By reformulating Wang tiles with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts. In the one dimensional case, we recover the decidability of the generalized domino problem by linking it to the trace characterization of nilpotent matrices. In the two-dimensional case, we provide extension of weak and strong aperiodicity respectively and show the equivalence of those generalized notions, extending the well known equivalence in the classical case. We also exhibit a quantum tileset being aperiodic while its underlying classical tile set is not, proving that quantum interference can suppress periodic patterns and paving the way to the investigation of a new kind of aperiodicity. Finally, we highlight the many new research directions opened by this generalization of Wang tiles, related to (quantum) cellular automata, condensed matter physics, symbolic dynamics and more.

We exhibit a strong connection between the matchgate formalism introduced by Valiant and the ZW-calculus of Coecke and Kissinger. This connection provides a natural compositional framework for matchgate theory as well as a direct combinatorial interpretation of the diagrams of ZW-calculus through the perfect matchings of their underlying graphs. We identify a precise fragment of ZW-calculus, the planar W-calculus, that we prove to be complete and universal for matchgates, that are linear maps satisfying the matchgate identities. Computing scalars of the planar W-calculus corresponds to counting perfect matchings of planar graphs, and so can be carried in polynomial time using the FKT algorithm, making the planar W-calculus an efficiently simulable fragment of the ZW-calculus, in a similar way that the Clifford fragment is for ZX-calculus. This work opens new directions for the investigation of the combinatorial properties of ZW-calculus as well as the study of perfect matching counting through compositional diagrammatical technics.

Abstract In this paper we study colorings (or tilings) of the two-dimensional grid ${\mathbb {Z}}^{2}$ ℤ2 . A coloring is said to be valid with respect to a setPofn×mrectangular patterns if alln×msub-patterns of the coloring are inP. A coloringcis said to be of low complexity with respect to a rectangle if there exist $m,n\in \mathbb {N}$ m,n∈ℕ and a setPofn×mrectangular patterns such thatcis valid with respect toPand |P|≤nm. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. If Nivat’s conjecture is true, all valid colorings with respect toPsuch that |P|≤mnmust be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that thenmbound is multiplicatively optimal for the decidability of the domino problem, as for allε> 0 it is undecidable to determine if there exists a valid coloring for a given $m,n\in \mathbb {N}$ m,n∈ℕ and set of rectangular patternsPof sizen×msuch that |P|≤ (1 +ε)nm. We prove a slightly better bound in the case wherem=n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 12).

Abstract We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

In this paper we study low-complexity colorings (or tilings) of the two-dimensional grid ℤ². A coloring is said to be of low complexity with respect to a rectangle if there exists m,n∈ℕ such that there are no more than mn different rectangular m× n patterns in it. Open since it was stated in 1997, Nivat’s conjecture states that such a coloring is necessarily periodic. Suppose we are given at most nm rectangular patterns of size n× m. If Nivat’s conjecture is true, one can only build periodic colorings out of these patterns - meaning that if the m× n rectangular patterns of the coloring are among these mn patterns, it must be periodic. The main contribution of this paper proves that there exists at least one periodic coloring build from these patterns. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Finally, we use our result to show that Nivat’s conjecture holds for uniformly recurrent configurations. The results also extend to other convex shapes in place of the rectangle.

We show that the domino problem is undecidable on orbit graphs of non-deterministic substitutions which satisfy a technical property. As an application, we prove that the domino problem is undecidable for the fundamental group of any closed orientable surface of genus at least 2.