A Necessary Relation Algebra for Mereotopology

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T ₀ topological space with an additional "contact relation" C defined by xCy x ØA (possibly) more general class of models is provided by the Region Connection Calculus (RCC) of Randell et al. We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the RCC, and hence, in any standard model of mereotopology. It follows that the expressiveness of the RCC in relational logic is much greater than the original 8 RCC base relations might suggest. We also interpret these 25 relations in the the standard model of the collection of regular open sets in the two-dimensional Euclidean plane.     

Division by zero in non-involutive meadows

Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study ‘non-involutive meadows’, i.e. variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.     

数学学业不良初中生的工作记忆特点：领域普遍性还是特殊性?

研究选取上海111名(男生48名, 女生63名, 平均年龄11.97岁)初中学生(数学学业不良学生55名与数学学业优秀学生56名), 根据国家数学课程标准, 将数学划分为数与代数、空间与几何两部分, 又从空间与几何领域中选取初一数学学业知识点—— 轴对称和中心对称图形, 分析在不同内容知识领域的解答过程中所涉及的工作记忆成分。结果发现：(1)数与代数学习需要中央执行系统、视觉-空间模板、语音环路三个成分的共同作用; 空间与几何学习主要受到视觉-空间模板和中央执行系统的影响, 但不存在语音环路的影响。(2)对轴对称与中心对称图形任务的成绩影响最大的是视觉-空间模板, 其次是中央执行系统, 语音环路对该任务作用不明显。两个研究说明工作记忆在初中不同年级的各类数学学业任务中具有不同的作用, 中央执行系统和视觉-空间模板相对更具有普遍性作用, 语音环路具有特殊性作用, 并且随着年龄增高, 视觉－空间模板对数学学业任务的作用更为突显。     

含有新算符的代数运算规则学习的有效样例设计

为了探索含有新算符的代数运算规则学习的有效样例设计方式,分别采用“转换标记法”和“解释法”设计“指-对数转换”运算和对数运算的样例,考察了初中三年级学生代数运算规则样例学习的迁移效果。结果显示：(1)采用“转换标记法”设计的样例可以明显提高“指-对数转换”规则样例学习的迁移效果;(2)采用“解释法”设计的运算样例,能够明显促进“对数运算规则”的样例学习迁移效果,并与被试的基础知识有关。     

Ordered domain algebras

We give a finite axiomatisation to representable ordered domain algebras and show that finite algebras are representable on finite bases.     

Infinitary Action Logic: Complexity, Models and Grammars

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. Presented by Jacek Malinowski     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

Interpolation in computing science: the semantics of modularization

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property . It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. Dedicated to the 50th anniversary of William Craig’s Interpolation Theorem.     

Simulating Turing machines on Maurer machines

In a previous paper, we used Maurer machines to model and analyse micro-architectures. In the current paper, we investigate the connections between Turing machines and Maurer machines with the purpose to gain an insight into computability issues relating to Maurer machines. We introduce ways to simulate Turing machines on a Maurer machine which, dissenting from the classical theory of computability, take into account that in reality computations always take place on finite machines. In one of those ways, multi-threads and thread forking have an interesting theoretical application.