关于XOR-CNF签名的枚举研究
On the enumeration of signatures of XOR-CNF's
AI总结：
针对易解类CNF的最大签名高效生成这一开放问题，完整刻画了XOR-CNF的所有、最小及最大签名的生成复杂度，补充了CNF签名枚举领域的相关复杂度结论。
摘要：
给定一个CNF公式$\varphi$，其子句为$C_1, \dots, C_m$，定义在变量集合$V$上，一个真值赋值$\mathbf{a} : V \to \{0, 1\}$会生成一个二进制序列$\sigma_\varphi(\mathbf{a})=(C_1(\mathbf{a}), \ldots, C_m(\mathbf{a}))$，称为$\varphi$的签名，其中若子句$C_i$在赋值$\mathbf{a}$下取值为1则$C_i(\mathbf{a})=1$，否则为0。签名及其相关生成问题在算法枚举领域催生了全新且颇具前景的研究方向。在近期一篇论文中，Bérczi等人得出了一个有趣的结论：尽管验证解的难度很高，但生成CNF的签名是易解的。他们还指出，由于一般情况下可满足性问题的难解性，寻找任意CNF的最大签名是困难的。他们的研究留下了一个待解决的问题：针对易解类的CNF（即可满足性问题可在多项式时间内求解的CNF），如何高效生成其最大签名。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的生成复杂度。
英文摘要：
Given a CNF formula $φ$ 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 $σ_φ(\mathbf{a})=(C_1(\mathbf{a}), \ldots, C_m(\mathbf{a}))$, called a signature of $φ$, 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, Bérczi 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
