Logical-Probabilistic Calculus: A Tool for Studying the Reliability and Safety of Structurally Complex Systems

The concept of the probabilistic logic is briefly described as an extension of inductive logic. A clear distinction is made between the probabilistic logic and logical-probabilistic calculus—the branch of mathematics that defines the rules of computation and operation with two-valued (true and false) statements. The logical-probabilistic calculus is based on the algebra of logic and rules of substitution of logical arguments in functions of the algebra of logic by their truth probabilities, and logical operations by arithmetical operations.     

Methods of Continuous Logic in Control

Continuous logic, its problems, and methods were outlined, and its basic operations were defined. The algebra of continuous logic was described, and its main functions of one, two, and three variables were listed. The laws of this logic were presented and contrasted with the laws of the discrete two-valued logic. Described were the problems of listing all continuous-logic functions of a given number of variables and representing them in a standard form. The difference between these forms and their counterparts in the two-valued logic was shown. Minimization procedures for the continuous-logic functions and their decomposition into functions of a smaller number of variables were described. The distinctions of these procedures from their counterparts in the two-valued logic were noted. The problems of analysis and synthesis of the continuous-logic functions were formulated, and methods for their solution were presented. The problem of synthesis was shown not to be necessarily solvable. The fundamentals of the continuous-logic differential and integral calculuses were presented. Any continuous-logic function was shown to have no-derivative points. The problem of completeness was described for the continuous logic together with the existing results and their distinctions from the discrete case. Numerous applications of the continuous logic to mathematics, engineering, economics, social sciences, and so on were described, and its perspectives were estimated.     

Extending the Continuous Logic to Uncertain Variables

Consideration was given to an extension of the continuous logic to uncertainty where the logic operations are performed over the intervals of possible values of real numbers and not over their precisely known values from a certain continuous set. Basic laws of the algebra of this logic were formulated. A possible application of the interval continuous logic to some problems of control was demonstrated.     

一种基于逻辑关系方程解的属性约简算法

粗糙集理论中所有的概念与运算都是通过代数学的等价关系和集合运算来定义的.在这种定义下,粗糙集理论的很多概念与运算的直观性较差.从逻辑代数的角度出发,建立了属性集与布尔矩阵以及逻辑关系方程之间的关系,给出了逻辑关系方程有解、有惟一解、有多个解的充分必要条件,在逻辑关系方程解的基础上给出了一种新的高效的属性约简算法.     

机会发现在多智能体系统中的形式化描述方法

针对多智能体系统(MAS)中执行决策的非确定性问题,研究结合时态/模态逻辑及机会发现理论,引入基于Kripke结构的复合逻辑 ,用于Agent的知识系统(全局知识、局部知识)及机会发现的形式化描述,为实现动态并发环境中MAS的系统协作行为建模及自动决策推理提供依据。给出 的结构及语义,证明了 的可判定性,且计算可在多项式级时间复杂度内实现。     

可分析集合、笛卡尔逻辑和命题的可推导性关系

本文在ZFC公理系统的基础上,首先提出可分析集合的概念且可表达为p={x|x∈p}。然后给出受囿变量的定义,引入笛卡尔逻辑以使逻辑的概念像算法的概念一样精确化而成为明确的数学对象,不仅足以适应代数和分析的要求,而且充分满足经典和非经典逻辑的需要,并探讨命题间的可推导性关系,包括:对立、排中、重言、归谬、反对、矛盾、存在、全称、独立、同一和不矛盾。进而,讨论一些逻辑运算和有关的逻辑问题,并进一步阐述可分析集合的几个基本关系和基本运算。     

集合代数是经典命题演算形式系统的语义解释

经典命题演算形式系统(CPC)中的公式只是一些形式符号,其意义是由具体的解释给出的.逻辑代数和集合代数都是布尔代数,都是CPC的解释.集合代数是CPC的集合语义,其中对联结词的解释就是集合运算;对形式公式的解释就是集合函数;对逻辑蕴涵.逻辑等价的解释就是集合包含和集合相等=.标准概率逻辑是在标准概率空间上建立的逻辑体系,命题表示随机事件,随机事件是集合,概率空间中的事件域是集合代数,概率逻辑就是CPC集合语义的实际应用.CPC完全适用于概率命题演算.     

