The concept of pseudoexhaustive testing is combined with that of qualifier preselection to provide easily testable designs of digital systems based on the algorithmic state machine (ASM) method. The highly structured implementations resulting from direct synthesis procedures with modern programmable devices are augmented by the incorporation of a serial shadow state register to provide built-in testability. A basic general test procedure is outlined and some refinements for particular ASMs are discussed. These can be incorporated at little extra cost or loss of generality.