Vesselin K. Vassilev

Principles in the Evolutionary Design of Digital Circuits—Part II

In a previous work it was argued that by studying evolved designs of gradually increasing scale, one might be able to discern new, efficient, and generalisable principles of design. These ideas are tested in the context of designing digital circuits, particularly arithmetic circuits. This process of discovery is seen as a principle extraction loop in which the evolved data is analysed both phenotypically and genotypically by processes of data mining and landscape analysis. The information extracted is then fed back into the evolutionary algorithm to enhance its search capabilities and hence increase the likelihood of identifying new principles which explain how to build systems which are too large to evolve.

Information Characteristics and the Structure of Landscapes

Various techniques for statistical analysis of the structure of fitness landscapes have been proposed. An important feature of these techniques is that they study the ruggedness of landscapes by measuring their correlation characteristics. This paper proposes a new information analysis of fitness landscapes. The underlying idea is to consider a fitness landscape as an ensemble of objects that are related to the fitness of neighboring points. Three information characteristics of the ensemble are defined and studied. They are termed: information content, partial information content, and information stability. The information characteristics of a range of landscapes with known correlation features are analyzed in an attempt to reveal the advantages of the information analysis. We show that the proposed analysis is an appropriate tool for investigating the structure of fitness landscapes.

The Advantages of Landscape Neutrality in Digital Circuit Evolution

The paper studies the role of neutrality in the fitness landscapes associated with the evolutionary design of digital circuits and particularly the three-bit binary multiplier. For the purpose of the study, digital circuits are evolved extrinsically on an array of logic cells. To evolve on an array of cells, a genotype-phenotype mapping has been devised by which neutrality can be embedded in the resulting fitness landscape. It is argued that landscape neutrality is beneficial for digital circuit evolution.

Towards the automatic design of more efficient digital circuits

This paper introduces a new methodology of evolving electronic circuits by which the process of evolutionary design is guaranteed to produce a functionally correct solution. The method employs a mapping to represent an electronic circuit on an array of logic cells that is further encoded within a genotype. The mapping is many-to-one and thus there are many genotypes that have equal fitness values. Genotypes with equal fitness values define subgraphs in the resulting fitness landscapes referred to as neutral networks. This is further used in the design of a neutral network that connects the conventional with other more efficient designs. To explore such a network a navigation strategy is defined by which the space of all functionally correct circuits can be explored. The paper shows that very efficient digital circuits can be obtained by evolving from the conventional designs. Results for several binary multiplier circuits such as the three and four-bit multipliers are reported. The evolved solution for the three-bit multiplier consists of 23 two-input logic gates that in terms of number of two-input gates used is 23.3% more efficient than the most efficient known conventional design. The logic operators required to implement this circuit are 14 ANDs, 9 XORs, and 2 inversions (NOT). The evolved four-bit multiplier consists of 57 two-input logic gates that is 10.9% more efficient (in terms of number of two-input gates used) than the most efficient known conventional design. The optimal size of the target circuits is also studied by measuring the length of the neutral walks from the obtained designs.

Scalability problems of digital circuit evolution evolvability and efficient designs

A major problem in the evolutionary design of combinational circuits is the problem of scale. This refers to the design of electronic circuits in which the number of gates required to implement the optimal circuit is too high to search the space of all designs in reasonable time, even by evolution. The reason is twofold: firstly, the size of the search space becomes enormous as the number of gates required to implement the circuit is increased, and secondly, the time required to calculate the fitness of a circuit grows as the size of the truth table of the circuit. This paper studies the evolutionary design of combinational circuits, particularly the three-bit multiplier circuit, in which the basic building blocks are small sub-circuits, modules inferred from other evolved designs. The structure of the resulting fitness landscapes is studied and it is shown that in general the principles of evolving digital circuits are scalable. Thus to evolve digital circuits using modules is faster, since the building blocks of the circuit are sub-circuits rather than two-input gates. This can also be a disadvantage, since the number of gates of the evolved designs grows as the size of the modules used.

