Symmetry and complexity

We examine the effect of symmetry on the complexity of Boolean functions and find a remarkably tight hierarchy. Generalizing the fact that all symmetric Boolean functions belong to (nonuniform) Z’CO, we find that the complexity of the class of Boolean functions admitting a given group of symmetries is essentially determined by a single parameter of that group. Let G be a permutation group acting on the set of n Boolean variables. Let 7(G) denote the set of Boolean functions on n variables which are invariant under G. Let ~ denote a sequence G. < Sym(Q.) of groups (/S2.[ = n). We say that the language L ~ {O, 1}* belongs to the symmetry class ~(~) if the indicator function of L n {O, l}n belongs to Y(Gn) for every n. Following Clote and Kranakis, we consider the parameter s(G), the number of orbits of G on the set {O, l}’”. We show that (a) all functions in F(G) are computable by circuits of size, polynomial in s(G) and depth, polynomial in log s(G); (b) there exist functions in 3(G) which cannot be computed by circuits of size < s(G)/(2 log s(G)). While part (b) is obtained by straightforward counting, it demonstrates that part (a) is tight. The result in particular confirms the following conjecture of Clote and Kranakis: if s(Gn ) is polynomially bounded (where G. < S.) then F(g) ~ NC (nonuniform). If in addition the groups are transitive, we prove that the left hand side is actually in TCO. oT& ~e~e=ch was pmti~y supported by NSF Gr~ts CCR 8710078 and CGR 9014562 1Department of Computer Science, University of Chicago, Chicago, Illinois 60637 2E~tv~s University, Budapest, Hungary H-lo88 31aci@cs.uchicago .edu 4beals@cs.uchicago. edu Permission to copy without fee all or part of this material is granted provided that the copies ara not made or distributed for diremt GommerGial advantaga, the ACM copyright notice and the title of the publication and its data appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or spacific permission. 24th ANNUAL ACM STOC 5/92/VICTORIA, B. C., CANADA ~ 1992 A(JM ().89791-51 2-7/92 JOO04/0438...$~ .50 The proof of the main result involves nontrivial elementary asymptotic structure theory of permutation groups; and a delicate analysis (under new circumstances) of algorithmic techniques developed largely by E. M. Luks in the context of graph isomorphism testing. In the context of isomorphism of sets under group action, uniform versions of our results are obtained.