Sheaf Cohomology and SAT Solver Difficulty A Categorical Perspective with Experimental Validation

Abstract

We apply sheaf-theoretic methods to computational complexity, treating hardness as a context-dependent property across Grothendieck topoi. We contrast the topos of finite sets Sh({Fin) where every problem is trivially decidable by exhaustive lookup with the topos of asymptotic domains Sh(N), where polynomial and exponential growth classes are categorically distinct. An essential geometric morphism connects these regimes, formalizing the intuition that finite instances of NP-hard problems are often tractable while the asymptotic distinction remains sharp. We introduce the myriad decomposition to relate this categorical perspective to existing theories in parameterized complexity. This formulation makes explicit the connection between the sheaf-theoretic view of global consistency and classical concepts like treewidth and fixed-parameter tractability, situating the framework within known computational boundaries. We partially validate the framework by computing sheaf-theoretic invariants on a sample of random 3-SAT instances across the phase transition. We find that these topological features specifically the dimension of the solution sheaf's global sections correlate with DPLL solver difficulty even after accounting for standard density measures. This suggests the framework captures structural information about computational hardness, providing a link between algebraic topology and algorithmic behavior. Code available at: https://github.com/DHDev0/Sheaf-Cohomology-and-SAT-Solver-Difficulty

Comments