Complexity of Boolean automata networks under block-parallel update modes

TL;DR

This work analyzes Boolean automata networks under block-parallel update schedules $BP_n$, showing that almost all classical decision problems about the resulting dynamics become ${\mathsf{PSPACE}}$-complete, while certain global properties such as bijectivity stay in ${\mathsf{coNP}}$. The authors build a general reduction framework based on Iter-CVP to simulate long substep horizons within a single step, using prime-based gadgets to drive complexity, and they derive PSPACE-complete results for image, preimage, fixed-point, limit-cycle, reachability, and constant-dynamics questions. They also establish nuanced bounds: bijectivity remains ${\mathsf{coNP}}$-complete and identity recognition exhibits a mix of lower bounds, while many questions about subdynamics are ${\mathsf{PSPACE}}$-hard or complete. The results reveal a trade-off where increased dynamical expressivity under block-parallel updates comes at a high computational cost for analysis, informing both theoretical understanding and practical modeling of complex regulatory-like systems. The findings motivate further structural analysis of update-mode-induced dynamics and potential identification of tractable subclasses or approximate methods for specific applications.

Abstract

Boolean automata networks (aka Boolean networks) are space-time discrete dynamical systems, studied as a model of computation and as a representative model of natural phenomena. A collection of simple entities (the automata) update their 0-1 states according to local rules. The dynamics of the network is highly sensitive to update modes, i.e., to the schedule according to which the automata apply their local rule. A new family of update modes appeared recently, called block-parallel, which is dual to the well studied block-sequential. Although basic, it embeds the rich feature of update repetitions among a temporal updating period, allowing for atypical asymptotic behaviors. In this paper, we prove that it is able to breed complex computations, squashing almost all decision problems on the dynamics to the traditionally highest (for reachability questions) class PSPACE. Despite obtaining these complexity bounds for a broad set of local and global properties, we also highlight a surprising gap: bijectivity is still coNP.

Figures (6)

• 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 configurtion space $\mathbb{B}^3$ (right). 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: 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 3: 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 4: Example graphs of out-degree at most one.
• 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 (32)

• Theorem 1: folklore
• proof
• Lemma 2
• proof
• Definition 3
• Lemma 4
• proof
• Example 5
• Theorem 6
• proof