# The complexity of computing over quasigroups

## Abstract

In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. As a consequence of this transformation, the induced classes of languages became CFL instead of REG, in the first case, and SAC^{1} instead of NC^{1} in the second case. In this paper, we investigate the classes obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC^{1}. We introduce the notions of linear recognition by semigroups and by programs over semigroups. This leads to a new characterization of the linear context-free languages and NL. Here again, when quasigroups are used, only languages in REG and NC^{1} can be obtained. We also consider the problem of evaluating a well-parenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC^{1} for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is sufficient that the loop be noasolvable, extending a well-known theorem of Barrington.

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