🤖 AI Summary
Addressing the classical undecidable problem of Wang tiling, this paper proposes a partially decidable framework integrating statistical mechanics and discrete dynamical systems theory. We construct mappings among finite tile sets and introduce effective entropy and temperature parameters to characterize the thermodynamic behavior of tile alphabets; dynamical phase transitions are identified via chaos criteria—including logistic map bifurcation analysis and Kendall Tau correlation. Results show that favorable thermodynamic behavior (low entropy, non-chaotic dynamics) strongly correlates with infinite planar tileability, whereas emergent chaotic dynamics signal the undecidable phase. The framework successfully distinguishes known tileable and non-tileable instances, establishing—for the first time—a quantitative link between thermodynamic properties and decidability. This yields a computationally tractable and interpretable paradigm for partial decidability in combinatorial undecidability problems.
📝 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 can 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 undecidable behavior is related to a transition from non-chaotic to chaotic regime in discrete dynamical systems of logistic type.