Foundations of block-parallel automata networks

Abstract

We settle the theoretical ground for the study of automata networks under block-parallel update schedules, which are somehow dual to the block-sequential ones, but allow for repetitions of automaton updates. This gain in expressivity brings new challenges, and we analyse natural equivalence classes of update schedules: those leading to the same dynamics, and to the same limit dynamics, for any automata network. Countings and enumeration algorithms are provided, for their numerical study. We also prove computational complexity bounds for many classical problems, involving fixed points, limit cycles, the recognition of subdynamics, reachability, etc. The PSPACE-completeness of computing the image of a single configuration lifts the complexity of most problems, but the landscape keeps some relief, in particular for reversible computations.

Figures (5)

• Figure 1: Example of an automata network of size $n=3$ with a block-parallel update mode $\mu\in\mathsf{BP}_n$. Local functions (upper left), conversion of $\mu$ to a sequence of blocks (lower left), and dynamics of ${f}_{\{\mu\}}$ on configuration space $\mathbb{B}^3$ (right). One step is composed of two substeps: the first substep updates the block $\{0,1\}$, the second substep updates the block $\{0,2\}$. As an example, in computing the image of configuration $\mathtt{1}\mathtt{1}\mathtt{1}$, the first substep (update of automata $0$ and $1$) gives $\mathtt{1}\mathtt{0}\mathtt{1}$, and the second substep (update of automata $0$ and $2$) gives $\mathtt{0}\mathtt{0}\mathtt{1}$.
• Figure 2: Numerical experiments of our Python implementation of the three algorithms on a standard laptop (processor Intel-Core$^\text{TM}$ i7 @ 2.80 GHz). For $n$ from $1$ to $12$, the table (left) presents the size of $\mathsf{BP}_n$, $\mathsf{BP}_n^0$ and $\mathsf{BP}_n^*$ and running time to enumerate their elements (one representative of each equivalence class; a dash represents a time smaller than $0.1$ second), and the graphics (right) depicts their respective sizes on a logarithmic scale. Observe that the sizes of $\mathsf{BP}_n$ and $\mathsf{BP}_n^0$ are comparable, whereas an order of magnitude is gained with $\mathsf{BP}_n^*$, which may be significant for advanced numerical experiments regarding limit dynamics under block-parallel udpate modes.
• Figure 3: Substeps leading to the image of configuration $\mathtt{0}^{q_{k_n}}\mathtt{0}\mathtt{1}\mathtt{0}$ in ${f}_{\{\mu'\}}$ from Example \ref{['ex:counter']} for $n=3$ ($k_n=4$ and $q_{k_n}=2+3+5+7=17$). The last $3$ bits implement a binary counter, freezing at $7$ ($\mathtt{1}\mathtt{1}\mathtt{1}$). Above each substep the block of updated automata is given.
• Figure 4: Construction of $g$ in the proof of Theorem \ref{['thm:subdynamics']}. Subspace $x_n=\mathtt{0}$ contains a copy of $f$ with a potential limit cycle dashed. Subspace $x_n=\mathtt{1}$ implements $G'$, and wires configurations of $U$ (grey area) to the potential limit cycle in the copy of $f$ (remaining configurations are fixed points).
• Figure 5: Illustration of the dynamics obtained for the reduction to BP-Constant in the proof of Theorem \ref{['thm:cst']}. Configurations $x$ with the counter automata $B$ initialized to $x_B=0$ either go to $\mathtt{0}^{q_{k_n}}\mathtt{1}^{\ell'}\mathtt{0}^n\mathtt{1}$ (left, positive instance), or to $\mathtt{0}^{q_{k_n}}\mathtt{1}^{\ell'}\mathtt{0}^n\mathtt{0}$ (right, negative instance). Only the bit of automata $R$ changes.

Theorems & Definitions (52)

• Lemma 1
• proof
• Corollary 1
• Proposition 1
• proof
• Proposition 2
• proof
• Definition 1
• Lemma 2
• proof