Auto-tuned Krylov methods on cluster of graphics processing unit

Exascale computers are expected to have highly hierarchical architectures with nodes composed by multiple core processors (CPU; central processing unit) and accelerators (GPU; graphics processing unit). The different programming levels generate new difficult algorithm issues. In particular when solving extremely large linear systems, new programming paradigms of Krylov methods should be defined and evaluated with respect to modern state of the art of scientific methods. Iterative Krylov methods involve linear algebra operations such as dot product, norm, addition of vectors and sparse matrix–vector multiplication. These operations are computationally expensive for large size matrices. In this paper, we aim to focus on the best way to perform effectively these operations, in double precision, on GPU in order to make iterative Krylov methods more robust and therefore reduce the computing time. The performance of our algorithms is evaluated on several matrices arising from engineering problems. Numerical experiments illustrate the robustness and accuracy of our implementation compared to the existing libraries. We deal with different preconditioned Krylov methods: Conjugate Gradient for symmetric positive-definite matrices, and Generalized Conjugate Residual, Bi-Conjugate Gradient Conjugate Residual, transpose-free Quasi Minimal Residual, Stabilized BiConjugate Gradient and Stabilized BiConjugate Gradient (L) for the solution of sparse linear systems with non symmetric matrices. We consider and compare several sparse compressed formats, and propose a way to implement effectively Krylov methods on GPU and on multicore CPU. Finally, we give strategies to faster algorithms by auto-tuning the threading design, upon the problem characteristics and the hardware changes. As a conclusion, we propose and analyse hybrid sub-structuring methods that should pave the way to exascale hybrid methods.     

An algorithm for identifying symmetric variables in the canonical OR-coincidence algebra system

To simplify the process for identifying 12 types of symmetric variables in the canonical OR-coincidence （COC） algebra system, we propose a new symmetry detection algorithm based on OR-NXOR expansion. By analyzing the relationships between the coefficient matrices of sub-functions and the order coefficient subset matrices based on OR-NXOR expansion around two arbitrary logical variables, the constraint conditions of the order coefficient subset matrices are revealed for 12 types of symmetric variables. Based on the proposed constraints, the algorithm is realized by judging the order characteristic square value matrices. The proposed method avoids the transformation process from OR-NXOR expansion to AND-OR-NOT expansion, or to AND-XOR expansion, and solves the problem of completeness in the dj-map method. The application results show that, compared with traditional methods, the new algorithm is an optimal detection method in terms of applicability of the number of logical variables, detection type, and complexity of the identification process. The algorithm has been implemented in C language and tested on MCNC91 benchmarks. Experimental results show that the proposed algorithm is convenient and efficient.     

逻辑系统的代数状态空间方法的基础、现状及其应用

逻辑系统指自变量只取有限个值的动态系统.包括2值的经典逻辑(或布尔逻辑)、k值逻辑、(一般)有限值逻辑.近年来,利用矩阵半张量积发展起来的逻辑动态系统的代数状态空间方法得到长足的进展和普遍的重视.同时,它被广泛应用于许多工程问题或理论研究中.它类似于Rn上由微分或差分方程描述的动态系统的Kalman状态空间方法,为逻辑系统的分析与控制设计提供了一个便捷的平台.本文首先对该方法作一简要介绍,然后,对该新兴学科分支的现状作一评述.最后,详细介绍该方法目前的应用以及其更广泛的应用前景.     

Nonlinear system identification using optimized dynamic neural network

In this paper, both off-line architecture optimization and on-line adaptation have been developed for a dynamic neural network (DNN) in nonlinear system identification. In the off-line architecture optimization, a new effective encoding scheme—Direct Matrix Mapping Encoding (DMME) method is proposed to represent the structure of neural network by establishing connection matrices. A series of GA operations are applied to the connection matrices to find the optimal number of neurons on each hidden layer and interconnection between two neighboring layers of DNN. The hybrid training is adopted to evolve the architecture, and to tune the weights and input delays of DNN by combining GA with the modified adaptation laws. The modified adaptation laws are subsequently used to tune the input time delays, weights and linear parameters in the optimized DNN-based model in on-line nonlinear system identification. The effectiveness of the architecture optimization and adaptation is extensively tested by means of two nonlinear system identification examples.     

