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

TL;DR

This paper investigates the computational complexity of solving linear congruences and nullspace problems modulo a constant, establishing their completeness for certain logspace modular counting classes and introducing new function classes.

Contribution

It introduces the class FUL_k and proves the completeness of linear algebra problems for coMod_k L for any constant k>1, extending previous results beyond prime moduli.

Findings

Linear algebra problems are complete for coMod_k L for all constant k>1.

Introduces the class FUL_k, low for Mod_k L and coMod_k L.

Explores relationships between FUL_k and verifiable function classes.

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>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.