A new measure of system diagnosis, t/s diagnosability, originally proposed by Friedman, is used to study the diagnosability of digital systems. This new measure incorporates the concept of possible replacement of fault-free units in system repair, whereas the previous measures have only considered the replacement of faulty units. Two categories of the new measure, one-step t/s diagnosability and sequential t/s diagnosability, are investigated. Two canonical classes of systems, single loop systems and DδAsystems (⌊t/2⌋ < A ≤ t), are examined based on these two categories of diagnosability. For each system class, the necessary and sufficient condition for one-step ts diagnosability is obtained and related to some previous results on t-fault diagnosability; also, an efficient one-step t/s repair procedure is presented. For both these system classes, optimal one-step t/s diagnosable system designs, which minimize the number of units and test links, are considered. Several sequential diagnosis strategies are presented for each system class. For all these diagnosis strategies optimal system designs are also considered. Finally, all the diagnosis strategies are compared and the tradeoff between the number of units replaced and the number of test iterations performed is discussed.