In a recent paper Sutner proved that the first-order theory of the phase-space SA=(QZ,⟶) of a one-dimensional cellular automaton A whose configurations are elements of QZ
Nearby in the stack
, for a finite set of states
Q
, and where
⟶
is the "next configuration relation", is decidable. He asked whether this result could be extended to a more expressive logic. We prove in this paper that this is actuallly the case. We first show that, for each one-dimensional cellular automaton
A
, the phase-space
SA
is an omega-automatic structure. Then, applying recent results of Kuske and Lohrey on omega-automatic structures, it follows that the first-order theory, extended with some counting and cardinality quantifiers, of the structure
SA
, is decidable. We give some examples of new decidable properties for one-dimensional cellular automata. In the case of surjective cellular automata, some more efficient algorithms can be deduced from results of Kuske and Lohrey on structures of bounded degree. On the other hand we show that the case of cellular automata give new results on automatic graphs.