NDLI: On the complexity of some problems on groups input as multiplication tables

Content Provider	IEEE Xplore Digital Library
Author	Barrington, D.M. Kadau, P. Lange, K.-J. McKenzie, P.
Copyright Year	2000
Description	Author affiliation: Massachusetts Univ., Amherst, MA, USA (Barrington, D.M.)
Abstract	The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G, and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investigate CGM for particular classes of groups. The problem for general groups is in SL (symmetric log space), but any kind of hardness result seems difficult because it would require constructing the entire multiplication table of a group. We introduce the complexity class FOLL, or FO(log log n), of problems solvable by uniform polysize circuit families of unbounded fan-in and depth O(log log n). Since parity is not in FOLL, no problem in FOLL can be complete for any class containing parity, such as NC/sup 1/, L, or SL. But FOLL is not known to be contained even in SL. We show that CGM for cyclic groups is in FOLL/spl cap/L and that CGM for abelian groups is in FOLL. We then partially extend our method to solvable groups, showing that CGM for groups of constant solvability class is in FOLL and that CGM for nilpotent groups can be solved by poly-size circuits of depth O((log log n)/sup 2/). (Thus the latter problem is provably not complete for any class containing parity) We also consider the problem of testing for various properties of a group input as a table: we prove that cyclicity and nilpotency can each be tested in FOLL/spl cap/L. Finally, we examine the implications of our results for the complexity of iterated multiplication, powering, and division of integers, in the context of the recent results of Chiu, Davida, and Litow.
Starting Page	62
Ending Page	69
File Size	250942
Page Count	8
File Format	PDF
ISBN	0769506747
ISSN	10930159
DOI	10.1109/CCC.2000.856736
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2000-07-04
Publisher Place	Italy
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	User-generated content Testing Algebra Circuits Polynomials
Content Type	Text
Resource Type	Article

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