Exact Schema Theory for Genetic Programming and Variable-Length Genetic Algorithms with One-Point Crossover

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 one-point crossover which overcome this problem. First, we give an exact formulation for the expected number of instances of a schema at the next generation in terms of microscopic quantities. Due 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.

[1]  Christopher R. Stephens,et al.  Schemata Evolution and Building Blocks , 1999, Evolutionary Computation.

[2]  P.A. Whigham,et al.  A Schema Theorem for context-free grammars , 1995, Proceedings of 1995 IEEE International Conference on Evolutionary Computation.

[3]  J. Doob.  Stochastic processes , 1953 .

[4]  Riccardo Poli,et al.  A Schema Theory Analysis of the Evolution of Size in Genetic Programming with Linear Representations , 2001, EuroGP.

[5]  Dan Boneh,et al.  On genetic algorithms , 1995, COLT '95.

[6]  Christopher R. Stephens,et al.  Effective Fitness as an Alternative Paradigm for Evolutionary Computation I: General Formalism , 2000, Genetic Programming and Evolvable Machines.

[7]  Gnter Rudolph.  Modes of stochastic convergence , 1997 .

[8]  David B. Fogel,et al.  Schema processing under proportional selection in the presence of random effects , 1997, IEEE Trans. Evol. Comput..

[9]  Prügel-Bennett,et al.  Analysis of genetic algorithms using statistical mechanics. , 1994, Physical review letters.

[10]  David B. Fogel,et al.  The Schema Theorem and the Misallocation of Trials in the Presence of Stochastic Effects , 1998, Evolutionary Programming.

[11]  L. Altenberg.  EMERGENT PHENOMENA IN GENETIC PROGRAMMING , 1994 .

[12]  Günter Rudolph,et al.  Convergence analysis of canonical genetic algorithms , 1994, IEEE Trans. Neural Networks.

[13]  Kenneth A. De Jong,et al.  Using Markov Chains to Analyze GAFOs , 1994, FOGA.

[14]  Riccardo Poli,et al.  Schema theorems without expectations , 1999 .

[15]  John H. Holland,et al.  Building Blocks, Cohort Genetic Algorithms, and Hyperplane-Defined Functions , 2000, Evolutionary Computation.

[16]  Christopher R. Stephens,et al.  Schemata as Building Blocks: Does Size Matter? , 2000, FOGA.

[17]  Riccardo Poli,et al.  Why Ants are Hard , 1998 .

[18]  John R. Koza,et al.  Genetic programming - on the programming of computers by means of natural selection , 1993, Complex adaptive systems.

[19]  Michael D. Vose,et al.  Modeling genetic algorithms with Markov chains , 1992, Annals of Mathematics and Artificial Intelligence.

[20]  Peter Nordin,et al.  Complexity Compression and Evolution , 1995, ICGA.

[21]  Peter A. Whigham,et al.  Grammatical bias for evolutionary learning , 1996 .

[22]  W. Langdon,et al.  Smooth uniform crossover, sub-machine code GP and demes: a recipe for solving high-order Boolean parity problems , 1999 .

[23]  Michael D. Vose,et al.  The simple genetic algorithm - foundations and theory , 1999, Complex adaptive systems.

[24]  Riccardo Poli,et al.  Solving High-Order Boolean Parity Problems with Smooth Uniform Crossover, Sub-Machine Code GP and Demes , 2000, Genetic Programming and Evolvable Machines.

[25]  Rafael A. Perez,et al.  The schema theorem considered insufficient , 1994, Proceedings Sixth International Conference on Tools with Artificial Intelligence. TAI 94.

[26]  Riccardo Poli.  Recursive Conditional Schema Theorem, Convergence and Population Sizing in Genetic Algorithms , 2000, FOGA.

[27]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[28]  Riccardo Poli,et al.  A Review of Theoretical and Experimental Results on Schemata in Genetic Programming , 1998, EuroGP.

[29]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[30]  William B. Langdon,et al.  Size Fair and Homologous Tree Crossovers for Tree Genetic Programming , 2000, Genetic Programming and Evolvable Machines.

