Reconstruction of Boolean Formulas in Conjunctive Normal Form

Abstract

A long-standing open problem in graph theory is to prove or disprove the graph reconstruction conjecture proposed by Kelly and Ulam in the 1940s. This conjecture roughly states that every graph on at least three vertices is uniquely determined by its vertex-deleted subgraphs. We adapt the idea of reconstruction for Boolean formulas in conjunctive normal form (CNFs) and formulate the reconstruction conjecture for CNFs: every CNF with at least four clauses is uniquely determined by its clause-deleted subformulas. Our main results can be summarized as follows. First, we prove that our conjecture is equivalent to a well-studied variation of the graph reconstruction conjecture, namely, the edge-reconstruction conjecture for hypergraphs. Second, we prove that the number of satisfying assignments of a CNF is reconstructible, i.e., this number can be computed from the clause-deleted subformulas. Third, we show that every CNF with m clauses over n variables is reconstructible if \(2^{m-1} > 2^n \cdot n!\).