Understanding the Role of Learning in the Evolution of Busy Beavers: a Comparison Between the Baldwin Effect and a Lamarckian Strategy Francisco B. Pereira 1,2 , Ernesto Costa 2 1 Instituto Superior de Engenharia de Coimbra, Quinta da Nora, 3030 Coimbra, Portugal 2 Centro de Informática e Sistemas da Universidade de Coimbra, Polo II, 3030 Coimbra, Portugal {xico, ernesto}@dei.uc.pt phone: +351 239790000 Abstract In this paper we study how individual learning interacts with an evolutionary algorithm in its search for good solutions to the Busy Beaver problem. Two learning strategies, the Baldwin Effect and Lamarckian learning, are compared with an extensive set of experiments. Results show that the Baldwin Effect is less sensitive to specific issues concerning the definition of the learning model and it is more effective in adjusting its learning power to maximise the search performance of the evolutionary algorithm. Some insight about the specific role that evolution and learning play during search is also presented. 1 INTRODUCTION Evolution and learning are the two major forces that promote the adaptation of individuals to the environment. Evolution, operating at the population level, includes all mechanisms of genetic changes that occur in organisms over generations. Learning operates at a different time scale. It gives to each individual the ability to modify its phenotype during its life in order to increase its adaptation to the environment and, hence, its chance to be selected for reproduction. In standard evolutionary computation (EC) optimisation, learning has usually been implemented as local search algorithms. These methods iteratively test several alternatives in the neighbourhood of the learning individual trying to discover better solutions. At the end of the learning process, the quality of an individual will be, not only the measure of its initial fitness, but also of its ability to improve, which leads to a better understanding of the fitness landscape. In our research we are interested in studying how learning and evolution may be combined in computer simulations. In this paper we use the Busy Beaver (BB) problem as the testbed to study the above-mentioned interactions. In 1962, Tibor Rado proposed this problem in the context of the existence of non-computable functions [13]. It can be defined as follows: suppose a Turing Machine (TM) with a two-way infinite tape and a tape alphabet={blank, 1}. The question Rado asked was: what is the maximum number of 1’s that can be written by a N-state halting TM when started on a blank tape? This number, which is a function of the number of states, is denoted by ∑(N). A TM that produces ∑(N) non-blanks cells is called a Busy Beaver. The BB is considered one of the most interesting theoretical problems and, since its proposal, has attracted the attention of many researchers. Some values for ∑(N) and the corresponding TMs are known today for small values of N. As the number of states increases, the problem becomes harder and, for N≥5, there are several candidates that set lower bounds on the value of ∑ (N). To prove that a particular candidate is the N-state BB we must perform an exhaustive search over the space of all N-state TMs and verify that no other machine produces a higher number of ones. This is extremely complex due to the halting problem. In the original setting, the problem was defined for 5-tuple TMs. One of the main variants consists in considering 4-tuple TMs. In the next section we present a formal definition of the BB problem for both variants. The search space of the BB problem possesses several characteristics, such as its dimension and its complexity, that make it extremely appealing to the EC field. We performed some empirical analysis on the topology of the landscape and verified that, in different areas of the search space, there are small groups of neighbour valid solutions to the BB problem. The size of these groups and the quality of the TMs that compose them varies but, nevertheless, they tend to be surrounded by large low fitness areas composed by invalid solutions. The combination of these factors makes the space highly irregular and very prone to premature convergence. The first attempt to apply EC techniques to the BB problem was reported by Terry Jones [6], who used a genetic algorithm to search for specific instances of the 5-tuple BB. In 1999, our research group obtained a remarkable success in our first effort to apply EC algorithms to the 4-tuple variant of the problem [8]. Several new lower bounds were set, leading to a large increase in the productivity of 6 and 7-state 4-tuple TMs. Following our research interests, in a previous work [12] we studied the influence that two different learning models had in the performance of an evolutionary algorithm when seeking for solutions to the 4-tuple BB. 884 ARTIFICIAL LIFE, ADAPTIVE BEHAVIOR, AND AGENTS