On the complexity of solving linear congruences and computing nullspaces modulo a constant

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>1. 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 (Buntrock et al, 1992). 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>1. 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>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) 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.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

A Note on Closure Properties of Logspace MOD Classes

Introduction The classes MOD k L for k 2 consist of those languages A, for which there exists a nondeterministic Turing machine M working on logarithmically bounded space such that, for all inputs x, x belongs to set A if and only if the number of accepting paths in the computation of M on x is not divisible by k. These classes were de ned and examined in [4]. Their importance stems from the fact that the complexity of a number of problems from linear algebra over Z=kZ is given by these classes (in the sense that they are complete in the respective classes); among these problems are singularity of matrices, inversion of matrices, iterated matrix product, etc. Buntrock et al. also examined structural properties of these classes. E.g., it was shown in [4] that for prime q, all the classes MOD q L are closed under Boolean operations, under logspace disjunctive or conjunctive reducibility, under join, and under NC reductions (Lemma 6 in [4]). In this short note we prove that MOD