Raymond T. Boute

Fuzzy versus quantitative association rules: a fair data-driven comparison

As opposed to quantitative association rule mining, fuzzy association rule mining is said to prevent the overestimation of boundary cases, as can be shown by small examples. Rule mining, however, becomes interesting in large databases, where the problem of boundary cases is less apparent and can be further suppressed by using sensible partitioning methods. A data-driven approach is used to investigate if there is a significant difference between quantitative and fuzzy association rules in large databases. The influence of the choice of a particular triangular norm in this respect is also examined.

The Euclidean definition of the functions div and mod

The definitions of the functions div and mod in the computer science literature and in programming languages are either similar to the Algol of Pascal definition (which is shown to be an unfortunate choice) or based on division by truncation (T-definition) or division by flooring as defined by Knuth (F-definition). The differences between various definitions that are in common usage are discussed, and an additional one is proposed, which is based on Euclids theorem and therefore is called the Euclidean definition (E-definition). Its distinguishing feature is that 0 ≤ D mod d < | d | irrespective of the signs of D and d. It is argued that the E- and F-definitions are superior to all other ones in regularity and useful mathematical properties and hence deserve serious consideration as the standard convention at the applications and language level. It is also shown that these definitions are the most suitable ones for describing number representation systems and the realization of arithmetic operations at the architecture and hardware level.

Systems semantics: principles, applications, and implementation

Systems semantics extends the denotational semantics of programming languages to a semantics for the description of arbitrary systems, including objects that are not computations in any sense. By defining different meaning functions, the same formal description may be used to denote different system properties, such as structure, behavior, component cost, and performance aspects (e.g., timing). The definition of these semantic functions also provides guidance in language design, in particular for the match between language constructs and the system concepts to be expressed. Aiming at compositionality ensures useful properties for formal manipulation. In this fashion, the meaning functions can be made sufficiently simple to serve not only as a direct implementation on a machine but also as rules for reasoning about systems in a transformational manner. As the applications show, however, compositionality can be ensured only through careful consideration of the characteristics of the flow of information inside the system. Two classes of application are discussed:Unidirectional systems, in particular digital systems without feedback (combinational) and with feedback (sequential), and a certain class of analog systems. Nonunidirectional systems, in particular two-port analog networks. The emphasis will be on the functional style of description and on formal reasoning (theorem proving, derivation of properties). Implementation and rapid prototyping strategies in various system description environments are also briefly discussed. These would permit the concepts of system semantics to be explored without the need for a complete implementation.

Functional declarative language design and predicate calculus: a practical approach

In programming language and software engineering, the main mathematical tool is de facto some form of predicate logic. Yet, as elsewhere in applied mathematics, it is used mostly far below its potential, due to its traditional formulation as just a topic in logic instead of a calculus for everyday practical use.The proposed alternative combines a language of utmost simplicity (four constructs only) that is devoid of the defects of common mathematical conventions, with a set of convenient calculation rules that is sufficiently comprehensive to make it practical for everyday use in most (if not all) domains of interest.Its main elements are a functional predicate calculus and concrete generic functionals. The first supports formal calculation with quantifiers with the same fluency as with derivatives and integrals in classical applied mathematics and engineering. The second achieves the same for calculating with functionals, including smooth transition between pointwise and point-free expression.The extensive collection of examples pertains mainly to software specification, language semantics and its mathematical basis, program calculation etc., but occasionally shows wider applicability throughout applied mathematics and engineering. Often it illustrates how formal reasoning guided by the shape of the expressions is an instrument for discovery and expanding intuition, or highlights design opportunities in declarative and (functional) programming languages.

Calculational semantics: Deriving programming theories from equations by functional predicate calculus

The objects of programming semantics, namely, programs and languages, are inherently formal, but the derivation of semantic theories is all too often informal, deprived of the benefits of formal calculation “guided by the shape of the formulas.” Therefore, the main goal of this article is to provide for the study of semantics an approach with the same convenience and power of discovery that calculus has given for many years to applied mathematics, physics, and engineering. The approach uses functional predicate calculus and concrete generic functionals; in fact, a small part suffices. Application to a semantic theory proceeds by describing program behavior in the simplest possible way, namely by program equations, and discovering the axioms of the theory as theorems by calculation. This is shown in outline for a few theories, and in detail for axiomatic semantics, fulfilling a second goal of this article. Indeed, a chafing problem with classical axiomatic semantics is that some axioms are unintuitive at first, and that justifications via denotational semantics are too elaborate to be satisfactory. Derivation provides more transparency. Calculation of formulas for ante- and postconditions is shown in general, and for the major language constructs in particular. A basic problem reported in the literature, whereby relations are inadequate for handling nondeterminacy and termination, is solved here through appropriately defined program equations. Several variants and an example in mathematical analysis are also presented. One conclusion is that formal calculation with quantifiers is one of the most important elements for unifying continuous and discrete mathematics in general, and traditional engineering with computing science, in particular.

Teaching formal methods

Formal Methods teaching at undergraduate level has been going on at Manchester for a good number of years. We have introduced various courses based on different approaches. We have experienced the usual problems. To combat these problems, our approaches and our course contents have evolved accordingly over the years. In this paper we briefly trace this evolution, and describe the latest course, on reasoning about simple imperative programs, for first-year students who are half-way through our introductory programming course.

Fundamentals of Hardware Description Languages and Declarative Languages

The description of natural and man-made objects, in particular hardware, involves two major aspects: behaviour and structure.

Integrating Formal Methods by Unifying Abstractions

Integrating formal methods enhances their power as an intellectual tool in modelling and design. This holds regardless of automation, but a fortiori if software tools are conceived in an integrated framework.

Concrete Generic Functionals

Generic design is not only pertinent to programs, but a fortiori to declarative formalisms as well, and the ensuing generalizations often benefit programming. In the considered framework, generality is obtained by (re)defining traditionally non-functional objects uniformly as functions. This results in intrinsic polymorphism, i.e., designing a collection of functionals without restriction on their arguments makes it intrinsically applicable to all objects captured by the uniformization.

Supertotal function definition in mathematics and software engineering

In engineering (including computing), mathematics and logic, expressions can arise that contain function applications where the argument is outside the functions domain. Such a situation need not represent a conceptual error, for instance, in conditional expressions, but it is traditionally considered a type error. Various solutions can be found in the literature based on the notion of partial function and/or a distinguished value undefined. However, these have rather pervasive effects, complicating function definition, sacrificing convenient algebraic laws of logical operators and/or Leibnizs rule, one of the most valuable assets in formal reasoning (especially in the calculational style). Other solutions have in common the realization that well-structured mathematical arguments are always guarded by conditions and that the value of A/spl rArr/B is not affected by domain violations in B in case-A. These solutions preserve Leibnizs rule and the standard meaning of the logical operators. In this second category, we propose the simplest possible solution, called supertotal function definition, and consisting of assigning the value false (or 0, depending on the preferred formalism) to any function application where the argument is outside the domain. This approach assumes the notion of function with which a domain is associated as a part of its specification. Ramifications regarding formal reasoning, use in software engineering (such as Parnass predicate calculus) and in mathematical formulation in general are discussed. The proposed solution justifies formal reasoning as usual, but with increased freedom in expressions regarding types of function arguments. Hence, it can be adopted in existing formalisms with very minor changes to the latter, As a bonus, this discussion includes a very simple new view on conditional expressions, yielding unusually powerful and convenient calculational properties. Finally, differences and advantages w.r.t. other approaches are pointed out.