Temporal association algebra: a mathematical foundation forprocessing object-oriented temporal databases

This paper describes an object-oriented temporal association algebra (called TA-algebra) which is intended to serve as a formal foundation for supporting a pattern-based query specification and processing paradigm. Different from the traditional table-and-attribute-based paradigm, the pattern-based paradigm views the intension of an object-oriented temporal database as a network of object classes interconnected by different association types and its extension as a network of associated temporal object instances. Consistent with this view, queries can be specified in terms of patterns of temporal object associations or nonassociations (i.e., linear, tree and network structures of object classes/objects with logical AND and OR branches). TA-algebra provides a set of algebraic operators for processing these patterns and allows the direct and/or indirect associations and/or nonassociations among temporal object instances to be more explicitly represented and maintained during processing than the traditional tabular representation of temporary or final query results. TA-algebra operators are based on time-interval and valid-time semantics and they preserve the closure property. The algebra is capable of operating on heterogeneous as well as homogeneous patterns of object associations. Both homogeneous and heterogeneous patterns are decomposed into a set of primitive temporal pattern instances for uniform treatment. This paper formally defines the TA-algebra operators and their mathematical properties. The applications of these operators in query decomposition and processing are illustrated by examples     

Coquand's calculus of constructions: A mathematical foundation for a proof development system

The calculus of constructions of Coquand, which is a version of higher order typed-calculus based on the dependent function type, is considered from the perspective of its use as the mathematical foundation for a proof development system. The paper considers formulations of the calculus, the underlying consistency of the formalism (i.e., the strong normalisation theorem), and the proof theory of adding assumptions for notions from logic and set theory. Proofs are not given, but references to them are.A preliminary version of this paper was presented at the Third Banff Higher Order Workshop, 23–28 September 1989.     

Algorithms for asynchronous parallel processing of object-orienteddatabases

Management of large quantities of complex data is essential in many advanced application areas. Object-oriented (OO) database management system have been developed to effectively model and process the complex domain knowledge. They have been shown to outperform some existing relational systems. The existing implementations of OO database management systems attempt to improve the efficiency of OO queries by explicitly capturing the relationships among objects. However, the execution of complex queries involving the retrieval of objects from many classes and relationships among them causes the existing system to operate inefficiently. In this paper, we present parallel algorithms for the processing of queries against a large OO database. The algorithms are based on a closed model of query processing pattern-based access instead of the conventional value-based access. During processing, the algorithms avoid the execution of time-consuming join operations by making use of the explicitly stored object associations. Generation of large quantities of temporary data is avoided by marking objects using their identifiers and by employing a two-phase query processing strategy. A query is processed by concurrent multiple waves, thereby improving parallelism avoiding the complexities introduced in their sequential implementation. The correctness and the performance of the parallel algorithms have been tested and analyzed by running parallel programs on a 32-node transputer based parallel machine designed and developed at the IBM Research Center at Yorktown Heights, New York. Benchmark queries of different semantic complexities are generated, and their performance is analyzed for various data and query parameters     

Using computer algebra systems in mathematical classrooms

Abstract This paper describes a research whose main focus is the use of Computer Algebra Systems (CAS) in mathematical classrooms and the didactical possibilities linked with its use. The possibilities of integrating Self-Regulated Learning (SRL) within the CAS environment are brought into focus. Forty-three Israeli students (mean age 13.3) were assigned to two learning algebraic groups. The first group received explicit meta-cognitive SRL with CAS (CAS + SRL); the second group was exposed to CAS without SRL (CAS). Empirical results from the experimental and case study designs revealed that (CAS + SRL) students outperformed (CAS) students on algebraic thinking and that (CAS + SRL) students regulated their learning more effectively.     

Solving some problems of algebra,analysis, and mathematical physics using computer algebra systems

We consider some specific features of solution of problems of algebra, analysis, and mathematical physics using computer algebra systems. Some methods and programs are described, and the strengths and weaknesses of computer algebra systems are discussed.Translated from Kibernetika, No. 2, pp. 23–29, March–April, 1991.     

