Atomic Gliders and CA as Language Generators (Extended Version)
Dana Fisman, Noa Izsak
TL;DR
This work treats cellular automata as genuine language-generating systems on unbounded bi-infinite grids, starting from regular initial configurations and evolving under a fixed-radius local rule. It introduces a glider-based generative semantics in which atomic, velocity-bearing gliders interact under a dominance relation to produce complex, structured languages, including non-regular and context-sensitive forms. The paper proves that even with regular initializations, CA can realize families such as $L=igl\{w_1^{e_1(n)}\cdots w_m^{e_m(n)}\mid n\in\mathbb{N}\bigr\}$ with linear exponents and shows a strict separation between glider-expressible languages and the broader CA-expressible class. It also connects these constructions to regular model checking and outlines implications for formal verification of linearly ordered multi-agent systems. Overall, the glider-centric, symbolic perspective provides a modular, verifiable framework to reason about unbounded CA dynamics and their language-theoretic properties.
Abstract
Cellular automata (CA) are well-studied models of decentralized parallel computation, known for their ability to exhibit complex global behavior from simple local rules. While their dynamics have been widely explored through simulations, a formal treatment of CA as genuine language generators remains underdeveloped. We formalize CA-expressible languages as sets of finite words obtained by projecting the non-quiescent segments of configurations reachable by one-dimensional, deterministic, synchronous CA over bi-infinite grids. These languages are defined with respect to sets of initial configurations specified by a regular language as in regular model checking. To capture structured dynamics, we propose a glider-based generative semantics for CA. Inspired by the classical notion of gliders, we define a glider as a one-cell entity carrying a symbol in a certain velocity under well defined interaction semantics. We show that despite the regularity of the initial configurations and the locality of the transition rules, the resulting languages can exhibit non-regular and even non-context-free structure. This positions regular-initialized CA languages as a surprisingly rich computational model, with potential applications in the formal analysis of linearly ordered MAS.