[Paper Review] Dynamics in parallel of double Boolean automata circuits

Mathilde Noual|arXiv (Cornell University)|Nov 17, 2010

Cellular Automata and Applications5 references3 citations

TL;DR

This paper investigates the dynamics of double Boolean automata circuits (dbac), focusing on networks with two interconnected cyclic circuits sharing a central node. It derives exact formulas for the number of attractors and periodic configurations in negative-positive and negative-negative dbac systems using number-theoretic functions like the Möbius function and Euler's totient function, showing that attractor counts depend only on total size $N = \ell + r$ and $\Delta = \gcd(\ell, r)$, not on individual sizes.

ABSTRACT

In this paper, we give some results concerning the dynamics of double Boolean automata circuits (dbac's for short), namely, networks associated to interaction graphs composed of two side-circuits that share a node. More precisely, we give formulas for the number of attractors of any period, as well as the total number of attractors of these networks.

Motivation & Objective

• To understand the periodic dynamics of double Boolean automata circuits (dbac), particularly when one or both side-circuits are negative.
• To characterize the number of attractors and periodic configurations in dbac systems with mixed or negative side-circuits.
• To show that attractor counts depend only on the total size $N = \ell + r$ and $\Delta = \gcd(\ell, r)$, not on $\ell$ or $r$ individually.
• To provide combinatorial formulas for attractor numbers using number-theoretic functions such as the Möbius function and Euler's totient function.

Proposed method

• Models dbac as networks with two side-circuits sharing a central node, where local transitions are defined by identity or negation functions.
• Uses the Möbius inversion formula to compute the number of configurations with exact period $p$, given by $\mathtt{C}_{p}^{\ast}(\ell,r) = \sum_{q|p} \mu(p/q) \cdot \mathtt{C}_{p}(\ell,r)$.
• Derives the number of $p$-attractors as $\mathtt{A}_{p}(\ell,r) = \frac{1}{p} \sum_{q|p} \mu(p/q) \cdot \mathtt{C}_{p}(\ell,r)$, where $\mathtt{C}_{p}(\ell,r)$ depends on $P(p/\Delta_p)^{\Delta_p}$ for negative-negative systems.
• Applies the function $P(k)$, defined as the number of binary strings of length $k$ with no two consecutive 1s, to compute configuration counts.
• Uses the totient function $\psi$ in the final formula for total attractors: $\mathtt{T}^{=}_{N,\Delta} = \frac{1}{N} \sum_{p|N} \psi(N/p) \cdot P(p/\Delta_p)^{\Delta_p}$.
• Performs computer simulations to validate and observe patterns in attractor counts across various $\ell, r$ values and $\gcd(\ell,r)$.

Experimental results

Research questions

• RQ1 How do attractor periods in double Boolean automata circuits depend on the sizes and signs of the side-circuits?
• RQ2 What is the exact number of periodic configurations and attractors in a dbac with one negative and one positive side-circuit?
• RQ3 Can the number of attractors in a doubly negative dbac be expressed solely in terms of $N = \ell + r$ and $\Delta = \gcd(\ell, r)$?
• RQ4 What is the role of the Möbius function in counting periodic configurations and attractors in such systems?
• RQ5 Under what conditions is the total number of attractors maximized for a given $N$?

Key findings

• For a doubly negative dbac $D_{\ell,r}$, the number of configurations of period $p$ depends only on $p$ and $\Delta_p = \gcd(\Delta, p)$, given by $C_p(\ell,r) = P(p/\Delta_p)^{\Delta_p}$, where $P(k)$ counts binary strings of length $k$ with no two consecutive 1s.
• The number of $p$-attractors in a doubly negative dbac is $\mathtt{A}_p^{=} = \frac{1}{p} \sum_{q|p} \mu(p/q) \cdot P(q/\Delta_q)^{\Delta_q}$.
• The total number of attractors in a doubly negative dbac is $\mathtt{T}^{=}_{N,\Delta} = \frac{1}{N} \sum_{p|N} \psi(N/p) \cdot P(p/\Delta_p)^{\Delta_p}$, which simplifies when $K = N/\Delta$ is prime.
• When $K = 2$, $\mathtt{T}^{=}_{N,N/2} = \frac{1}{N} \sum_{q|N/2, \gcd(q,2)=1} \psi(q) \cdot 2^{N/(2q)}$, and when $K = 3$, $\mathtt{T}^{=}_{N,N/3} = \frac{1}{N} \sum_{q|N/3, \gcd(q,3)=1} \psi(q) \cdot 3^{N/(3q)}$, with $P(2)=2$ and $P(3)=3$.
• Computer simulations suggest that $\mathtt{T}^{=}\!(\ell,r)$ is maximized when $\ell = r$, and for $N$ not divisible by 3, it is maximized when $\Delta = N/2$; when $N$ is divisible by 3, it is maximized when $\Delta = N/3$, though these are open conjectures.
• The total number of attractors in a negative-negative dbac depends only on $N = \ell + r$ and $\Delta = \gcd(\ell, r)$, not on individual values of $\ell$ and $r$, as confirmed by simulations and theoretical formulas.