The remarkable advances in SAT solving achieved in the last years have allowed to use this technology in many real-world applications of Artificial Intelligence, such as planning, formal verification, and scheduling, among others. Interestingly, these industrial SAT problems are commonly believed to be easier than classical random SAT formulas, but estimating their actual hardness is still a very challenging question, which in some cases even requires to solve them. In this context, realistic pseudo-industrial random SAT generators have emerged with the aim of reproducing the main features shared by the majority of these application problems. The study of these models may help to better understand the success of those SAT solving techniques and possibly improve them.

In this work, we present a model to estimate the temperature of real-world SAT instances. This temperature represents the degree of distortion into the expected structure of the formula, from highly structured benchmarks (more similar to real-world SAT instances) to the complete absence of structure (observed in the classical random SAT model). Our solution is based on the Popularity-Similarity (PS) random model for SAT, which has been recently presented to reproduce two crucial features of application SAT benchmarks: scale-free and community structures. The PS model is able to control the hardness of the generated formula by introducing some randomizations in the expected structure. Our solution is a first step towards a hardness oracle based on the temperature of SAT formulas, which may be able to estimate the cost of solving real-world SAT instances without solving them.