  The satisfiability and optimization of finite-dimensional Boolean formulas
are studied using percolation theory, rare region arguments, and boundary
effects. In contrast with mean-field results, there is no satisfiability
transition, though there is a logical connectivity transition. In part of the
disconnected phase, rare regions lead to a divergent running time for
optimization algorithms. The thermodynamic ground state for the NP-hard
two-dimensional maximum-satisfiability problem is typically unique. These
results have implications for the computational study of disordered materials.
