Particle Complexity of Universal Finite Number-Conserving Cellular Automata

A number-conserving cellular automaton (NCCA) is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of particle-based modeling of the physical conservation law of mass. In this paper we focus on the case we call finite NCCA when states are non-negative integers, and the total sum is finite. In spite of the strong constraint, we constructed a radius 1 universal FNCCA by simulating register machines with two registers. We also consider the particle complexity in the case of large (but finite) radius, and constructed a universal FNCCA with only five particles.