(Un)decidability of injectivity and surjectivity in one-dimensional sand automata

Abstract

Extension of sand pile models, one-dimensional sand automata are an intermediate discrete dynamical system between one dimensional cellular automata and two-dimensional cellular automata. In this paper, we shall study the decidability problem of global behavior of this system. In particular, we shall focus on the problem of injectivity and surjectivity which have the property of being decidable for one-dimensional cellular automata and undecidable for two-dimensional one. We prove the following quite surprising property that surjectivity is undecidable whereas injectivity is decidable. For completeness, we also study these properties on some classical restrictions of configurations (finite, periodic and bounded ones). © 2009 Springer Berlin Heidelberg.