Edward K. Blum

Approximation theory and feedforward networks

Abstract Approximation of real functions by feedforward networks of the usual kind is shown to be based on the fundamental principle of approximation by piecewise-constant functions. This principle underlies a simple construction given for three-layer networks and suggests possible difficulties in determining two-layer networks.

Algebraic specification of modules and their basic interconnections

Abstract An algebraic specification concept for modules in software engineering is introduced which includes, in addition to a parameter and body part, explicit import and export interfaces. This concept integrates the main ideas of parameterized specifications for abstract data types and the information-hiding concept required for modules in software engineering. The concept is carefully motivated and defined with formal syntax and semantics within the framework of algebraic specifications. The basic constructions for combining modules are composition, actualization, extension, and union of modules with shared submodules. In this paper, composition and union are studied in detail. Both constructions are shown to be compositional. This means that the semantics of a combined module can be expressed in terms of the semantics of the components. To show the practical significance, specifications for the modules of an airport-schedule system and corresponding Ada packages are presented as an example.

Original Contribution: Stability of fixed points and periodic orbits and bifurcations in analog neural networks

We consider some neural networks which have interesting oscillatory dynamics and analyze stability and bifurcation properties. The neurons are of the sigmoidal type (i.e., analog elements which have states in a real interval, X). The dynamics is discrete-time and synchronous. Thus, for an m-neuron network, it is given by iteration of a map F:X^m -> X^m. The study of such discrete-time continuum-state systems is motivated both by the difference equations which arise in numerical simulation of differentiable neural networks and independently by the possibility of constructing VLSI circuits with clocking techniques to implement such neural networks having prescribed fixed-points or periodic orbits.

Discrete-time versus continuous-time models of neural networks

Abstract In mathematical modeling, very often discrete-time (DT) models are taken from, or can be viewed as numerical discretizations of, certain continuous-time (CT) models. In this paper, a general criterion, the asymptotic consistency criterion, for these DT models to inherit the dynamical behavior of their CT counterparts is derived. Detailed instances of this criterion are established for several classes of neural networks.

Algebraic Theory of Module Specification with Constraints

The concept for modules in software engineering based on equational algebraic specifications is extended by a suitable notion of constraints. This allows to have loose specifications with constraints for parameter, export and import interfaces of module specifications without loosing executability of the body specification. Correctness of such module specifications ensures that data types satisfying the import constraints are transformed into ones satisfying the export constraints. Operations on module specifications like composition, actualization and union are extended to the case with constraints. They are shown to preserve correctness and to be compositional w.r.t. the semantics. Moreover these operations on module specifications satisfy algebraic laws comparable to those of R-modules in algebra.

The Semantics of Shared Submodules Specifications

After reviewing the concept of module specification with import and export interfaces introduced by H. Ehrig for the modular development of software systems, precise definitions of submodule and union of modules specifications are given along with some basic results on their compatibility and semantics. The notion of amalgamated sum is used for the semantics of unions of modules and some connections are made with parametrized specifications. The results are restricted to the basic algebraic case.

SEMANOL (73) a metalanguage for programming the semantics of programming languages

SummarySEMANOL is a practical programming system for writing readable formal specifications of the syntax and semantics of programming languages. SEMANOL is based on a theory of semantics which embraces algorithmic (operational) and extensional (input/output) semantics. Specifications for large contemporary languages have been constructed in the formal language, SEMANOL (73), which is a readable high-level notation. A SEMANOL (73) specification can be executed (by an existing interpreter program); when given a program from the specified language, and its input, the execution of the SEMANOL (73) specification produces the programs output. The demonstrated executability of SEMANOL (73) provides important practical advantages. This paper includes discussions of the theory of semantics underlying SEMANOL, the syntax and semantics of the SEMANOL (73) language, the use of the SEMANOL (73) language in the SEMANOL method for describing programming languages, and the contrast between the Vienna definition method (VDL) and SEMANOL.

Consistency of local dynamics and bifurcation of continuous-time dynamical systems and their numerical discretizations

Numerical integration methods for solving differential equations naturally give rise to difference equations which in many cases can be subsequently converted into iterative maps. In this paper we study some consistency problems of local dynamics and bifurcation between the continuous-time (CT) dynamical systems defined by the differential equations and the discrete-time (DT) dynamical systems resulting from numerical methods of solving the differential equations. We first formulate the concepts of dynamical and bifurcational consistencies, and then present qualitative and quantitative results on the discretization step size and bifurcation parameter for general one-step methods of order p and specific methods like the Euler, backward Euler, explicit and implicit Runge-Kutta methods, so that the local dynamics and low-dimensional bifurcations (e.g., the saddle-node and Hopf bifurcations) of the CT systems are inherited exactly by the DT systems.

A Difference in Expressive Power Between Flowcharts and Recursion Schemes

We show the existence of a single interpretation for which no flowchart produces the same results as a particular recursion scheme.

Relative complexity of algebras

A simple algebraic model is proposed for measuring the relative complexity of programming systems. The appropriateness of this model is illustrated by its use as a framework for the statement and proof of results dealing with coding-independent limitations on the relative complexity of basic algebras.