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This work deals with a problem of synthesis of the artificial neural networks using the evolutional scanning method. The basic task to be solved is to create a symbolic regression algorithm on principles of analytic programming, which will be capable of performing a convenient neural network synthesis. The main motivation here is the computerization of such synthesis and discovering so far unknown solutions. INTRODUCTION A shape of the Artificial Neural Network (ANN) which successfully solves concrete problem depends on many factors. A designer who deals with ANN designing pays attention to complexion of problem in the first instance, for example a number of inputs and outputs. However some other aspects such as ANN topology or neurons transfer function are considered in the same time. Concurrently, final shape of ANN is heavily influenced by learning algorithm which is going to be implemented. (Bose and Liang 1996) ANN designing operates with list of rules and heuristics to ensure correct application of chosen learning algorithm for created ANN. This technique generates relatively small set of possible ANN. Nevertheless, if the rules of classic ANN theory are disobeyed and personal experience is replace by clever algorithm, scanned set which includes suitable ANN becomes infinite. This paper shows how symbolic regression can be used as tool for successful scanning of such huge sets in order to find and to learn appropriate ANN. The particular method of symbolic regression in this case Analytic Programming (Zelinka and Oplatkova 2003), an algorithm developed by our faculty on a longterm basis. This algorithm proves a potential to find among a set of functions F, the class of functions f* (uneducated neural network), which is feasible to be reduced-optimized by a suitable educational method to the specific function f (educated neural network) having the quality to solve the particular problem successfully. Such attempt enables to further enhance the set F by functions of a non-neuronal character and carry a research on so far into quite unexplored territory between two purely neural and purely mathematical frameworks in such way. The goal of this research is to build-up a symbolic regression algorithm, which will, using the evolutionary scanning method, successfully find (synthesize) suitable solutions f ∈ F of the particular problem P. EVOLUTIONARY SCANNING Clause: Let there be a set of all neural networks with forward running propagation ANNall = {ANN1, ANN2, ..., ANNi, ...} and a set of all functions Fall = {f1, f2, ..., fk, ...}. Then for each ANNi ∈ ANNall exists a function fk ∈ Fall, alternatively a set of functions Fk ⊂ Fall such, that holds ANNi ⇔ fk, alternatively ANNi ⇔ Fk. Kolmonogorov theorem (Bose and Liang 1996) further shows also validity of the inverse clause: For every continuous function fk ∈ Fall exists ANNi ∈ ANNall such, that holds fk ⇔ ANNi. Task: Design an algorithm, which will by the means of the symbolic regression methods, evolutionary scan a set Fall in order to find: a) fk ⇔ ANNi b) fk, whose at least some subfunctions {f1, f2, ...} ⇔ {ANNn, ANNm, ...} which solve the particular problem P with global error ET < ξ, where ξ is the user defined biased tolerance threshold. Proceedings 21st European Conference on Modelling and Simulation Ivan Zelinka, Zuzana Oplatkova, Alessandra Orsoni ©ECMS 2007 ISBN 978-0-9553018-2-7 / ISBN 978-0-9553018-3-4 (CD) Figures 1: Principle of the Evolutionary scanning ANALYTIC PROGRAMMING Algorithm referred to as analytic programming (AP) proved itself by providing the solution to a variety of problems of the symbolic regression as: synthesis of trigonometric functions (Zelinka 2005a), polynomial functions (Zelinka and Oplatkova 2003), functions of boolean parity (Zelinka and Oplatkova 2004), functions of boolean symmetry (Zelinka at al. 2004), solution of differential equations (Zelinka 2002b) and optimization of the artificial ant tour (Zelinka and Oplatkova 2006). Those accomplishments have led to the motivation to utilise an AP also in the ANN synthesis. AP has been built-up on the sets of functions, operators and so called terminals, which are mostly invariables or independent variables. These mathematical objects form a set, out of which the AP strives to synthesize a suitable solution – so called General Function Set (GFS). Members of GFS are then the principal functions gf. (That is, we can easily understand the operators and terminals as functions.) GFS = {gf1, ... gfi, ... gfn}. The basic principle of AP is based on the handling with Discrete Sets (Discrete Set Handling, DSH). DSH forms the interface between the Evolutionary Algorithm (Evolutionary Algorithm, EA) (Zelinka, 2004), a problem, which should be solved symbolically. Thanks to that, almost any EA can be applied in the AP. In such a way the EA in effect works as an engine actuating the AP. From the EA nature further results to be rational to define the Cost Function (CF), whose value is necessary to be minimized. Note: AP does not use as terminals particular, however, general constants K1 to Kn, whose particular value is determined by means of nonlinear interlacing as closely as possible near the assessment of the relevant CF. SYNTHESIS OF NEURAL NETWORKS To accomplish by the means of an AP the successfully synthesize of the ANN, it is necessary at first to appropriately choose the elements of GFS and correctly define the CF, applicable for the problem being solved. To fulfill both those steps is, however, necessary to comprehend the relationship of equivalence between the ANN and functions – namely an AP itself does not understand the ANN conception, it is capable to synthesise only functions, eventually algorithms. The relation between an ANN and mathematical functions The connection between an ANN and mathematical functions is often neglected in the professional literature, at the same time each ANN can be expressed as to its equivalent class of functions f*. The ANN synthesis process is thus a process of searching of a suitable f* in the set of class of functions F, which contains the classes accrued by combinations of (basic) base functions b of a set B. The constitution of elements of B then differs according to individual types of ANN. The process of ANN education can then be expressed as searching for convenient values of invariables, which differentiate particular functions f in f* from each other. The resulting function f is then equivalent to the searched for ANN. Note: Let's introduce an agreement – if generally speaking about the neural networks, which are already successfully educated and/or it is not important if they are or are not educated at the moment, it would be prudent to continue using the ANN shortcut. If however one wishes, wish to accent, that the matter, that it is so far uneducated network the short form ANN* can be utilized.

Synthesis Of Artificial Neural Networks By The Means Of Evolutionary Scanning – Preliminary Study