INVARIANT SUBSETS OF THE SEARCH SPACE AND THE UNIVERSALITY OF A GENERALIZED GENETIC ALGORITHM

INVARIANT SUBSETS OF THE SEARCH SPACE AND THE UNIVERSALITY OF A GENERALIZED GENETIC ALGORITHM BORIS MITAVSKIY Abstract In this paper we shall give a mathematical description of a general evolutionary heuristic search algorithm which allows to see a very special property which slightly generalized binary genetic algorithms have comparing to other evolutionary computation techniques It turns out that such a generalized genetic algorithm, which we call a binary semi-genetic algorithm, is capable of encoding virtually any other reasonable evolutionary heuristic search technique Introduction In this paper we shall describe a mathematical framework which allows to see some special properties which binary genetic algorithms have comparing to other evolutionary computation techniques It turns out that a slightly generalized version of a binary genetic algorithm can encode virtually any reasonable heuristic search algorithm (see corollary 5 and corollary 53 This seems interesting at least from a philosophical point of view, for it says something about the special properties of the reproduction mechanisms occurring in nature Moreover, it has been pointed out in [7] that such generalizations may actually be useful for practical purposes In fact, the binary embedding theorem and its corollaries (see theorem 5 and corollary 5 provide both, sufficient and necessary conditions when a given evolutionary algorithm can be embedded into (encoded by a binary semi-genetic algorithm These conditions depend only on the nature of the family of the reproduction transformations, and are completelndependent of any particular structure on the search space Theorem 5 classifies all such encodings in terms of the invariant subsets of the search space This may be useful for practical purposes, to simulate a given evolutionary heuristic search algorithm on a computer By an evolutionary heuristic search algorithm we mean a heuristic search technique used to solve optimization problems which mimics the basic natural evolution cycle: the natural selection, (or the survival of the fittest reproduction, and mutation The precise mechanism is outlined in the following sections Notation Ω is a finite set, called a search space f : Ω (0, is a function, called a fitness function The goal is to find a maximum of the function f F q is a collection of q-ary operations on Ω Intuitively F q can be thought of as the collection of reproduction operators: some q parents produce one offspring In nature q =, for every child has two parents, but in the artificial setting there

BORIS MITAVSKIY seems to be no special reason to assume that every child has no more than two parents M is a collection of unary operations on Ω Intuitively these are asexual reproduction, or mutation operators 3 how does a heuristic search algorithm work? A population P = x x x m Evaluation: Individuals of P are evaluated: with x i Ω is selected randomly x x x m f(x f(x f(x m Selection: A new population y P y = f(x is obtained where = x with probability Σ m l= f(x l In other words, all of the individuals of P are these of P, and the expectation of the number of occurrences of anndividual of P in P is proportional to the number of occurrences of that individual in P times the individual s fitness value In particular, the fitter the individual is, the more copies of that individual are likely to be present in P On the other hand, the individuals having relatively small fitness value are not likely to enter into P at all This is designed to imitate the natural survival of the fittest principle Partition: The individuals of P are partitioned into pairwise disoint tuples for mating according to some probabilistic rule: For instance the tuples could be Reproduction: y m Q = Q = Q = q q q

INVARIANT SUBSETS OF THE SEARCH SPACE AND THE UNIVERSALITY OF A GENERALIZED GENETIC ALGORITHM3 Replace every one of the selected q -tuples Q = Q = T ( T ( T q (,,, q,, q,, q,, q with the q -tuples for some randomly selected q -tuples of transformations (T, T,, T q (F q q This gives us a new population z P z = Mutation: Finally, with small probability we replace z i with F (z i for some randomly chosen F M This, once again, gives us a new population P w = Upon completion of mutation start all over with the initial population P The cycle is repeated a certain number of times depending on the problem z m w w m 4 a couple special heuristic search algorithms: The search space of every one of the following heuristic search algorithms is S = {0, } n Binary Genetic Algorithm: For every subset M {,,, n}, let L M (a, b = (x, x,, x i,, x n if i M where a = (a, a,, a n and b = (b,, b n S and x i = b i otherwise Let F M = {L M M {,,, n}} play the role of F from the previous section Example: With n = 5 and M = {, 3, 4}, M = {, 3, 5} we have ( 0 0 0 0 ( LM ((, 0, 0,,, (,, 0, 0, L M ((, 0, 0,,, (,, 0, 0, { a i = ( 0 0 0 0 The genetic crossover transformations are classified by the following property: If both parents have a in the i th position then the offspring also has a in the i th position Likewise, if both parents have a 0 in the i th position then the offspring

