Algebraic semantics of ER-models in the context of the calculus of relations: I: Static view

Entity-Relationship modeling is a popular technique for data modeling. Despite its popularity and wide spread use, it lacks a firm semantic foundation. We propose a translation of an ER-model into the formalism provided by the calculus of relations, suggesting that this calculus does provide suitable mechanisms for establishing a formal semantics of entity-relationship modeling. This paper deals with the techniques necessary for the translation and provides a static view of an ER-model.     

Evolution of object behavior using context relations

A collection of design patterns was described by E. Gamma et al. (1994). Each pattern ensures that a certain system aspect can vary over time, for example the operations that can be applied to an object or the algorithm of a method. The patterns are described by constructs such as the inheritance and reference relations, attempting to emulate more dynamic relationships. As a result, the design patterns demonstrate how awkward it is to program natural concepts of evolution when using a traditional object oriented language. We present a new relation between classes: the context relation. It directly models dynamic evolution, and it is meaningful at both the design and implementation level. At the design level we extend the Unified Modeling Language (UML) to include the context relation as a new form of arrow between classes. At the implementation level we present a small extension of Java. The context relation introduces a new form of dynamic binding that serves as a replacement to delegation. We demonstrate how the context relation can be used to easily model and program numerous design patterns     

The complexity of tensor calculus

Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for $\bigoplusP$, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitzs theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts $\bigoplusLOGCFL$ and $\bigoplusL$, and several other counting classes. Finally, the known inclusions $\NP/\poly \subseteq \bigoplusP/\poly$, $\LOGCFL/\poly \subseteq \bigoplusLOGCFL/\poly$, and $\NL/\poly \subseteq \bigoplusL/\poly$, which have scattered proofs in the literature (Valiant & Vazirani 1986; Gál & Wigderson 1996), are shown to follow from the new characterizations in a single blow. As an intermediate tool, we define and make use of the natural notion of an algebraic Turing machine over a semiring $ \mathcal{S}$.     

Unification of nonlinear filtering in the context of binary logical calculus,part II: Gray-scale filters

This second part of a two-part study concerning the logical structure of nonlinear filters treats gray-scale filters. The algebraic framework of threshold decomposition is dedscribed in terms of the appropriate underlying commuting diagram, along with the manner in which generalized stack filters fall out of the framework when it is interpreted in the context of logical calculus. Relationships between representations for morphological and generalized stack filters are expressed in cellular logic.     

Unification of nonlinear filtering in the context of binary logical calculus,part I: Binary filters

The mathematical structure of nonlinear filtering is expressed in the context of binary logic. This first part of a two-part study concerns the binary setting. Operator properties, such as antiextensivity and idempotence, are expressed in finite logical expressions, as are the Matheron representation for morphological filters and its extension to translation-invariant operators, thereby giving simplicity to both operational properties and representations and also exposing the manner in which logic methods can be used for filter design and analysis. The second part of the study treats gray-scale filters.     

The Cartan matrix and enumerative calculus

Let G be a compact connected Lie group and P⊂G be the centralizer of a one-parameter subgroup in G. We explain a program that reduces integration along a Schubert variety in the flag manifold G/P to the Cartan matrix of G.As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in Zhao and Duan (J. Symbolic Comput. 33 (2002) 507).     

The lambda-context calculus (extended version)

We present the Lambda Context Calculus. This simple lambda-calculus features variables arranged in a hierarchy of strengths such that substitution of a strong variable does not avoid capture with respect to abstraction by a weaker variable. This allows the calculus to express both capture-avoiding and capturing substitution (instantiation). The reduction rules extend the ‘vanilla’ lambda-calculus in a simple and modular way and preserve the look and feel of a standard lambda-calculus with explicit substitutions.Good properties of the lambda-calculus are preserved. The LamCC is confluent, and a natural injection into the LamCC of the untyped lambda-calculus exists and preserves strong normalisation.We discuss the calculus and its design with full proofs. In the presence of the hierarchy of variables, functional binding splits into a functional abstraction λ (lambda) and a name-binder и (new). We investigate how the components of this calculus interact with each other and with the reduction rules, with examples. In two more extended case studies we demonstrate how global state can be expressed, and how contexts and contextual equivalence can be naturally internalised using function application.     

