Given a CNF formula $\varphi$ with clauses $C_1, \dots, C_m$ over a set of variables $V$, a truth assignment $\mathbf{a} : V \to \{0, 1\}$ generates a binary sequence $\sigma_\varphi(\mathbf{a})=(C_1(\mathbf{a}), \ldots, C_m(\mathbf{a}))$, called a signature of $\varphi$, where $C_i(\mathbf{a})=1$ if clause $C_i$ evaluates to 1 under assignment $\mathbf{a}$, and $C_i(\mathbf{a})=0$ otherwise. Signatures and their associated generation problems have given rise to new yet promising research questions in algorithmic enumeration. In a recent paper, B\'erczi et al. interestingly proved that generating signatures of a CNF is tractable despite the fact that verifying a solution is hard. They also showed the hardness of finding maximal signatures of an arbitrary CNF due to the intractability of satisfiability in general. Their contribution leaves open the problem of efficiently generating maximal signatures for tractable classes of CNFs, i.e., those for which satisfiability can be solved in polynomial time. Stepping into that direction, we completely characterize the complexity of generating all, minimal, and maximal signatures for XOR-CNFs.

翻译：给定一个CNF公式$\varphi$，其子句为$C_1, \dots, C_m$，变量集为$V$，一个真值赋值$\mathbf{a} : V \to \{0, 1\}$生成一个二元序列$\sigma_\varphi(\mathbf{a})=(C_1(\mathbf{a}), \ldots, C_m(\mathbf{a}))$，称为$\varphi$的签名，其中若子句$C_i$在赋值$\mathbf{a}$下取值为1，则$C_i(\mathbf{a})=1$，否则$C_i(\mathbf{a})=0$。签名及其关联的生成问题为算法枚举领域带来了新颖且富有前景的研究问题。在近期的一篇论文中，Bérczi等人有趣地证明了生成CNF的签名是可行的，尽管验证解的难度很大。他们还指出，由于一般情况下的可满足性问题的难解性，任意CNF的最大签名生成问题是困难的。他们的研究留下了这样一个未解决问题：对于可解类CNF（即可在多项式时间内求解可满足性的CNF），如何高效生成最大签名。沿着这一方向，我们完整刻画了XOR-CNF的所有签名、最小签名和最大签名生成的复杂度。