4 BORIS MITAVSKIY also has a 0 in the i th position If, on the other hand, the alleles of the i th gene don t coincide, then the i th allele could be either a 0 or a It turns out, that if we relax the condition on the preservation of genes, so to speak, by half, meaning that If both parents have a in the i th position then the offspring also has a in the i th position, but, in any other case, there is no requirement on the i th gene: it could be either a 0 or a, then one obtains a very special evolutionary heuristic search algorithm described below In section 5 we shall see that such an evolutionary algorithm is virtually universal, since it describes virtually any other reasonable heuristic search algorithm (see theorem 5, corollary 5 and corollary 53 Binary Semi-Genetic Algorithm: Definition 4 Fix m and u = (u, u,, u n S Define a semi-crossover transformation F u m : S m S as follows: For any given matrix a a a n a a a n P = a m a m a mn in S m F u m (P = x = (x, x, x n S where { a i if k m a i = a ki = x i = u i otherwise In other words, F u m preserves the i th gene if it is equal to in all of the rows of P, and replaces it with u i otherwise Denote by F m = {F u m u S} the family of all semi-crossover transformations Example: With n = 5 and u = (0,,, 0,, u = (0,, 0, 0, we have ( ( ( 0 0 Fu ((, 0, 0,,, (,, 0, 0, 0 = 0 0 F u ((, 0, 0,,, (,, 0, 0, 0 0 Notice, that if is present in the i th position of both parents, then it remains in the i th position of both offsprings There are absolutely no other restrictions, though 5 the binary embedding theorem Question: Under which conditions can a given heuristic search algorithm be encoded by a binary semi-genetic or, better yet, by a binary genetic algorithm? The main idea behind answering the question above is to observe that the families of invariant subsets naturally determine the corresponding families of transformations fixing them The rigorous machiners fully developed in the appendix of [5], and is also available upon request from the author Let Γ denote a family of transformations from Ω m into Ω Let Λ Γ = {S S Ω, T (S m S T Γ } denote the family of all invariant subsets under the action of Γ

INVARIANT SUBSETS OF THE SEARCH SPACE AND THE UNIVERSALITY OF A GENERALIZED GENETIC ALGORITHM5 Under certain slightly technical conditions on the family of transformations Γ (these conditions are satisfied by both, the family of all crossover transformations and the family of all semi-crossover transformations All of the rigorous details can be found in [5], and are also available upon request from the author the family of transformations Γ = {T x Ω m a transformation T x Γ such that T (x = T x (x} is the largest family of transformations such that Λb Γ = Λ Γ As we have seen in the section 3, a given evolutionary heuristic search algorithm is entirely determined by the families of its reproduction transformations This motivates the following definition: Definition 5 A heuristic k-tuple Ω = (Ω, Γ, Γ, Γ k is a k + -tuple where Ω denotes an arbitrary set while Γ i is ust a family of transformations from Ω m i into Ω and m < m < m i < < m k We say that the k-tuple of integers (m, m,, m k is the arity of the heuristic k-tuple (Ω, Γ, Γ, Γ k We also say that the collection Λ Ω = i k Λ Γ i is the collection of Ω-invariant subsets For x Ω, denote by Sx Ω the smallest element of Λ Ω containing x (Notice that Λ Ω is closed under arbitrarntersections so that Sx Ω = K Λ Γi, x K K In section 4 we have described the binary semi-genetic algorithm by the following heuristic k-tuple: Definition 5 Let S = {0, } We shall say that (S, F m, F m,, F mk is a semi-genetic heuristic k-tuple of dimension n, where m < m < < m k The following definition provides the means for the comparison of the various heuristic k-tuples An encoding of Ω by Φ is simply a mapping δ : Ω Φ w Ω δ(w is ust the code of w in Φ If the mapping δ : Ω Φ is one-to-one, then one can completely recover any w Ω from its code δ(w In other words, Ω is completeldentified with the subset δ(ω Φ If Ω = (Ω, Γ, Γ, Γ k and Φ = (Φ, Θ, Θ,, Γ k are two heuristic k-tuples of the same arity, a natural way to compare Ω with Φ is to construct an encoding mapping δ : Ω Φ which respects the reproduction transformations This motivates the following definition: Definition 53 Given two heuristic k-tuples Ω = (Ω, Γ, Γ,, Γ k and Φ = (Φ, Θ, Θ,, Θ k of the same arity, ( see definition 5 a morphism δ : Ω Φ is ust a function δ : Ω Φ which respects the reproduction transformations, meaning that i k and T Γ i F Θ i such that w, w,, w mi Ω we have F (δ(w, δ(w,, δ(w mi = δ(t (w, w,, w mi We say that a morphism δ : Ω Φ is an embedding if the underlying function δ : Ω Φ is one-to-one The binary embedding theorem establishes an explicit one-to-one correspondence between the set of all embeddings of a given heuristic k-tuple into binary semi-genetic algorithms and a certain collection of ordered n-tuples of Ω-invariant subsets Definition 54 Fix any heuristic k-tuple Ω = (Ω, Γ, Γ,, Γ k We say that collection Υ n = {I I = (I, I,, I n I Λ Ω, x, y Ω with x y n

