Lower Bounds on Inverse Cellular Automata via Proof Complexity

TL;DR

This paper investigates the computational complexity of inverse cellular automata, providing simpler proofs of co-NP-completeness and establishing lower bounds on their proof size using propositional proof complexity techniques.

Contribution

It offers a simplified proof of the co-NP-completeness of inverse cellular automata injectivity and derives proof complexity lower bounds via propositional proof systems.

Findings

Deciding injectivity is co-NP-complete.

A simpler reduction from UNSAT is provided.

Lower bounds on proof size are established using proof complexity methods.

Abstract

We study the complexity of inverse cellular automata on configurations of bounded size. Deciding injectivity in this setting is co-NP-complete by a theorem of Durand. We give a simpler proof of this theorem by a direct reduction from UNSAT to this problem, avoiding more complicated intermediate constructions. We also show that one direction of the reduction can be formalized in the weak theory of bounded arithmetic $V^0$. Durand's coNP-completeness result allows one to view inverse cellular automata acting on bounded size configurations as propositional proofs, cf. Cavagnetto, and we prove lower bounds on their size. The proof uses known lower bounds for bounded-depth Frege systems together with the Paris--Wilkie translation of arithmetic proofs into propositional proofs, which allows us to transfer proof complexity lower bounds to our setting.