Partial decidability protocol for the Wang tiling problem from statistical mechanics and chaotic mapping
Fabrizio Canfora, Marco Cedeno
TL;DR
The paper tackles the undecidable Wang tiling problem by introducing a partial decidability framework that maps tile alphabets to thermodynamic-like quantities (entropy, temperature, partition function). It presents two protocols: Protocol I relies on monotonic growth of entropy with tile region size, while Protocol II analyzes discrete maps derived from tiling counts to distinguish good versus bad alphabets, interpreted through chaos theory. Kendall's Tau is employed as a robust metric to quantify monotonicity, enabling a quantitative separation between alphabets that likely tile the plane and those that do not. The approach is validated against known alphabets exhibiting periodic, self-similar, and irregular tilings, and it reveals a link between the good-to-bad transition and a shift from regular to chaotic discrete dynamics. While not resolving decidability, the work offers practical heuristics with implications for complexity, statistical physics interpretations, and potential extensions beyond two-dimensional tilings.
Abstract
We introduce a partial decidability protocol for the Wang tiling problem (which is the prototype of undecidable problems in combinatorics and statistical physics) by constructing a suitable mapping from tilings of finite squares of different sizes. Such mapping depends on the initial family of Wang tiles (the alphabet) with which one would like to tile the plane. This allows to define effective entropy and temperature associated to the alphabet (together with the corresponding partition function). We identify a subclass of good alphabets by observing that when the entropy and temperature of a given alphabet are well-behaved in the thermodynamical sense then such alphabet is a good candidate to tile the infinite two-dimensional plane. Our proposal is tested successfully with the known available good alphabets (which produce periodic tilings, aperiodic but self-similar tilings as well as tilings which are neither periodic nor self-similar). Our analysis shows that the Kendall Tau coefficient is able to distinguish alphabets with a good thermodynamical behavior from alphabets with bad thermodynamical behavior. The transition from good to bad behavior is related to a transition from non-chaotic to chaotic regime in discrete dynamical systems of logistic type.