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Abstract

Abstract:We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k 2. These problems are known to be complete for the logspace modular counting classes Mod k L = coMod k L in special case that k is prime [7]. By considering variants of standard logspace function classes -related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k -we show that these problems of linear algebra are also complete for coMod k L for any constant k 2.Our results are obtained by defining a class of functions FUL k which are low for Mod k L and coMod k L for k 2, using ideas similar to those used in the case of k prime in Ref. [7] to show closure of Mod k L under NC 1 reductions (including Mod k L oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL k for arbitrary moduli k to the class F·coMod k L of functions whose output symbols are verifiable by coMod k L algorithms; and consider what consequences such a comparison may have for oracle closure results of the form Mod k L Mod k L = Mod k L for composite k. † This work was supported by the EC project QCS.