Process algebra with guards: Combining Hoare logic with process algebra

We extend process algebra with guards, comparable to the guards in guarded commands or conditions in common programming constructs such as if — then — else — fi and while — do — od.The extended language is provided with an operational semantics based on transitions between pairs of a process and a (data-)state. The data-states are given by a data environment that also defines in which data-states guards hold and how atomic actions (non-deterministically) transform these states. The operational semantics is studied modulo strong bisimulation equivalence. For basic process algebra (without operators for parallelism) we present a small axiom system that is complete with respect to a general class of data environments. Given a particular data environmentL we add three axioms to this system, which is then again complete, provided weakest preconditions are expressible andL is sufficiently deterministic.Then we study process algebra with parallelism and guards. A two phase-calculus is provided that makes it possible to prove identities between parallel processes. Also this calculus is complete. In the last section we show that partial correctness formulas can easily be expressed in this setting. We use process algebra with guards to prove the soundness of a Hoare logic for linear processes by translating proofs in Hoare logic into proofs in process algebra.Supported by ESPRIT Basic Research Action no. 3006 (CONCUR) and by RACE project no. 1046 (SPECS).Supported by RACE project no. 1046 (SPECS).     

Practical Reflection for Sequent Logics

It is well-known that adding reflective reasoning can tremendously increase the power of a proof assistant. In order for this theoretical increase of power to become accessible to users in practice, the proof assistant needs to provide a great deal of infrastructure to support reflective reasoning. In this paper we explore the problem of creating a practical implementation of such a support layer.Our implementation takes a specification of a logical theory (which is identical to how it would be specified if we were simply going to reason within this logical theory, instead of reflecting it) and automatically generates the necessary definitions, lemmas, and proofs that are needed to enable the reflected meta-reasoning in the provided theory.One of the key features of our approach is that the structure of a logic is preserved when it is reflected. In particular, all variables, including meta-variables, are preserved in the reflected representation. This also allows the preservation of proof automation—there is a structure-preserving one-to-one map from proof steps in the original logic to proof step in the reflected logic.To enable reasoning about terms with sequent context variables, we develop a principle for context induction, called teleportation.This work is fully implemented in the MetaPRL theorem prover.     

How to draw a sphere.2. Coordinate systems

For pt. 1 see ibid., 1995. The author continues his discussion about how to render a single arbitrarily transformed sphere. In part 1 he described what the inner loop is going to look like and he listed a whole bunch of vector matrix algebra operations and the geometric constructions they represent. He now starts to derive the inner loop of the rendering algorithm in order to see all the matrix algebra in action. Most of the time, however, will be spent picking the correct coordinate system to make the inner loop fast and accurate. Even if you aren't going to write special purpose sphere rendering programs, this will clearly illustrate the workings of the homogeneous perspective transform and the geometric interpretations of transformation matrices     

N-tuple algebra-based probabilistic logic

The concept of “probabilistic logic” known in artificial intelligence needs a more through substantiation. A new approach to constructing probabilistic logic based on the n-tuple algebra developed by the author is proposed. A brief introduction is given to the n-tuple algebra and its properties that provide efficient paralleling of algorithms for solving problems of logical analysis of systems in computer implementation are generalized. Methods for solving direct and inverse problems of probabilistic simulation of logical systems are considered.     

DATALOG与关系代数

本文旨在通过描述DATALOG和关系代数的概念，以及DATALOG与关系代数的区别，经过从关系代数到逻辑规则和从逻辑到关系的论述，并比较DATALOG和关系代数，说明DATALOG是一种基于逻辑的数据模型，是PROLOG语言的数据库版本，关系代数是过程化的语言，关系代数表达式与安全、非递归、带有非操作的DATALOG程序的表达能力是相同的。     

Designing dependable logic controllers using algebraic specifications

Formal methods can strongly contribute to improve dependability of logic controllers during design, by providing means to avoid flaws due to designers’ omissions or specifications misinterpretations. This article presents a formal synthesis method that is aimed at obtaining the control laws of a logic system from specifications given in natural language. The formal framework that underlies the method is a Boolean algebra for logic discrete event systems. The operations and relations of this algebra enable to represent controller specifications formally, to detect inconsistencies within specifications and to generate control laws from a consistent specifications set. The scalability of this method is clearly demonstrated with the help of the case study of an experimental manufacturing line.     

开放环境下多比特量子计算机的相干控制模型

研究了开放环境下多比特量子计算系统的相干控制建模问题.基于开放量子系统的数学模型,选取适当的矩阵基将描述多比特量子计算机的复矩阵动态控制模型转化为实向量空间上的控制模型,并给出计算相应的结构系数的方法.这些工作提供了进一步研究控制律设计的基础.