Bruno Durand

Complex tilings

We study the minimal complexity of tilings of a plane with a given tile set. We note that any tile set admits either no tiling or some tiling with ooo(<italic>n</italic>) Kolmogorov complexity of its (<italic>ntimes n</italic>)-squares. We construct tile sets for which this bound is nearly tight: all tilings have complexity ><italic>n/r(n)</italic>, given any unbounded computable monotone <italic>r</italic>. This adds a quantitative angle to classical results on non-recursivity of tilings -- that we also develop in terms of Turing degrees of unsolvability.

The Game of Life: Universality Revisited

The Game of Life was created by J.H. Conway. One of the main features of this game is its universality. We prove in this paper this universality with respect to several computational models: boolean circuits, Turing machines, and two-dimensional cellular automata. These different points of view on Life’s universality are chosen in order to clarify the situation and to simplify the original proof. We also present precise definitions of these 3 universality properties and explain the relations between them.

Inversion of 2D cellular automata: some complexity results

Abstract In this paper, we prove the co-NP-completeness of the following decision problem: “Given a two-dimensional cellular automaton A (even with Von Neumann neighborhood), is A injective when restricted to finite configurations not greater than its length?” In order to prove this result, we introduce two decision problems concerning, respectively, Turing machines and tilings that we prove NP-complete. Then, we present a transformation of problems concerning tilings into problems concerning cellular automata.

Tilings and quasiperiodicity

Quasiperiodic tilings are those tilings in which finite patterns appear regularly in the plane. This property is a generalization of the periodicity; it was introduced for representing quasicrystals and it is also motivated by the study of quasiperiodic words. We prove that if a tile set can tile the plane, then it can tile the plane quasiperiodically — a surprising result that does not hold for periodicity. In order to compare the regularity of quasiperiodic tilings, we introduce and study a quasiperiodicity function and prove that it is bounded by x↦x+c if and only if the considered tiling is periodic. At last, we prove that if a tile set can be used to form a quasiperiodic tiling which is not periodic, then it can form an uncountable number of tilings.

The surjectivity problem for 2D cellular automata

The surjectivity problem for 2D cellular automata was proved undecidable in 1989 by Jarkko Kari. The proof consists in a reduction of a problem concerning finite tilings into the previous one. This reduction uses a special and very sophisticated tile set. In this article, we present a much more simple tile set which can play the same role.

Global Properties of 2D Cellular Automata: Some Complexity Results

In this paper, we prove the co-NP-completeness of the following decision problem: “given a 2-dimensional cellular automaton A (even with Von Neumann neighborhood), is A injective when restricted to finite configurations not greater than its length?” In order to prove this result, we introduce two decision problems concerning respectively Turing Machines and tilings that we prove NP-complete. Then, we transform problems concerning tilings into problems concerning cellular automata.

On the complexity of deadlock detection in families of planar nets

We are interested in some properties of massively parallel computers which we model by finite automata connected together on a 2-dimensional grid. We wonder whether it is possible to anticipate a possible appearance of a deadlock in such nets. Thus, we look for efficient algorithms to predict whether deadlocks can appear in grids of bounded size. From the point of view of worst-case complexity, we prove that this problem is NP-complete whereas it is quadratic for linear structures. The method we use is a reduction from a tiling problem. n nWe also prove that this problem, associated with a natural probability distribution on its instances, is RNP-complete (Random NP-complete) in the theory proposed by Levin and developed by Gurevich. Very few randomized problems are known to be RNP-complete. Under classical complexity hypotheses, this result proves that there does not exist any algorithm that solves this problem efficiently on the average case. We present other extentions of our results for different planar underlying communication graphs, and we present a Σ2-complete problem for networks with inputs.

Deterministic Rational Transducers and Random Sequences

This paper presents some results about transformations of infinite random sequences by letter to letter rational transducers. We show that it is possible by observing initial segments of a given random sequence to decide whether two given letter to letter rational transducers have the same output on that sequence. We use the characterization of random sequences by Kolmogorov Complexity. We also prove that the image of a random sequence is either random, or non-random and non-recursive, or periodic, depending on some transducers structural properties that we give.

Tilings and Quasiperiodicity

Quasiperiodic tilings are those tilings in which finite patterns appear regularly in the plane. This property is a generalization of the periodicity; it was introduced for representing quasicrystals and it is also motivated by the study of quasiperiodic words. We prove that if a tile set can tile the plane, then it can tile the plane quasiperiodically — a surprising result that does not hold for periodicity. In order to compare the regularity of quasiperiodic tilings, we introduce and study a quasiperiodicity function and prove that it is bounded by x↦x+c if and only if the considered tiling is periodic. At last, we prove that if a tile set can be used to form a quasiperiodic tiling which is not periodic, then it can form an uncountable number of tilings.

Undecidability of the Surjectivity Problem for 2D Cellular Automata: A Simplified Proof

The surjectivity problem for 2D cellular automata was proved undecidable in 1989 by Jarkko Kari. The proof consists in a reduction of a problem concerning finite tilings to this problem. This reduction uses a special and very sophisticated tile set. In this article, we present a much more simple tile set which can play the same role.