On the enumeration of signatures of XOR-CNF's

2024-02-28

arXiv.org

AI Summary

This paper investigates the efficient enumeration of signatures—sequences of truth values assigned to clauses under satisfying assignments—for XOR-CNF formulas. While satisfiability for XOR-CNF is tractable (in P), the enumeration complexity of its signatures remained unexplored. We provide the first complete characterization: full, minimal, and maximal signature enumeration are all solvable in polynomial time. Our approach leverages the equivalence between XOR-CNF and systems of linear equations over GF(2), combining Gaussian elimination, affine solution space decomposition, and extremal combinatorial enumeration techniques. This circumvents the NP-hardness inherent in maximal signature enumeration for general CNF formulas. Our results fill a fundamental gap in the theory of signature enumeration for tractable CNF classes and establish a new paradigm for structure-aware, satisfiability-driven enumeration algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Given a CNF formula $varphi$ with clauses $C_1, dots, C_m$ over a set of variables $V$, a truth assignment $mathbf{a} : V o {0, 1}$ 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.

Problem

• Enumerate signatures of XOR-CNF formulas efficiently
• Characterize complexity of generating minimal and maximal signatures
• Solve open problem for tractable CNF classes

Innovation

• Efficiently enumerates XOR-CNF signatures
• Characterizes complexity for maximal signatures
• Focuses on tractable XOR-CNF classes