An Expressive Coalgebraic Modal Logic for Cellular Automata

Cellular automata provide models of parallel computation based on cells, whose connectivity is given by an action of a monoid on the cells. At each step in the computation, every cell is decorated with a state that evolves in discrete steps according to a local update rule, which determines the next state of a cell based on its neighbour’s states. In this paper, we consider a coalgebraic view on cellular automata, which does not require typical restrictions, such as uniform neighbourhood connectivity and uniform local rules. Using the coalgebraic view, we devise a behavioural equivalence for cellular automata and a modal logic to reason about their behaviour. We then prove a Hennessy-Milner style theorem, which states that pairs of cells satisfy the same modal formulas exactly if they are identified under cellular behavioural equivalence.

💡 Research Summary

The paper presents a novel coalgebraic framework for modeling cellular automata (CA) that lifts many of the traditional restrictions such as uniform neighbourhoods and uniform local update rules. The authors start by observing that the spatial structure of a CA can be captured by the action of an arbitrary monoid M on a set of cells X. This action is encoded as a coalgebra c : X →