Robbins algebra: Conditions that make a near-boolean algebra boolean

Some problems posed years ago remain challenging today. In particular, the Robbins problem, which is still open and which is the focus of attention in this paper, offers interesting challenges for attack with the assistance of an automated reasoning program; for the study presented here, we used the program OTTER. For example, when one submits this problem, which asks for a proof that every Robbins algebra is a Boolean algebra, a large number of deduced clauses results. One must, therefore, consider the possibility that there exists a Robbins algebra that is not Boolean; such an algebra would have to be infinite. One can instead search for properties that, if adjoined to those of a Robbins algebra, guarantee that the algebra is Boolean. Here we present a number of such properties, and we show how an automated reasoning program was used to obtain the corresponding proofs. Additional properties have been identified, and we include here examples of using such a program to check that the corresponding hand-proofs are correct. We present the appropriate input for many of the examples and also include the resulting proofs in clause notation.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Equivalences for a biological process algebra

This paper investigates Bio-PEPA, the stochastic process algebra for biological modelling developed by Ciocchetta and Hillston. It focuses on Bio-PEPA with levels where molecular counts are grouped or concentrations are discretised into a finite number of levels. Basic properties of well-defined Bio-PEPA systems are established after which equivalences used for the stochastic process algebra PEPA are considered for Bio-PEPA, and are shown to be identical for well-defined Bio-PEPA systems. Two new semantic equivalences parameterised by functions, called g-bisimilarity and weak g-bisimilarity are introduced. Different functions lead to different equivalences for Bio-PEPA. Congruence is shown for both forms of g-bisimilarity under certain reasonable conditions on the function and the use of these equivalences are demonstrated with a biologically-motivated example where two similar species are treated as a single species, and modelling of alternative pathways in the MAPK kinase signalling cascade.     

FLASc: a formal algebra for labeled property graph schema

Automated Software Engineering - Mailing lists are a major communication channel for supporting developer coordination in open-source software projects. In a recent study, researchers explored...     

A prefix operator for a switching algebra

A new notation for a NAND operator is proposed, based on the prefix Polish notation following the style of Lukasiewicz. There is a direct one-to-one correspondence between the operators in the resulting system and gates in a circuit realization if identical subfunctions are recognized. An axiom set for the Boolean algebra based on this operator is given, and the axiomatic derivation of the algebra is demonstrated. A number of results concerning circuit manipulation, redundant sets of gate inputs, and necessary and sufficient conditions for the relocation of certain gate inputs are presented, and their use demonstrated by a number of examples.     

Geometric algebra: a computational framework for geometrical applications. 2

Every vector space with an inner product has a geometric algebra, whether or not you choose to use it. This article shows how to call on this structure to define common geometrical constructs, ensuring a consistent computational framework. The goal is to show you that this can be done and that it is compact, directly computational, and transcends the dimensionality of subspaces. We do not use geometric algebra to develop new algorithms for graphics, but hope to demonstrate that one can automatically take care of some of the lower level algorithmic aspects, without tricks, exceptions, or hidden degenerate cases by using geometric algebra as a language.     

The stream-based service-centred calculus: a foundation for service-oriented programming

We give a formal account of stream-based, service-centered calculus (SSCC), a calculus for modelling service-based systems, suitable to describe both service composition (orchestration) and the protocols that services follow when invoked (conversation). The calculus includes primitives for defining and invoking services, for isolating conversations (called sessions) among clients and servers, and for orchestrating services. The calculus is equipped with a reduction and a labelled transition semantics related by an equivalence result. SSCC provides a good trade-off between expressive power for modelling and simplicity for analysis. We assess the expressive power by modelling van der Aalst workflow patterns and an automotive case study from the European project Sensoria. For analysis, we present a simple type system ensuring compatibility of client and service protocols. We also study the behavioural theory of the calculus, highlighting some axioms that capture the behaviour of the different primitives. As a final application of the theory, we define and prove correct some program transformations. These allow to start modelling a system from a typical UML Sequence Diagram, and then transform the specification to match the service-oriented programming style, thus simplifying its implementation using web services technology.     

Coinduction for preordered algebra

We develop a combination, called hidden preordered algebra, between preordered algebra, which is an algebraic framework supporting specification and reasoning about transitions, and hidden algebra, which is the algebraic framework for behavioural specification. This combination arises naturally within the heterogeneous framework of the modern formal specification language CafeOBJ. The novel specification concept arising from this combination, and which constitutes its single unique feature, is that of behavioural transition. We extend the coinduction proof method for behavioural equivalence to coinduction for proving behavioural transitions.     

Model-driven development: a metamodeling foundation

Metamodeling is an essential foundation for MDD, but there's little consensus on the precise form it should take and role it should play. The authors analyze the underlying motivation for MDD and then derive a concrete set of requirements that a supporting infrastructure should satisfy. They discuss why the traditional "language definition" interpretation of metamodeling isn't a sufficient foundation and explain how it can be extended to unlock MDD's full potential.     

Cadabra: a field-theory motivated symbolic computer algebra system

Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions, this often leads to cumbersome input formats, unexpected side-effects, or the need for a lot of special-purpose code. This makes a direct translation of problems from paper to computer and back needlessly time-consuming and error-prone. A prototype computer algebra system is presented which features -like input, graph data structures, lists with Young-tableaux symmetries and a multiple-inheritance property system. The usefulness of this approach is illustrated with a number of explicit field-theory problems.