[31]  Peter Nordin,et al.  Genetic programming - An Introduction: On the Automatic Evolution of Computer Programs and Its Applications , 1998 .

[32]  W. M. Spears,et al.  Aggregating models of evolutionary algorithms , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[33]  Riccardo Poli,et al.  Schema Theory for Genetic Programming with One-Point Crossover and Point Mutation , 1997, Evolutionary Computation.

[34]  Riccardo Poli,et al.  General Schema Theory for Genetic Programming with Subtree-Swapping Crossover , 2001, EuroGP.

[35]  Riccardo Poli,et al.  Exact Schema Theorem and Effective Fitness for GP with One-Point Crossover , 2000, GECCO.

[36]  M. Degroot,et al.  Probability and Statistics , 2021, Examining an Operational Approach to Teaching Probability.

[37]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[38]  Riccardo Poli,et al.  A schema theory analysis of mutation size biases in genetic programming with linear representations , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[39]  Riccardo Poli,et al.  On the Search Properties of Different Crossover Operators in Genetic Programming , 2001 .

[40]  R. Poli,et al.  Exact GP schema theory for headless chicken crossover and subtree mutation , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).

[41]  B. W.,et al.  Size Fair and Homologous Tree Genetic Programming Crossovers , 1999 .

[42]  Riccardo Poli,et al.  Smooth Uniform Crossover with Smooth Point Mutation in Genetic Programming: A Preliminary Study , 1999, EuroGP.

[43]  Lee Altenberg,et al.  The Schema Theorem and Price's Theorem , 1994, FOGA.

[44]  ProgrammingJustinian P. RoscaComputer.  Analysis of Complexity Drift in Genetic , 1997 .

[45]  John J. Grefenstette,et al.  Deception Considered Harmful , 1992, FOGA.

[46]  Riccardo Poli,et al.  An Experimental Analysis of Schema Creation, Propagation and Disruption in Genetic Programming , 1997, ICGA.

[47]  W. Langdon,et al.  Analysis of Schema Variance and Short Term Extinction Likelihoods , 2001 .

[48]  Jonathan E. Rowe,et al.  Population Fixed-Points for Functions of Unitation , 1998, FOGA.

[49]  David B. Fogel,et al.  Evolution-ary Computation 1: Basic Algorithms and Operators , 2000 .

[50]  Christopher R. Stephens,et al.  Effective Degrees of Freedom in Genetic Algorithms and the Block Hypothesis , 1997, ICGA.

[51]  Riccardo Poli,et al.  Exact Schema Theorems for GP with One-Point and Standard Crossover Operating on Linear Structures and Their Application to the Study of the Evolution of Size , 2001, EuroGP.

[52]  Christopher R. Stephens,et al.  Effective Fitness as an Alternative Paradigm for Evolutionary Computation II: Examples and Applications , 2001, Genetic Programming and Evolvable Machines.

[53]  Una-May O'Reilly,et al.  The Troubling Aspects of a Building Block Hypothesis for Genetic Programming , 1994, FOGA.

[54]  José Carlos Príncipe,et al.  A Markov Chain Framework for the Simple Genetic Algorithm , 1993, Evolutionary Computation.

[55]  R. Poli.  Why the schema theorem is correct also in the presence of stochastic effects , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[56]  Darrell Whitley,et al.  A genetic algorithm tutorial , 1994, Statistics and Computing.

[57]  Riccardo Poli,et al.  Hyperschema Theory for GP with One-Point Crossover, Building Blocks, and Some New Results in GA Theory , 2000, EuroGP.

[58]  P. Nordin,et al.  Explicitly defined introns and destructive crossover in genetic programming , 1996 .

[59]  David E. Goldberg,et al.  Genetic Algorithms and Walsh Functions: Part II, Deception and Its Analysis , 1989, Complex Syst..

[60]  Wolfgang Banzhaf,et al.  Genetic Programming: An Introduction , 1997 .