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Adding blocked clauses to a CNF formula can substantially speed up SAT-solving, both in theory and practice. In theory, the addition of blocked clauses can exponentially reduce the length of the shortest refutation for a formula [17, 19]. In practice, it has been recently shown that the runtime of CDCL solvers decreases significantly for certain instance families when blocked clauses are added as a preprocessing step [10,22]. This fact is in contrast to, but not in contradiction with, prior results showing that Blocked- Clause Elimination (BCE) is sometimes an effective preprocessing step [14,15]. We suggest that the practical role of blocked clauses in SAT-solving might be richer than expected. Concretely, we propose a theoretical study of the complexity of Blocked-Clause Addition (BCA) as a preprocessing step for SAT-solving, and in particular, consider the problem of adding the maximum number of blocked clauses of a given arity k to an input formula F. While BCE is a confluent process, meaning that the order in which blocked clauses are eliminated is irrelevant, this is not the case for BCA: adding a blocked clause to a formula might unblock a different clause that was previously blocked. This order-sensitivity turns out to be a crucial obstacle for carrying out BCA efficiently as a preprocessing step. Our main result is that computing the maximum number of k-ary blocked clauses that can be added to an input formula F is NP-hard for every k ≥ 2. 

A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position (i.e., no three points on a common line) contains an empty convex hexagon. Heule and Scheucher solved this problem with a combination of geometric insights and automated reasoning techniques by constructing CNF formulas ϕ_n, with O(n⁴) clauses, such that if ϕ_n is unsatisfiable then every set of n points in general position must contain an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17 300 CPU hours of parallel computation. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers ideas in discrete computational geometry and SAT encoding techniques by introducing a framework that connects geometric objects to propositional assignments. We see this as a key step towards the formal verification of other SAT-based results in geometry, since the abstractions we use have been successfully applied to similar problems. Overall, we hope that our work sets a new standard for the verification of geometry problems relying on extensive computation, and that it increases the trust the mathematical community places in computer-assisted proofs. 

Radio 2-colorings of graphs are a generalization of vertex colorings motivated by the problem of assigning frequency channels in radio networks. In a radio 2-coloring of a graph, vertices are assigned integer colors so that the color of two vertices u and v differ by at least 2 if u and v are neighbors, and by at least 1 if u and v have a common neighbor. Our work improves the best-known bounds for optimal radio 2-colorings of small hypercube graphs, a combinatorial problem that has received significant attention in the past. We do so by using automated reasoning techniques such as symmetry breaking and Cube and Conquer, obtaining that for n = 7 and n = 8, the coding-theory upper bounds of Whittlesey et al. (1995) are not tight. Moreover, we prove the answer for n = 7 to be either 12 or 13, thus making a substantial step towards answering an open problem by Knuth (2015). Finally, we include several combinatorial observations that might be useful for further progress, while also arguing that fully determining the answer for n = 7 will require new techniques.