On the enumeration of signatures of XOR-CNF's

Abstract

Given a CNF formula $\phi$ 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 ${\sigma }_{\phi }(\mathbf{a})=(C_1(\mathbf{a}),\dots ,C_m(\mathbf{a}))$, called a signature of $\phi$, 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.

22 pages, 5 figures