🤖 AI Summary
This work addresses the challenge of satisfiability in the existential theory of the reals (∃ℝ), particularly the realizability of configurations in discrete geometry, by introducing a novel paradigm termed Satisfiability Modulo Realizability. The approach encodes geometric problems as SAT instances over abstract order types and integrates diversity-driven sampling, partial realizability feedback, and a new flip-based heuristic to efficiently guide the solver away from non-realizable regions during search. Leveraging this framework, the study resolves a long-standing open problem in discrete geometry by proving that the largest point set containing no empty convex hexagon or convex heptagon has size 23, thereby substantially advancing the computational frontier in this domain.
📝 Abstract
Problems complete for the existential theory of the reals ($\exists \mathbb{R}$) arise throughout discrete geometry. We introduce satisfiability modulo realizability, a SAT-based approach for solving satisfiable instances of $\exists \mathbb{R}$ whose solutions correspond to realizable geometric configurations. Our method encodes an underapproximation of a geometric problem as a SAT instance over abstract order types. Since almost all abstract order types are unrealizable, naive search is infeasible. We guide the search toward realizable order types using diversity-driven sampling, partial realizability feedback, and a novel flippability heuristic that passes only limited information between components. We apply our method to discrete geometry problems and resolve an open problem by showing that the largest set of points avoiding empty convex hexagons and convex heptagons is of size 23.