ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 302–312.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
K.A. Popkov, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 62–81.
Complete Fault Detection Tests of Length 2 for Logic Networks
under Stuck-at Faults of Gates
K. A. Popkov
Keldysh Institute of Applied Mathematics, Miusskaya pl. 4, Moscow, 125047 Russia
Received October 6, 2017
Abstract—We consider the problem of the synthesis of the logic networks implementing Boolean
functions of n variables and allowing short complete fault detection tests regardingarbitrary stuck-at
faults at the outputs of gates. We prove that there exists a basis consisting of two Boolean functions
of at most four variables in which we can implement each Boolean function by a network allowing
such a test with length at most 2.
Keywords: logic network, stuck-at fault, complete fault detection test
In this article, the problem is considered of the synthesis of the easily tested circuits implementing
given Boolean functions. A logical approach to testing the design of electric circuits was proposed by
Yablonskii and Chegis . This approach is also applicable for testing logic networks (see [2, 3, 4]).
Let there be a logic network S with one output implementing the Boolean function f(x
).Undertheinﬂuence of some source of faults one or more gates of S cangotosome
faulty state. In result, instead of the original function f (x
) the network S implements some Boolean
) that is, in general, distinct from f. Every such function g(x
) obtained for all possible
admissible faults of the gates of a network S is called a fault function of S.
We introduce the following deﬁnitions [20, 21, 13]: The fault detection test for a network S is
understood to be a set T of the value vectors of the variables x
such that, for every fault function
) of S distinct from f(x
), there is a vector ˜σ in T on which f (˜σ) = g(˜σ). We refer as the diagnostic
test for S to the set T of the value vectors of the variables x
such that T is the fault detection
test and, moreover, for every two diﬀerent fault functions g
) and g
) of S thereissomevector˜σ
in T on which g
(˜σ) = g
(˜σ). The number of the value vectors in T is called the length of T .Asatrivial
diagnostic (and fault detection) test of length 2
for S we can always take the set of all binary vectors of
length n. A test is called complete if however many gates in a network S can be faulty, and single if only
onegateinS can be faulty. Usually the single tests are considered for the irredundant networks ;
i.e. for the networks such that every admissible fault of an arbitrary gate leads to a fault function that is
distinct from the original function implemented by the given network.
We will call each set of Boolean functions a (schematic) basis.
Let us ﬁx the type of faults of gates, let B be an arbitrary functionally complete basis, and let T be
a complete fault detection test for some logic network S in the basis B. We now introduce the notation:
(T ) denote the length of T , while
where the minimum is taken over all complete fault detection tests T for S;put