11 research outputs found
Exhaustive Generation of Linear Orthogonal Cellular Automata
We consider the problem of exhaustively visiting all pairs of linear cellular automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of coprime polynomials over a finite field having the same degree and a nonzero constant term. While previous research showed how to count all such pairs for a given degree and order of the finite field, no practical enumeration algorithms have been proposed so far. Here, we start closing this gap by addressing the case of polynomials defined over the field \F_2, which corresponds to binary CA. In particular, we exploit Benjamin and Bennett's bijection between coprime and non-coprime pairs of polynomials, which enables us to organize our study along three subproblems, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. In the course of this investigation, we unveil interesting connections with algebraic language theory and combinatorics, obtaining an enumeration algorithm and an alternative derivation of the counting formula for this problem
Exhaustive Generation of Linear Orthogonal Cellular Automata
We consider the problem of exhaustively visiting all pairs of linear cellular
automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal
Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of
coprime polynomials over a finite field having the same degree and a nonzero
constant term. While previous research showed how to count all such pairs for a
given degree and order of the finite field, no practical enumeration algorithms
have been proposed so far. Here, we start closing this gap by addressing the
case of polynomials defined over the field \F_2, which corresponds to binary
CA. In particular, we exploit Benjamin and Bennett's bijection between coprime
and non-coprime pairs of polynomials, which enables us to organize our study
along three subproblems, namely the enumeration and count of: (1) sequences of
constant terms, (2) sequences of degrees, and (3) sequences of intermediate
terms. In the course of this investigation, we unveil interesting connections
with algebraic language theory and combinatorics, obtaining an enumeration
algorithm and an alternative derivation of the counting formula for this
problem.Comment: 9 pages, 1 figure. Submitted to the exploratory track of AUTOMATA
2023. arXiv admin note: text overlap with arXiv:2207.0040
An Enumeration Algorithm for Binary Coprime Polynomials with Nonzero Constant Term
We address the enumeration of coprime polynomial pairs over \F_2 where both polynomials have a nonzero constant term, motivated by the construction of orthogonal Latin squares via cellular automata. To this end, we leverage on Benjamin and Bennett's bijection between coprime and non-coprime pairs, which is based on the sequences of quotients visited by dilcuE's algorithm (i.e. Euclid's algorithm ran backward). This allows us to break our analysis of the quotients in three parts, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. For (1), we show that the sequences of constant terms form a regular language, and use classic results from algebraic language theory to count them. Concerning (2), we remark that the sequences of degrees correspond to compositions of natural numbers, which have a simple combinatorial description. Finally, we show that for (3) the intermediate terms can be freely chosen. Putting these three obeservations together, we devise a combinatorial algorithm to enumerate all such coprime pairs of a given degree, and present an alternative derivation of their counting formula
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
The generic limit set of cellular automata
In this article, we consider a topological dynamical system. The generic
limit set is the smallest closed subset which has a comeager realm of
attraction. We study some of its topological properties, and the links with
equicontinuity and sensitivity. We emphasize the case of cellular automata, for
which the generic limit set is included in all subshift attractors, and discuss
directional dynamics, as well as the link with measure-theoretical similar
notions
On the impact of treewidth in the computational complexity of freezing dynamics
An automata network is a network of entities, each holding a state from a
finite set and evolving according to a local update rule which depends only on
its neighbors in the network's graph. It is freezing if there is an order on
states such that the state evolution of any node is non-decreasing in any
orbit. They are commonly used to model epidemic propagation, diffusion
phenomena like bootstrap percolation or cristal growth. In this paper we
establish how treewidth and maximum degree of the underlying graph are key
parameters which influence the overall computational complexity of finite
freezing automata networks. First, we define a general model checking formalism
that captures many classical decision problems: prediction, nilpotency,
predecessor, asynchronous reachability. Then, on one hand, we present an
efficient parallel algorithm that solves the general model checking problem in
NC for any graph with bounded degree and bounded treewidth. On the other hand,
we show that these problems are hard in their respective classes when
restricted to families of graph with polynomially growing treewidth. For
prediction, predecessor and asynchronous reachability, we establish the
hardness result with a fixed set-defiend update rule that is universally hard
on any input graph of such families
The generic limit set of cellular automata
In this article, we consider a topological dynamical system. The generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions