Method of synthesis of easily testable circuits admitting single fault detection tests of constant length

Abstract

It is constructively proved that any nonconstant Boolean function may be realized (over an arbitrary complete basis of gates) by a testable combinational circuit admitting (under single inverse or arbitrary constant faults at outputs of gates) a single fault detection test set whose cardinality does not exceed 4.Keywords: circuit of functional elements, single fault detection test, constant fault at the output of gate, Shannon function, length of a test, easily testable circuit.

Lower bounds for lengths of single tests for Boolean circuits

“…Since the functions , 0, 1 are distinct, the test must contain a tuplẽ1 such that (̃1) = and a tuplẽ2 such that (̃2) = . Note that (̃2) = , (7) since if (̃2) = then the stuck-at fault of the type at the output of the gate can not be detected on the input̃2 and = (̃2) = (̃2) = , which is impossible. If (̃1) = then the stuck-at fault of the type of the gate can not be detected using inputs̃1,̃2, thus the test must contain one more tuple, hence | | ⩾ 3, as it was required to prove.…”

“…The result from [5] was further generalized by S. S. Kolyada for the case of an arbitrary functionally complete finite basis ( [6]). Kolyada's result was improved by D. S. Romanov who proved in [7] that for any functionally complete basis it holds that ; 0,1 , ( ) ⩽ 4 (however the paper cited used a different definition of an irredundant circuit). For the case of complete fault detection tests N. P. Redkin in [8,9] showed that for any complete finite basis 2 it holds that…”