The classification of minimally unsatisfiable 2-CNFs -- a fundamental study

Abstract

Conjunctive normal forms where every clause has length at most two are called 2-CNFs. We study minimally unsatisfiable 2-CNFs (2-MUs), that is, unsatisfiable 2-CNFs where removing any clause destroys unsatisfiability, and obtain their full classification up to isomorphism. The main tool is the implication digraph: we show that for 2-MUs these digraphs are "weak double cycles" (WDCs), big cycles of small cycles with possible overlaps. Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point-free anti-automorphism reversing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs, reducing the classification of 2-MUs to the classification of a well-structured class of digraphs. We obtain a variety of applications for 2-MUs of deficiency k (the difference between the number of clauses and the number of variables): the smoothing of skew-symmetric WDCs corresponds exactly to the canonical normal form obtained by 1-singular Davis-Putnam reduction, and the resulting homeomorphism types are in one-to-one correspondence with binary bracelets of length k. The automorphism group of any 2-MU of deficiency k is a subgroup of the dihedral group with 4k elements. The isomorphism problem for 2-MUs is decidable in quadratic time, and the number of isomorphism types of 2-MUs for fixed k is Theta(n^(3k-1)). The article is addressed to both the logic and the graph theory communities, with complete proofs provided throughout.