Graph Based Crossover – A Case Study with the Busy Beaver Problem Francisco B. Pereira ISEC Qta. da Nora, 3030 Coimbra, Portugal xico@dei.uc.pt Penousal Machado ISEC machado@dei.uc.pt Ernesto Costa CISUC Dep. de Eng. Informática, Univ. of Coimbra, Polo II, 3030 Coimbra, Portugal ernesto@dei.uc.pt Amílcar Cardoso CISUC amilcar@dei.uc.pt Abstract The success of the application of evolutionary approaches depends, to a large extent, on problem representation and on the used genetic operators. In this paper we introduce a new graph based crossover operator and compare it with classical two-point crossover. The study was carried out using a theoretical hard problem known as Busy Beaver. This problem involves the search for the Turing Machine that produces the maximum number of ones when started on a blank tape. Experimental results show that, in this domain, the new graph-based operator provides a clear advantage over two-point crossover. 1 INTRODUCTION Genetic operators play a particular important role in Evolutionary Computation. The task of recombination operators is to promote the exchange of genetic material between individuals. Typically, the operators used depend on the problem being solved and on the chosen representation. When using linear representations the most common operators are single-point, two-point and uniform crossover [Mitchell, 1996]. If a structured representation (e.g. trees) is adopted, it is advantageous to define recombination operators suitable for this structure [Angeline, 1994]. An example of such an operator is the standard GP crossover that exchanges sub-trees between individuals [Koza, 1992]. For some problems the natural representation is a graph. The main goal of this paper is to introduce a graph-based crossover operator and study its application to this type of problems. The application domain is the Busy Beaver (BB) problem [Rado, 1962] which involves the representation of Turing Machines (TMs). Tibor Rado introduced BB in the context of the existence of non- computable functions. It can be described as follows: Suppose a TM with a two way infinite tape and tape alphabet = {blank, 1}. The question Rado asked was: What is the maximum number of 1s that can be written by an N-State halting TM when started on a blank tape? This number, which is function of the number of states, is denoted by ∑(N). A machine that produces ∑(N) non-blank cells is called a Busy Beaver. ∑(N) happens to be non-computable. Some values for ∑(N), and the corresponding TM’s are known today for small values of N (e.g. ∑(1)=1, ∑(2)=4, ∑(3)=6, ∑(4)=13). As the number of states increases the problem becomes harder, and, for N>=5, there are candidates (or contenders) which set lower bounds on the value of ∑(N). This is partially due to the fact that there is neither a general, nor a particular theory about the structure of a BB. To prove that a machine is the N-state BB, we must perform an exhaustive search over the space of all N-state TMs and prove that no other machine produces a higher number of ones. This becomes extremely complex due to the halting problem. The BB is one of the most interesting theoretical problems. It has attracted the attention of many researchers and several contests were organized trying to produce the best candidates. The used techniques perform a partial search on the solution space, looking for TMs that produce the best lower bound for the value of ∑(N). Some of the best contenders were obtained by [Marxen and Buntrock, 1990] (e.g., he established that ∑(5) >= 4098). His approach involves enumeration and simulation of all N-state TMs, using several techniques to reduce the number of inspected machines, accelerate simulation and decide non-termination. In the original setting, the problem was defined for 5-tuple TMs. This type of machines, given a current state and symbol, write a new symbol, enter a new state and move the read/write head left or right. One of the main variants consists in considering 4-tuple TMs. These machines, during the transition to a new state, either write a new symbol to the tape or move the head (the actions are not allowed simultaneously). In [Pereira et al, 1999] we used an evolutionary approach to the BB problem, and set new best lower bounds for