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It has been shown that the number of tests required to detect all faults in a one-dimensional unilateral combinational iterative array consisting of p cells will, in general, be proportional to p. In this paper we consider properties of such systems that enable them to be tested with a fixed constant number of tests independent of p, the number of cells in the system. Such systems are referred to as C-testable. Necessary and sufficient conditions on the basic cell state table are derived for an iterative system to be C-testable. It is shown that an arbitrary N-state cell table can be augmented by the addition of, at most, one row and less than [log2 N]2 columns (for N ≥ 2) so as to be C-testable.