🤖 AI Summary
Determining dynamical properties of automata networks (AN) and succinct graph representations (SGR) suffers from high computational complexity, yet tight lower bounds—especially for succinct systems—remain largely unestablished.
Method: We introduce the first general circuit gadget constructible in deterministic logarithmic space (O(log n)), integrating finite model theory with dynamical systems theory to build a logspace meta-reduction framework.
Contribution/Results: We establish the first universal Rice-type undecidability lower bound for succinct dynamical systems. Specifically, we prove that deciding any nontrivial dynamical property over AN or SGR is NL-hard (and thus at least as hard as nondeterministic logspace), with reductions provably constrained to deterministic logspace. This yields the first tight, unified lower-bound tool for classifying the complexity of succinct dynamical systems, resolving a long-standing gap in succinct computation theory.
📝 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.