Smoothness, ruggedness and neutrality of fitness landscapes: from theory to application

The theory of fitness landscapes has been developed to provide a suitable mathematical framework for studying the evolvability of a variety of complex systems. In evolutionary computation the notion of evolvability refers to the efficiency of evolutionary search. It has been shown that the structure of a fitness landscape affects the ability of evolutionary algorithms to search. Three characteristics specify the structure of landscapes. These are the landscape smoothness, ruggedness and neutrality. The interplay of these characteristics plays a vital role in evolutionary search. This has motivated the appearance of a variety of techniques for studying the structure of fitness landscapes. An important feature of these techniques is that they characterize the landscapes by their smoothness and ruggedness, ignoring the existence of neutrality. Perhaps, the reason for this is that the role of neutrality in evolutionary search is still poorly understood.In this chapter some recent results on the spectral properties of the algebraic structures of fitness landscapes are summarized to provide a basis for studying the landscape structure. This approach is further employed to introduce an information analysis that characterizes the structure of fitness landscapes in terms of their smoothness, ruggedness and neutrality. The findings are finally applied in a study of the fitness landscapes generated by evolving digital circuits using an idealized model of a field-programmable gate array. The landscapes of this engineering problem are quite different from many recently studied landscapes that tend to be defined over simplified combinatorial and optimization problems. The difference originates from the genotype representation that is a configuration defined over two completely different alphabets. This makes the study of the corresponding landscapes much more involved. It is shown that the circuit evolution landscapes are products of subspaces with different characteristics. They are landscapes with vast neutrality and sharply differentiated plateau.

Digital circuit evolution and fitness landscapes

We study the fitness landscapes generated by evolving digital circuits using an idealised model of a field-programmable gate array. It appears that the fitness landscapes of this engineering problem are quite different from many recently studied landscapes, often defined over simplified combinatorial and optimisation problems. The difference stems from the genotype representation which allows us to evolve the functionality and connectivity of an array of logic cells. Here, the genotypes are sequences which are defined over two completely different alphabets. We propose a model for studying the structure of these landscapes and measure correlation characteristics of the landscapes. It is furthermore shown that the evolutionary search can be improved when the results of the analysis are taken into account.

On the nature of two-bit multiplier landscapes

The two-bit multiplier is a simple electronic circuit, small enough to be evolvable, and practically useful for the implementation of many digital systems. In this paper we study the structure of the two-bit multiplier fitness landscapes generated by circuit evolution on an idealised model of a field-programmable gate array. The two-bit multiplier landscapes are challenging. The difficulty in studying these landscapes stems from the genotype representation which allows us to evolve the functionality and connectivity of an array of logic cells. Here, the genotypes are simply strings defined over two completely different alphabets. This makes the study of the corresponding landscapes much more involved. We outline a model for studying the two-bit multiplier landscapes and estimate the amplitudes derived from the Fourier transform of these landscape. We show that the two-bit multiplier landscapes can be characterised in terms of subspaces, determined by the interactions between the genotype partitions.

The Evolution of Computation in Co-evolving Demes of Non-uniform Cellular Automata for Global Synchronisation

We study the evolution of computation performed by non-uniform cellular automata in which global information processing appears at two different levels of self-organisation. In our model, the first level of self-organisation is characterised by interactions among cellular macrostructures or computational demes which compete for room in a finite grid of cells. This level is related to the formation, evolution and extinction of macrostructures, and it is designed in a completely local manner. The second level of self-organisation refers to the interactions among the cells within the demes. The model, derived from the cellular programming approach, allows global computation to occur as a result of many local interactions among computational demes of interacting cells. The study reveals some of the mechanisms by which co-evolving demes of non-uniform cellular automata perform non-trivial computation, such as the synchronisation tasks.

Co-evolving demes of non-uniform cellular automata for synchronisation

Emergent computation refers to systems in which global information processing appears as a result of the interactions among many components, each of which may be a system that exhibits an ability for emergent computation at a different level of self-organisation. In this paper we employ a modification of cellular programming to evolve cellular machines for synchronisation. This allows global computation to occur by many local interactions among computational demes of interacting cells. The computational machine, derived from the non-uniform cellular automata model, consists of a grid of cells which are co-evolved in isolated demes. We describe experiments which show that demes can be co-evolved to perform non-trivial computation. We also analyse the mechanisms of computation within the different synchronising demes. Our results not only show that the co-evolution of demes is possible, but that they can attain high computational performance through co-operative action.