The Complexity of Computing over Quasigroups

In 7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the context-free languages and the class SAC 1. In this paper, we investigate the classes of languages 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 groupoids and by programs over groupoids, and characterize the linear context-free languages and NL. Here again, when quasigroups are used, only regular languages and languages in NC 1 can be obtained. We also consider the problem of evaluating a well-parenthesized expression over a nite loop (a quasi-group with an identity). This problem is in NC 1 for any nite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is suucient that the loop be nonsolvable, extending a well-known theorem of Barrington ((3]).