On the Expressive Power of GF(2)-Grammars

GF(2)-grammars, recently introduced by Bakinova et al. (“Formal languages over GF(2)”, LATA 2018), are a variant of ordinary context-free grammars, in which the disjunction is replaced by exclusive OR, whereas the classical concatenation is replaced by a new operation called GF(2)-concatenation: \(K \odot L\) is the set of all strings with an odd number of partitions into a concatenation of a string in K and a string in L. This paper establishes several results on the family of languages defined by these grammars. Over the unary alphabet, GF(2)-grammars define exactly the 2-automatic sets. No language of the form \(\{a^n b^{f(n)} \,\mid \, n \geqslant 1\}\), with uniformly superlinear f, can be described by any GF(2)-grammar. The family is not closed under union, intersection, classical concatenation and Kleene star, non-erasing homomorphisms. On the other hand, this family is closed under injective nondeterministic finite transductions, and contains a hardest language under reductions by homomorphisms.