The Laplace transform in discrete fractional calculus

We develop properties of the Laplace Transform in a discrete, fractional calculus setting, giving a precise treatment to convergence along the way. The end goal is to apply the Laplace Transform Method to solve a fractional initial value problem.     

The Lambek calculus enriched with additional connectives

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.I would like to thank Professor Johan van Benthem for much encouragement and a number of helpful suggestions. Thanks are also due to Dirk Roorda, who commented on an earlier draft of the paper, and to my advisor Professor Stanley Peters.     

Bonding calculus

We present the bonding calculus, a calculus in which it is easy to handle covalent bonds between molecules. Our purpose is to use bonding calculus to model the dynamics of the interactions in biochemical systems. We provide an operational semantics by means of a transition system, and use a known software platform to both simulate the chemical reactions described naturally in bonding calculus and verify their specific properties.     

The situatedness of economic relations

This article seeks to challenge the position of neoclassical economic theory, and approaches to the social sciences based on a natural scientific view of rationality. A series of culturally diverse vignettes suggest a more practical approach.     

The configuration of inter-organizational relations

Based on recent literature on the impact of inter-organizational information systems (IOS) on industrial organization, this paper proposes a framework for research into the interdependencies between IOS and coordination strategy. The elements of the framework are: (1) market structure; (2) governance forms; (3) relationship dimensions and (4) enterprise resources. It is suggested that these four elements at the same time affect and are affected by coordination strategy which focuses on inter-organizational relations. The research proposition is that inter-organizational arrangements have to be interpreted as complex, multi-layer configurations of organizational parameters. The composition and flexible adaptation of these configurations – rather than the selection of one efficient governance form – and the related design of IOS are the major tasks of coordination strategy. The design parameters are related to the elements of the framework: (1) the position of the enterprise in inter-organizational relations, be it a network, groups of networks or markets; (2) combinations of governance forms with different partners; (3) combinations of governance forms within a given network of relations referring to the different layers in inter-organizational relations, namely the institutional, operational and technical layers and (4) combinations of (inter-) organizational arrangements for the use and development of resources. EUROSELECT, a horizontal network of European grocery wholesalers, is used as an example to illustrate how different contingencies affect the inter-organizational design and why the complex configuration of relations is chosen.     

The weakest precondition calculus: Recursion and duality

An extension of Dijkstra's guarded command language is studied, including unbounded demonic choice and a backtrack operator. We consider three orderings on this language: a refinement ordering defined by Back, a new deadlock ordering, and an approximation ordering of Nelson. The deadlock ordering is in between the two other orderings. All operators are monotonic in Nelson's ordering, but backtracking is not monotonic in Back's ordering and sequential composition is not monotonic for the deadlock ordering. At first sight recursion can only be added using Nelson's ordering. We show that, under certain circumstances, least fixed points for non-monotonic functions can be obtained by iteration from the least element. This permits the addition of recursion even using Back's ordering or the deadlock ordering in a fully compositional way. In order to give a semantic characterization of the three orderings that relates initial states to possible outcomes of the computation, the relation between predicate transformers and discrete power domains is studied. We consider (two versions of) the Smyth power domain and the Egli-Milner power domain.     

The weakest precondition calculus: Recursion and duality

An extension of Dijkstra's guarded command language is studied, including unbounded demonic choice and a backtrack operator. We consider three orderings on this language: a refinement ordering defined by Back, a new deadlock ordering, and an approximation ordering of Nelson. The deadlock ordering is in between the two other orderings. All operators are monotonic in Nelson's ordering, but backtracking is not monotonic in Back's ordering and sequential composition is not monotonic for the deadlock ordering. At first sight recursion can only be added using Nelson's ordering. We show that, under certain circumstances, least fixed points for non-monotonic functions can be obtained by iteration from the least element. This permits the addition of recursion even using Back's ordering or the deadlock ordering in a fully compositional way. In order to give a semantic characterization of the three orderings that relates initial states to possible outcomes of the computation, the relation between predicate transformers and discrete power domains is studied. We consider (two versions of) the Smyth power domain and the Egli-Milner power domain.