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In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k − 1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices. 1

We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomial-time solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomial-time randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “black-box groups ” under minimal assumptions. In a black-box group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”

In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing _only_ on composite numbers that can actually be split in expected polynomial time. Therefore, factoring would be easy if Rabin's test systematically failed with a 25% probability on each composite integer (which, of course, it does not). The second observation is more fundamental because is it _not_ restricted to primality testing: it has consequences for the entire field of probabilistic algorithms. The failure probability when using a probabilistic algorithm for the purpose of testing some property is compared with that when using it for the purpose of obtaining a random element hopefully having this property. More specifically, we investigate the question of how reliable Rabin's test is when used to _generate_ a random integer that is probably prime, rather than to _test_ a specific integer for primality. Key words: factorization, false witnesses, primality testing, probabilistic algorithms, Rabin's test.

Givengenerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients. 1.

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹-hard under AC0-reductions. NC¹-completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's test so good. It turns out that a single iteration fails with a non-negligible probability on a composite number of the form 4j +3 only if this number happens to be easy to split. The second observation is much more fundamental because is it not restricted to primality testing: it has profound consequences for the entire field of probabilistic algorithms. There we ask the question: how good is Rabin's algorithm? Whenever one wishes to produce a uniformly distributed random probabilistic prime with a given bound on the error probability, it turns out that the size of the desired prime must be taken into account. 1. Introduction In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's te...

We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also