Computational Complexity of Predicting Periodicity in the Models of Lorentz Lattice Gas Cellular Automata

Some diffusive and recurrence properties of Lorentz Lattice Gas Cellular Automata (LLGCA) have been expensively studied in terms of the densities of some of the left/right static/flipping mirrors/rotators. In this paper, for any combination S of these well known scatters, we study the computational complexity of the following problem which we call PERIODICITY on the S-model: given a finite configuration that distributes only those scatters in S, whether a particle visits the starting position periodically or not. Previously, the flipping mirror model and the occupied flipping rotator model have been shown unbounded, i.e. the process is always diffusive [17]. On the other hand, PERIODICITY is shown PSPACEcomplete in the unoccupied flipping rotator model [21]. In this paper, we show that PERIODICITY is PSPACE-compete in any S-model that is neither occupied, unbounded, nor static. Particularly, we prove that PERIODICITY in any unoccupied and bounded model containing flipping mirror is PSPACEcomplete.