Universality of a concept here means wide conceptual and practical usefulness in mathematics and applications. The function concept owes its universality to simplicity, generality, and powerful algebraic properties. Advantages proven in the sciences at large significantly benefit computing science as well. Universality critically depends on the definitional choices. The first half of this article shows that a “function” in the sense prevalent throughout the sciences, namely, as fully specified by its domain and its values , entails the characteristics that most contribute to universality. This link is clarified by some less well-understood aspects, including the role of function types as partial specifications, the ramifications of having composition defined for any pair of functions, and unification by capturing various notions not commonly seen as functions. Simple but representative examples are given in diverse areas, mostly computing. When a codomain appears at all in basic textbooks, it mostly involves a self-contradicting definition, corrected by the labeled function variant. Either way, it severely reduces universality, especially for composition. Yet, the axiomatization of category theory common in theoretical computing science offers no other choice. The second half explores how waiving one axiom generalizes category theory to include a wider variety of concepts, primarily the conventional function variant. It is also shown how this can be done unobtrusively for typical categorical notions, such as products, coproducts, functors, natural transformations, adjunctions, Galois connections, and auxiliary concepts, illustrated by example definitions and technical comments. Allowing the familiar function variant renders category theory more appealing to a wider group of scientists. A lesson for mathematics in general is Rogaway’s maxim: “Your definitional choices should be justified!”.

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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.

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.

In programming language and software engineering, the main mathematical tool isde factosome 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.

In engineering (including computing), mathematics and logic, expressions can arise that contain function applications where the argument is outside the function's 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 Leibniz's 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 Leibniz's 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 Parnas's 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.

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 Euclid's 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.

The input/output transformation effected by digital systems can be considered as concrete realizations of abstract mathematical functions. The mappings between abstract functions and concrete realizations, if kept explicit throughout the formulation, constitute the necessary 'handles' (embodied by function definitions) for transformational reasoning about digital systems. Deductive reasoning can be factored out and reduced considerably. This is demonstrated by a functional recast of the major parts of digital systems theory. Since the emphasis of this study is on the method (transformational reasoning) rather than on new system concepts, examples are chosen from familiar areas. However, some new results are obtained.>

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.

On the basis of a general sampling formula it is shown that, for a system ofnband-limited signals, FDM and band-limited TDM are equivalent in the sense that one can be derived from the other by a simple linear transformation. The transformation matrix is the same as for expanding an arbitrary polyphase system withnphases into its symmetric components. It is further shown that this equivalence also holds for nonideal sampling waveforms as they appear in certain multiplexing and demultiplexing systems used in practice.

A new concept, called distinguishing set or D-set is presented. Its use yields a considerable reduction in the length of checking sequences. Arbitrarily chosen examples have indicated a reduction of 30-50 percent. It is shown that the sequences of a distinguishing set actually constitute the optimum, i.e., minimum length, for state identification through input-output observations only.

An algorithmic procedure for designing optimal and near-optimal checking sequences for output faults is presented. For the specific cases where minimum length cannot be guaranteed, the algorithm also determines an upper bound on the excess length of the resulting sequence. Several extensions of the method are discussed, such as the application of output checking sequences for diagnosing purposes. The possibilities of this approach in the search for algorithms that yield optimal checking sequences for more general classes of faults are illustrated by applying the method in an ad hoc fashion and obtaining a complete checldng experiment.

In this paper the problem of numerical encoding of pictures consisting of regions of differing contrast is discussed. For this purpose we define the containment code, a special case of the more general property code presented in [1]. The containment code leads in a natural way to procedures for following the boundaries of the regions of the picture. This permits a compression of data into a linear array that can be stored for later processing or for picture reconstruction. The boundary following algorithms presented lend themselves well to hardware implementation as can be seen from an explicit sequential machine description. Related codes such as the boundary index code and the directional code are obtained from the containment code. Procedures are described for extracting various topological features such as curve length, chord length, area, and moments.