Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions (that is, conjunction and disjunction) is replaced with the operations in the two-element field GF(2) (conjunction and exclusive OR). Union is thus replaced with symmetric difference, whereas concatenation gives rise to a new GF(2)-concatenation operation, which is notable for being invertible. All operations preserve regularity, and for a pair of languages recognized by an m-state and an n-state DFA, their GF(2)-concatenation is recognized by a DFA with $m\cdot 2^n$ states, and this number of states is in the worst case necessary. Similarly, the state complexity of GF(2)-inverse is $2^n+1$. Next, a new class of formal grammars based on GF(2)-operations is defined, and it is shown to have the same computational complexity as ordinary grammars with union and concatenation: in particular, simple parsing in time $O(n^3)$, fast parsing in the time of matrix multiplication, and parsing in NC².

研究了形式语言上联合和连接操作的变体，其中定义中的布尔逻辑（即合取和析取）被两元素字段 GF(2) 中的操作（合取和异或）替换。因此，并集被对称差分取代，而连接产生了一种新的 GF(2)-连接操作，该操作以可逆而著称。所有操作都保持规律性，并且对于由m状态和n状态 DFA识别的一对语言，它们的 GF(2)-concatenation 被 DFA 识别$米\cdot 2^n$状态，并且这个状态数量在最坏的情况下是必要的。类似地，GF(2)-inverse 的状态复杂度为$2^n+1$. 接下来，定义了一类新的基于 GF(2) 运算的形式文法，它被证明具有与普通文法相同的计算复杂度，特别是简单的时间解析$○(n^3)$，在矩阵乘法时快速解析，在 NC ²中解析。