Circuit metaconstruction in logspace for Rice-like complexity lower bounds in ANs and SGRs

TL;DR

This paper addresses the problem of establishing Rice-like complexity lower bounds for properties of automata networks (ANs) and succinct graph representations (SGRs) by developing a logspace metareduction from SAT to circuits encoding the dynamics of q-uniform ANs. It extends prior metatheorems to the logspace setting, detailing an explicit circuit-construction that transforms a SAT instance into a q-uniform AN (or NAN) whose dynamics satisfies a fixed MSO formula ψ iff the SAT instance is satisfiable. The key result states that for any fixed q ≥ 2 and any q-non-trivial (resp. q-arborescent) MSO formula ψ, the problems ψ-q-AN-dynamics (resp. ψ-q-NAN-dynamics) are NP-hard or coNP-hard under logspace reductions, thereby achieving uniform, constant-size metaconstructions applicable to a broad class of dynamical systems. This work demonstrates how Rice-like complexity lower bounds can be realized in logspace and sets the stage for transferring these metatheorems to other computational models and graph-logic settings, highlighting potential extensions to different graph parameters and enriched signatures.

Abstract

A new proof technique combining finite model theory and dynamical systems has recently been introduced to obtain general complexity lower bounds on any question one may formulate on the dynamics (seen as a graph) of a given automata network (AN). ANs are abstract finite dynamical systems of interacting entities whose evolution rules are encoded as circuits, hence the study also applies to succinct graph representations (SGRs). In this article, we detail the construction of circuits to obtain general complexity lower bounds (metareduction) and show that the reduction is feasible in logarithmic space.

Figures (5)

• Figure 1: Example deterministic AN of size $n=2$ on alphabet $\llbracket q\rrbracket=\{0,1,2\}$ with $q=3$. Circuit of the function $F:\llbracket q\rrbracket^n \to \llbracket q\rrbracket^n$ (left) and transition digraph dynamics $\mathcal{G}_{F}$ on configuration space $\llbracket q\rrbracket^n$ (right). A configuration is encoded on four bits, where inputs $0000$ to $0001$ correspond respectively to configurations $00$ to $22$.If we write a configuration of this AN $x_1 x_2 \in \llbracket q\rrbracket^2$, a way to define the local functions is $F_1(x_1 x_2) = x_1$, $F_2(x_1 x_2)= (x_2 \mod 2) +1$.
• Figure 2: Notation for the ports of $G_j$ for $j\in\{0,1,4\}$.
• Figure 3: Allocation of the configurations of $F$ within the copies of $G_0, G_1, G_2, G_3, G_4$, for the purpose of constructing the dynamics $\mathcal{G}_{F}$ given by Equation \ref{['eq:dynaF']}. Recall that $G_{01}$ denotes $G_0$ or $G_1$ which have the same set of vertices (configurations). Observe that $P'_3$, which are both primary and secondary ports that are merged in all copies, appear at the very end of the configuration space of $F$.
• Figure 4: Circuit of $F$, computing $F(c) = c'$. For sake of clarity, some wires are not drawn, but are replaced with references in blue (for the wires coming from $c$ and $i$ especially). We label in red the output of some subcircuits, these notations will be used in their formal description.
• Figure 5: Circuit of $F$, computing $F(c, c') = b$. For clarity, some wires are not drawn, but are replaced with references in blue (relative to $c$), green (relative to $c'$) and red.

Theorems & Definitions (6)

• Theorem 1
• Example 1
• Remark 1
• Proposition 1
• Remark 2
• Remark 3