The complexity of functions on lattices

Low complexity functions and convex sets in $\mathbb{Z}^k$

“…For |A| = 4 all exceptions to (PP) were classified. A discussion of limitations to the periodicity principle ends [14]. In [15] the authors prove (PP) for all (2 × n)-blocks in Z 2 .…”

“…[9,7,6,8]) and occur in many branches of mathematics, computer science and physics (cf. [1,16,10], and [2,14] for more references).…”

“…In [14] and [15] the authors studied the periodicity principle of Morse and Hedlund for more general configurations. First of all it was observed that for 1-dimensional functions f condition (1) can be replaced by |P f (A)| ≤ |A| (2) for some configuration A and again periodicity follows [14,Theorem 1].…”

“…First of all it was observed that for 1-dimensional functions f condition (1) can be replaced by |P f (A)| ≤ |A| (2) for some configuration A and again periodicity follows [14,Theorem 1]. Further, it was shown in [14] that for functions f : Z k → {0, 1} with k ≥ 2 the periodicity principle (PP), which says that (2) implies periodicity of f , is true if |A| ≤ 3. For |A| = 4 all exceptions to (PP) were classified.…”

Modelling landscape complexity for land use management in Rio de Janeiro, Brazil

“…The complexity of lattices has been examined in many ways (Adamatzky, 1998;Bettati et al, 1998;Sander and Tijdeman, 2000;Felsner et al, 2000;Beimel et al, 2001;Cheung and Vogel, 2005), and the computational complexity of lattices in particular (Brunetti et al, 2001;Machta, 1992;Nourine and Raynaud, 1999).…”