Abstract— This paper analyses the performance of a genetic algorithm using a new concept, namely a fractional-order dynamic fitness function, for the synthesis of combinational logic circuits. The experiments reveal superior results in terms of speed and convergence to achieve a solution. Keywords— Circuit Design, Fractional-Order Systems, Genetic Algorithms, Logic Circuits. I. INTRODUCTION n the last decade genetic algorithms (GAs) have been applied in the design of electronic circuits, leading to a novel area of research called Evolutionary Electronics (EE) or Evolvable Hardware (EH) [1]. EE considers the concept for automatic design of electronic systems, employing search algorithms to develop good designs. One decade ago Sushil and Rawlins [2] applied GAs to the combinational circuit design problem while John Koza [3] adopted genetic programming. In the sequence of this work, Coello, Christiansen and Aguirre [4] presented a computer program that automatically generates high-quality circuit designs. Miller, Thompson and Fogarty [5] applied evolutionary algorithms for the design of arithmetic circuits. Kalganova, Miller and Lipnitskaya [6] proposed a new technique for designing multiple-valued circuits. In order to solve complex systems, Torresen [7] proposed the method of increased complexity evolution. The idea is to evolve a system gradually as a kind of divide-and-conquer method. The evolved functions are the basic blocks adopted in further evolution of a larger and more complex system. A major bottleneck in the evolutionary design of electronic circuits is the problem of scale. This refers to the very fast growth of the number of gates, used in the target circuit, as the number of inputs of the evolved logic function increases. This results in a huge search space that is difficult to explore even with evolutionary techniques. Another related obstacle is the time required to calculate the fitness value of a circuit [8]. A Manuscript received November 16, 2004. Cecília Reis is with the Electrical Engineering Department, Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal (e-mail: cecilia@dee.isep.ipp.pt). J. A. Tenreiro Machado is with the Electrical Engineering Department, Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal (e- mail: jtm@dee.isep.ipp.pt). J. Boaventura Cunha is with the Engineering Department, University of Trás-os-Montes and Alto Douro, Vila Real, Portugal (e-mail: jboavent@utad.pt). possible method to solve this problem is to use building blocks either than simple gates. The idea of using memory to achieve better fitness function performances was first introduced by Sano and Kita [9]. Their goal was the optimization of systems with randomly fluctuating fitness function and they developed a Genetic Algorithm with Memory-based Fitness Evaluation (MFEGA). The key ideas of the MFEGA are based on storing the sampled fitness values into memory as a search history, introducing a simple stochastic model of fitness values to be able to estimate fitness values of points of interest using the history for selection operation of the GA. Following this line of research, and looking for better performance GAs, this paper proposes a GA for the design of combinational logic circuits using fractional-order dynamic fitness functions. The area of Fractional Calculus (FC) deals with the operators of integration and differentiation to an arbitrary (including noninteger) order and is as old as the theory of classical differential calculus [11-12]. The theory of FC is a well-adapted tool to the modeling of many physical phenomena, allowing the description to take into account some peculiarities that classical integer-order models simply neglect. Nevertheless, the application of FC has been scarce until recently, but the advances on the theory of chaos motivated a renewed interest in this field. Bearing these ideas in mind this article is organized as follows. Section 2 describes the adopted GA as well as the fractional-order dynamic fitness functions. Section 3 presents the simulation results and finally, section 4 outlines the main conclusions and addresses perspectives towards future developments. II. THE ADOPTED GENETIC ALGORITHM In this section we present the GA developed in the study, in terms of the circuit encoding as a chromosome, the genetic operators and the static and dynamic fitness functions. A. Problem Definition To design combinational logic circuits a GA strategy is adopted. The circuits are specified by a truth table and the goal is to implement a functional circuit with the least possible complexity. Two sets of logic gates have been defined, as shown in Table 1, being Gset a the simplest one (i.e., a RISC- like set) and Gset b a more complex gate set (i.e., a CISC-like set). For each gate set the GA searches the solution space, based on a simulated evolution aiming the survival of the fittest strategy. In general, the best individuals of any population Synthesis of Logic Circuits Using Fractional- Order Dynamic Fitness Functions Cecília Reis, J. A. Tenreiro Machado and J. Boaventura Cunha I World Academy of Science, Engineering and Technology 1 2005 55