Continuous-time orbit problems are decidable in polynomial-time

We place the continuous-time orbit problem in P$\mathsf{P}$, sharpening the decidability result shown by Hainry [7].     

基于软件的环境量化模型

在一些特殊领域中需要建立一定的实验环境对软件性能进行测试,因此实验环境与实际环境之间的近似程度对软件的性能起到关键作用。为建立环境之间的近似度量,在进程代数理论基础上,根据软件与环境的交互程度,利用拓扑度量和论域理论中的偏序关系,建立实验环境之间近似程度的量化模型。根据软件与环境之间的部分交互,建立实验环境之间近似程度的度量模型。通过实例对度量模型进行说明,并证明度量模型的代数性质。     

格蕴涵代数的软LI-理想

借助软集合,提出了格蕴涵代数的软LI-理想的概念,研究了软集运算下的一些诸如交、且等基本性质,并给出了格蕴涵代数的软LI-理想的一些等价刻画。此外,研究了格蕴涵代数的软LI-理想与格蕴涵代数的软格理想之间的关系。得到了格蕴涵代数的软LI-理想像与原像的性质。     

软BCK代数

提出了软BCK代数的概念,研究了软BCK代数的并,交和乘积运算,讨论了这些运算的代数性质。     

粗糙集信息观与代数观联系研究

粗糙集的信息观和代数观是通过不同的方式来定义知识的,两种观点在绝对约简中完全等价,但是在相对约简中,两种观点的关系并不明确。文章研究了两种观点在相对约简中的关系,证明了两种观点在相容决策表中完全相同,而在不相容决策表中,信息观的现有定理存在错误,也就是说信息观不适用于不相容决策表。     

Ready to preorder: The case of weak process semantics

Recently, Aceto, Fokkink and Ingólfsdóttir proposed an algorithm to turn any sound and ground-complete axiomatisation of any preorder listed in the linear time-branching time spectrum at least as coarse as the ready simulation preorder, into a sound and ground-complete axiomatisation of the corresponding equivalence—its kernel. Moreover, if the former axiomatisation is ω-complete, so is the latter. Subsequently, de Frutos Escrig, Gregorio Rodríguez and Palomino generalised this result, so that the algorithm is applicable to any preorder at least as coarse as the ready simulation preorder, provided it is initials preserving. The current paper shows that the same algorithm applies equally well to weak semantics: the proviso of initials preserving can be replaced by other conditions, such as weak initials preserving and satisfying the second τ-law. This makes it applicable to all 87 preorders surveyed in “the linear time-branching time spectrum II” that are at least as coarse as the ready simulation preorder. We also extend the scope of the algorithm to infinite processes, by adding recursion constants. As an application of both extensions, we provide a ground-complete axiomatisation of the CSP failures equivalence for BCCS processes with divergence.     

Automatic Construction and Verification of Isotopy Invariants

We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely, generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on subblocks. In addition, given the complexity of the theorems that verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving, and satisfiability-solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extend the previously known enumeration results. This work was previously beyond the capabilities of automated reasoning techniques. The author’s work was supported by EPSRC MathFIT grant GR/S31099.     

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss–Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10⁻³).     

基于偏序简化的并发系统验证

构件交互风格和交互协议的描述与验证是基于构件的分布式系统开发的基础和关键,而构件交互协议是一种典型的分布式并发系统.传统的方法难以解决系统建模和验证中的所谓的状态爆炸问题.偏序简化是应用迹的概念,对模型进行化简并且对模型进行死锁验证.但这样的验证重点放在了Petri网模型上,而没有涉及进程代数模型,所验证的只是模型是否有死锁状态.而以通信系统演算CCS为代表的进程代数,因其概念简洁,可用的数学工具丰富,在分布式并发系统的规范、分析、设计和验证方面获得了广泛应用.对此,提出将偏序规约应用于进程代数模型,给出基于进程代数模型的偏序简化算法,并提出利用进程代数模型偏序简化算法来验证安全性的方法.     

Inductive semimodules and the vector modules over them

The notion of an inductive semimodule over an ordered *-semiring is introduced and some related properties are investigated. Inductive semimodules are extensions of several important algebraic structures such as Kleene modules, Kleene algebras and inductive *-semirings. We prove that an inductive semimodule over an ordered *-semiring K is a Kleene module if and only if K is a Kleene algebra. Moreover, we establish that the vector module of an inductive semimodule over an ordered Conway semiring is again an inductive semimodule over the matrix semiring. Consequently, in an inductive semimodule over an ordered Conway semiring, least solutions to linear inequation systems can be denoted by linear expressions, avoiding the least fixed point operator. In addition, we also introduce a related notion called weak inductive semimodules, and propose several open problems on them.