One­way and two­way cellular language acceptors with restricted non­ determinism are investigated. The number of nondeterministic state transitions is regarded as limited resource which depends on the length of the input. We center our attention to real­time, linear­time and unrestricted­time computations. A speed­up result that allows any ... linear­time computation to be sped­up to real­time is proved. The relationships to deterministic arrays are considered. For an important subclass a characterization in terms of deterministic language families and [Epsilon]­free homomorphisms is given. Finally we prove strong closure properties of languages acceptable with a constant number of nondeterministic transitions.