6 BORIS MITAVSKIY such that either (x I and y / I or vise versa: (y I and x / I } is a family of separating n-tuples Theorem 5 Fix a heuristic k-tuple Ω = (Ω, Γ, Γ,, Γ k We now have the following biection φ : Υ n Ϝ n Ω which is defined explicitly as follows: Given an ordered n-tuple of sets from Λ Ω, call it I = (I, I,, I n Υ n, (see definition{ 54 let φ(i = δ I where δ I (x = (x, x,, x n S = {0, } n with if x I x = x Ω 0 otherwise Proof Due to space limitation, a detailed argument is available upon request from the author It turns out that the conditions under which a given heuristic k-tuple can be embedded into a binary semi-genetic heuristic k-tuple are rather mild and naturally occurring as the following two corollaries demonstrate: Corollary 5 Given a heuristic k-tuple Ω = (Ω, Γ, Γ,, Γ k, Ω, the following are equivalent: ( Ω can be embedded into an n-dimensional semi-genetic heuristic k-tuple for some n ( x, y Ω with x y we have either x / Sy Ω (see definition?? or vise versa: y / Sx Ω (3 x, y Ω with x y we have Sx Ω Sy Ω (Another way to say this, is that the map sending x to Sx Ω is one-to-one Moreover, if an embedding exists for some n, then there exists one for n = Ω We also must have n log Ω Proof One simply shows that x, y Ω with x y we have either x / Sy Ω or y / Sx Ω if and onlf Ω -tuple S = (Sx Ω, Sx Ω,, Sx Ω Ω where {x i } n i= is an enumeration of all the elements of Ω is separating ( i e S Υ n, see definition 54 if and onlf Υ n which, in turn, according to theorem 5, happens if and onlf Ω can be embedded into an n-dimensional semi-genetic heuristic k-tuple for some n This establishes the equivalence of and Clearly, implies 3 To see the converse, we show that Not implies Not 3 Indeed, if x Sy Ω and y / Sx Ω, then, by minimality, (see definition 5 we have Sx Ω Sy Ω and Sy Ω Sx Ω, so that Sx Ω = Sy Ω Due to space limitations, a detailed argument is available upon request from the author Corollary 53 Given a heuristic k-tuple Ω = (Ω, Γ, Γ,, Γ k, if k and for every T Γ, T is idempotent ( in other words, x Ω T (x, x,, x = x then Ω can be embedded into a binary semi-genetic heuristic k-tuple of dimension less than or equal to Ω Proof The desired conclusion follows immediately from corollary 5 by observing that x, y Ω with x y we have Sx Ω = {x} so that x {x} = Sx Ω while y / {x} = Sx Ω

