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47
The Evolution of Size and Shape
, 1999
"... Introduction The rapid growth of programs produced by genetic programming (GP) is a well documented phenomenon [Koza, 1992; Blickle and Thiele, 1994; Nordin and Banzhaf, 1995; McPhee and Miller, 1995; Soule et al., 1996; Greeff and Aldrich, 1997; Soule, 1998] . This growth, often referred to as "co ..."
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Cited by 87 (40 self)
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Introduction The rapid growth of programs produced by genetic programming (GP) is a well documented phenomenon [Koza, 1992; Blickle and Thiele, 1994; Nordin and Banzhaf, 1995; McPhee and Miller, 1995; Soule et al., 1996; Greeff and Aldrich, 1997; Soule, 1998] . This growth, often referred to as "code bloat", need not be correlated with increases in the fitness of the evolving programs and consists primarily of code which does not change the semantics of the evolving program. The rate of growth appears to vary depending upon the particular genetic programming paradigm being used, but exponential rates of growth have been documented [Nordin and Banzhaf, 1995] . Code bloat occurs in both tree based and linear genomes [Nordin, 1997; Nordin and Banzhaf, 1995; Nordin et al., 1997] and with automatically defined functions [Langdon, 1995] . Recent research suggests that code bloat will occur in most fitness based search techniques which allow variable length solutions [Langdon, 1998b; Langdo
Size fair and homologous tree genetic programming crossovers. Genetic Programming And Evolvable Machines
"... Size fair and homologous crossover genetic operators for tree based genetic programming are described and tested. Both produce considerably reduced increases in program size and no detrimental e ect on GP performance. GP search spaces are partitioned by the ridge in the number of program v. their si ..."
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Cited by 78 (21 self)
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Size fair and homologous crossover genetic operators for tree based genetic programming are described and tested. Both produce considerably reduced increases in program size and no detrimental e ect on GP performance. GP search spaces are partitioned by the ridge in the number of program v. their size and depth. A ramped uniform random initialisation is described which straddles the ridge. With subtree crossover trees increase about one level per generation leading to subquadratic bloat in length. 1
General Schema Theory for Genetic Programming with SubtreeSwapping Crossover
 In Genetic Programming, Proceedings of EuroGP 2001, LNCS
, 2001
"... In this paper a new, general and exact schema theory for genetic programming is presented. The theory includes a microscopic schema theorem applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. A more macroscopic schema ..."
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Cited by 45 (28 self)
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In this paper a new, general and exact schema theory for genetic programming is presented. The theory includes a microscopic schema theorem applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. A more macroscopic schema theorem is also provided which is valid for crossover operators in which the probability of selecting any two crossover points in the parents depends only on their size and shape. The theory is based on the notions of Cartesian node reference systems and variablearity hyperschemata both introduced here for the first time. In the paper we provide examples which show how the theory can be specialised to specific crossover operators and how it can be used to derive an exact definition of effective fitness and a sizeevolution equation for GP. 1
An Overview of Evolutionary Algorithms: Practical Issues and Common Pitfalls
 Information and Software Technology
, 2001
"... An overview of evolutionary algorithms is presented covering genetic algorithms, evolution strategies, genetic programming and evolutionary programming. The schema theorem is reviewed and critiqued. Gray codes, bit representations and realvalued representations are discussed for parameter optimi ..."
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Cited by 34 (0 self)
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An overview of evolutionary algorithms is presented covering genetic algorithms, evolution strategies, genetic programming and evolutionary programming. The schema theorem is reviewed and critiqued. Gray codes, bit representations and realvalued representations are discussed for parameter optimization problems. Parallel Island models are also reviewed, and the evaluation of evolutionary algorithms is discussed.
Exact Schema Theory for Genetic Programming and Variablelength Genetic Algorithms with OnePoint Crossover
, 2001
"... A few schema theorems for Genetic Programming (GP) have been proposed in the literature in the last few years. Since they consider schema survival and disruption only, they can only provide a lower bound for the expected value of the number of instances of a given schema at the next generation rathe ..."
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Cited by 30 (16 self)
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A few schema theorems for Genetic Programming (GP) have been proposed in the literature in the last few years. Since they consider schema survival and disruption only, they can only provide a lower bound for the expected value of the number of instances of a given schema at the next generation rather than an exact value. This paper presents theoretical results for GP with onepoint crossover which overcome this problem. Firstly, we give an exact formulation for the expected number of instances of a schema at the next generation in terms of microscopic quantities. Thanks to this formulation we are then able to provide an improved version of an earlier GP schema theorem in which some (but not all) schema creation events are accounted for. Then, we extend this result to obtain an exact formulation in terms of macroscopic quantities which makes all the mechanisms of schema creation explicit. This theorem allows the exact formulation of the notion of effective fitness in GP and opens the way to future work on GP convergence, population sizing, operator biases, and bloat, to mention only some of the possibilities.
