Unconventional complexity classes in unconventional computing (extended abstract)

TL;DR

The paper examines how deterministic unconventional computing models—specifically cellular automata and membrane systems—can realize intermediate complexity classes beyond $P$ and $PSPACE$ by tuning geometry and communication. It shows that hyperbolic CA can simulate alternating TM computations to achieve $PSPACE$ in polynomial time, while expanding/shrinking CA yield problems related to $NP$ through truth-table reductions. In membrane computing, division-enabled P systems reach $PSPACE$ in polynomial time, whereas one-way communication yields $P^{NP}$ and shallow/bidirectional constraints yield $P^{ obreak}P$ or $P^{#P}$, providing concrete models for these oracle-augmented classes. Overall, the work highlights how spatial structure and directed communication govern the computational power of deterministic unconventional models and invites broader generalization to other topologies and computation paradigms.

Abstract

Many unconventional computing models, including some that appear to be quite different from traditional ones such as Turing machines, happen to characterise either the complexity class P or PSPACE when working in deterministic polynomial time (and in the maximally parallel way, where this applies). We discuss variants of cellular automata and membrane systems that escape this dichotomy and characterise intermediate complexity classes, usually defined in terms of Turing machines with oracles, as well as some possible reasons why this happens.