A Physical Analogy between Molecular Ordering and SAT-to-Ising Annealing

TL;DR

This paper establishes a thermodynamic analogy between molecular ordering and SAT problem solving, demonstrating how simulated annealing can lead to logical crystallization akin to physical crystallization.

Contribution

It introduces a novel mapping of SAT problems to Ising models and empirically shows the analogy through simulated annealing, linking computational coherence to physical thermodynamics.

Findings

Logical configurations resemble low-energy crystalline states.

Rapid 'first-order' crystallization of satisfiable solutions observed.

Backbone rigidity does not strongly influence physical ordering.

Abstract

As temperature drops, molecular systems may undergo spontaneous ordering, moving from random behavior to orderly structure. This research demonstrates a direct analogy between this type of thermodynamic ordering in molecular systems and the development of coherent logic in computationally complex problem sets. We have proposed a mapping of Boolean SAT problem instances to pairwise Ising Hamiltonian models. Using simulated annealing, we then applied phenomenal cooling to the system through thermal evolution from high entropy random assignment to lower entropy, ordered assignments (the energy minima) using molecular cooling analogs. This indicated that there was a rapid "first-order" or "logical crystallization" of satisfiable logical configurations. The degree of backbone rigidity did not strongly correlate with the level of physical ordering observed in the system; thus, it appears that there is primarily a local alignment of constraint satisfaction occurring in the system. Thus, we have provided empirical evidence that satisfiable logical configurations are analogous to the low energy crystalline states observed in molecular systems and provide evidence for a unified thermodynamic view of computational coherence and complexity.