Exact schema theorem and effective fitness for GP with onepoint crossover
 Proceedings of the Genetic and Evolutionary Computation Conference, pages 469476, Las Vegas
, 2000
"... This paper extends recent results in the GP schema theory by formulating a proper exact schema theorem for GP with onepoint crossover. This gives an exact expression for the expected number of instances of a schema at the next generation in terms of macroscopic quantities. This result allows the ex ..."
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Cited by 28 (16 self)
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This paper extends recent results in the GP schema theory by formulating a proper exact schema theorem for GP with onepoint crossover. This gives an exact expression for the expected number of instances of a schema at the next generation in terms of macroscopic quantities. This result allows the exact formulation of the notion of effective fitness in GP. 1
Exact Schema Theorems for GP with OnePoint and Standard Crossover Operating on Linear Structures and their Application to the Study of the Evolution of Size
 IN GENETIC PROGRAMMING, PROCEEDINGS OF EUROGP 2001, LNCS
, 2001
"... In this paper, firstly we specialise the exact GP schema theorem for onepoint crossover to the case of linear structures of variable length, for example binary strings or programs with arity1 primitives only. Secondly, we extend this to an exact schema theorem for GP with standard crossover app ..."
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Cited by 26 (16 self)
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In this paper, firstly we specialise the exact GP schema theorem for onepoint crossover to the case of linear structures of variable length, for example binary strings or programs with arity1 primitives only. Secondly, we extend this to an exact schema theorem for GP with standard crossover applicable to the case of linear structures. Then we study, both mathematically and numerically, the schema equations and their fixed points for infinite populations for both a constant and a lengthrelated fitness function. This allows us to characterise the bias induced by standard crossover. This is very peculiar. In the case of a constant fitness function, at the fixedpoint, structures of any length are present with nonzero probability. However, shorter structures are sampled exponentially much more frequently than longer ones.
Hyperschema Theory for GP with OnePoint Crossover, Building Blocks, and Some New Results in GA Theory
 Genetic Programming, Proceedings of EuroGP 2000
, 2000
"... Two main weaknesses of GA and GP schema theorems axe that they provide only information on the expected value of the number of instances of a given schema at the next generation E[m(H,t + 1)], and they can only give a lower bound for such a quantity. This paper presents new theoretical results o ..."
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Cited by 23 (17 self)
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Two main weaknesses of GA and GP schema theorems axe that they provide only information on the expected value of the number of instances of a given schema at the next generation E[m(H,t + 1)], and they can only give a lower bound for such a quantity. This paper presents new theoretical results on GP and GA schemata which laxgely overcome these weaknesses. Firsfly, unlike previous results which concentrated on schema survival and disruption, our results extend to GP recent work on GA theory by Stephens and Waelbroeck, and make the effects and the mechanisms of schema creation explicit. This allows us to give an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. Thanks to this formulation we are then able to provide in improved version for an eaxlier GP schema theorem in which some schema creation events axe accounted for, thus obtaining a tighter bound for E[m(H, t + 1)]. This bound is a function of the selection probabilities of the schema itself and of a set of lowerorder schemata which onepoint crossover uses to build instances of the schema. This result supports the existence of building blocks in GP which, however, axe not necessaxily all short, loworder or highly fit. Building on eaxlier work, we show how Stephens and Waelbroeck 's GA results and the new GP results described in the paper can be used to evaluate schema vaxiance, signaltonoise ratio and, in general, the probability distribution of re(H, t + 1). In addition, we show how the expectation operator can be removed from the schema theorem so as to predict with a known probability whether re(H, t + 1) (rather than Elm(H, t + 1)]) is going to be above a given threshold.
Schema Theory Analysis of Mutation Size Biases in Genetic Programming With Linear Representations
, 2001
"... Understanding operator bias in evolutionary computation is important because it is possible for the operator's biases to work against the intended biases induced by the fitness function. In recent work we showed how developments in GP schema theory can be used to better understand the biases induced ..."
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Cited by 21 (16 self)
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Understanding operator bias in evolutionary computation is important because it is possible for the operator's biases to work against the intended biases induced by the fitness function. In recent work we showed how developments in GP schema theory can be used to better understand the biases induced by the standard subtree crossover when genetic programming is applied to variable length linear structures. In this paper we use the schema theory to better understand the biases induced on linear structures by two common GP subtree mutation operators: FULL and GROW mutation. In both cases we find that the operators do have quite specific biases and typically strongly oversample shorter strings.