A new dimension sensitive property for cellular automata

In this paper we study number-decreasing cellular automata. They form a super-class of standard number-conserving cellular automata. It is well-known that the property of being number-conserving is decidable in quasi-linear time. In this paper we prove that being number-decreasing is dimension sensitive, i.e. it is decidable for one-dimensional cellular automata and undecidable for dimension 2 or greater. There are only few known examples of dimension sensitive properties for cellular automata and this denotes some rich panel of phenomena in this class.