INVARIANT SUBSETS OF THE SEARCH SPACE AND THE UNIVERSALITY OF A GENERALIZED GENETIC ALGORITHM7 6 Conclusions and Future Work In a classical binary genetic algorithm crossover works by swapping the alleles, while in the generalized version it works by preserving only the good allele ( = and may or may not preserve the 0 gene (see Definition 4 It seems interesting to know that such an algorithm is almost universal in the sense of Corollary 5 and Corollary 53 Notice that the conditions of Corollary 53 are quite natural to assume They basically say that two or more identical individuals produce the offspring which is identically the same as the parent individual Corollary 5 also shows that the dimension of the embedding can always be made less than or equal to the size of the underlying set, Ω It can be shown that, in general, the dimension can not be reduced any further, but the author conectures, that, due to the rigidity of the collection of m-fixable family of subsets (see Appendix A of [5] for the definitions and machinery The material is also available upon request from the author, under some mild conditions, the dimension may be reduced drastically This provides at least one possible direction for the future research Another natural question to ask is the following: Under which conditions can a given heuristic search algorithm be encoded by a classical (not necessarily binary genetic algorithm? It turns out that the conditions involve some basic Abstract Algebra: In fact, a given heuristic k-tuple Ω can be encoded by a genetic algorithm (not necessarily a binary one if and onlf there exists a way to enlarge a set Ω to a superset Ψ so that there exists a ring structure on Ψ with comaximal ideals I, I, I n for which n = I = 0 and n any union of cosets of I intersected with Ω is in Λ Ω The proof of this fact involves Chinese Remainder Theorem (see, for instance, Dummitt and Foote [4] together with a few other technical facts (due to space limitations, these are available upon request from the author used in ways similar to their usage in the proof of Theorem 5 An alternative approach has been developed by Nicholas J Radcliffe [6] Notice, however that Radcliffe s work relies on the notion of a formae which is less general than Mitavskiy s notion of the m-fixable family of subsets described in detail in [5] In particular there is no way to use Radcliiffe s formae to describe the family of semi-genetic crossover operators, while the family of m-fixable subsets describes absolutely any family of m-ary reproduction transformations on an arbitrary, representation independent search space (see Appendix A of [5] for details Also available upon request from the author This type of theorems will be studied in my future research 7 Acknowledgements I want to thank Professor John Holland for the helpful discussions and for the encouragement I ve received from him to write this paper I also want to thank my thesis advisor, Professor Andreas Blass for the numerous helpful advisor meetings which have stimulated some of the ideas for this and for my future work Finally I would like to thank my fellow graduate student of mathematics, Ronald Walker for a few very helpful discussions, and the University of Michigan Complex Systems Group for the suggestions regarding the organization of this paper References [] Antonisse, J A new interpretation of Schema Notation that Overturns the Binary Encoding Constraint Procedings of the Third International Conference on Genetic Algorithms, Ed J D Schaffer, Morgan Kaufmann, San Francisco, 989, pp 86-97

8 BORIS MITAVSKIY [] Zbigniew Michalewicz Genetic Algorithms + Data Structures = Evolution Programs, Berlin; New York: Springer-Verlag, 996 [3] Michael D Vose Generalizing the Notion of a Schema in Genetic Algorithms Artificial Intelligence 50(3 385-396 (99 [4] David S Dummit, Richard M Foote Abstract Algebra, Prentice-Hall, Inc (99 [5] Boris Mitavskiy Crossover Invariant Subsets of the Search Space for Genetic Algorithms and Possible Generalizations, Evolutionary Computation, Submitted in May of 00 [6] Nicholas J Radcliffe, 994 The Algebra of Genetic Algorithms Annals of Math and Artificial Intelligence, 0: 339-384 [7] Watson, R A and Pollack, J B (000 Recombination Without Respect: Schema Combination and Disruption in Genetic Algorithm Crossover, Proceedings of the 000 Genetic and Evolutionary Computation Conference, Whitly D, et al(eds, Morgan Kaufmann, 000 pp -9 Department of Mathematics, University of Michigan, Ann Arbor, MI, 4809 E-mail address: bmitavsk@umichedu