order logical inference or probabilistic inference. It also includes much simpler manipulations commonly used to build large learning systems. For instance, we can build an optical character recognition system by first training a character segmenter, an isolated character recognizer, and a language model, using appropriate labelled training sets. Adequately concatenating these modules and fine tuning the resulting system can be viewed as an algebraic operation in a space of models. The resulting model answers a new question, that is, converting the image of a text page into a computer readable text. This observation suggests a conceptual continuity between algebraically rich inference systems, such as logical or probabilistic inference, and simple manipulations, such as the mere concatenation of trainable learning systems. Therefore, instead of trying to bridge the gap between machine learning systems and sophisticated “all-purpose” inference mechanisms, we can instead algebraically enrich the set of manipulations applicable to training systems, and build reasoning capabilities from the ground up.     

基于双阶样条的代数双曲B样条升阶及割角算法

考虑代数双曲B样条曲线的升阶问题,从理论上证明了曲线的升阶可以理解为控制顶点的割角过程.为了实现代数双曲B样条曲线的升阶,文中构造了一组基函数——双阶代数双曲B样条基函数,这组基函数并不具有统一的阶数,而具有"双阶"性质.代数双曲B样条基函数与双阶样条基函数之间的变换公式可以导出曲线升阶的割角算法.