Energy models for delay testing

We present a new formulation of the delay testing problem as an energy minimization problem. Two important applications have motivated this work. First, it can be used to efficiently generate robust and nonrobust tests for path delay faults in scan and hold type of sequential circuits. Second, It allows the design of a special class of delay fault testable circuits, called (k,K)-circuits, that have polynomial-time test generation complexity. For the new formulation, the relationship between input and output signal states of a logic gate for an arbitrary pair of input vectors is expressed through an energy function. The minimum-energy states of this function correspond to signal values that are consistent with the gate´s logic function. The function also implicitly includes the information about the potential hazards due to arbitrary delay distributions in the circuit. The energy function for the circuit is the summation of the individual gate energy functions. To derive tests for a given delay fault, this function is suitably modified such that any minimum-energy state is guaranteed to be a test. The specific modifications to the energy function depend on the type (robust or nonrobust, with or without hazards) of delay test desired. For (k, K)-circuits, we show that the energy function can be minimized in polynomial-time. For general circuits, where the problem still has an exponential complexity, the recently proposed transitive closure based test generation technique is very effective in generating tests. This approach efficiently determines a delay test or establishes that no test is possible for the given delay fault. We report experimental results on various sequential benchmark circuits (full-scan versions) showing the feasibility and practicality of the new methods