Propositional Argumentation Systems and Symbolic Evidence Theory

7.2 Diagnostics in Digital Circuits 107
(a2 and a6 and a12) (a4 and a6 and a11)
? (dsp tuberculosis) ? (dsp bronchitis)
0.1924 0.5857
From symbolic support the doctor gets configurations of assumptions that allow
him to explain or deduce the hypothesis. In contrast, numerical degrees of support
help to judge and compare the results quantitatively. Here, bronchitis is almost
three times as credible as tuberculosis.
7.2 Diagnostics in Digital Circuits
Let S = {C 1 , . . . , C n } describe a technical system consisting of n different
components. As shown in Example 3.2 of Section 3.2, such systems can
easily be described using the formalism of symbolic hints (Kohlas et al.,
1995). In the case of digital circuits, i.e. if the system consists only of binary
variables, then it is also appropriate to use the formalism of assumption–
based systems. This section proposes a general method to model digital
circuits as assumption–based systems, including the case of multiple failure
modes. The aim is to find possible explanations for an incorrect system
behaviour.
If C is a component of a technical system S, then it determines the relation
between its input– and output–variables. Let I = {I 1 , . . . , I r } be the
set of input–variables and O = {O 1 , . . . , O s } the set of output–variables
connected with C. If S is a digital circuit, then all variables connected
with C are Boolean variables with two possible values 0 and 1. Let P =
{in 1 , . . . , in r , out 1 , . . . , out s } be the set of propositional symbols representing
the Boolean variables connected with C. in 1 , for example, stands for
I 1 = 1 and ∼in 1 for I 1 = 0. The behaviour of C can then be expressed
as logical formula m 0 ∈ L P . For example, if C has two inputs and one
output, and if C is a module producing the logical and of its inputs, then
m 0 = out 1 ↔ in 1 ∧ in 2 represents the correct behaviour of C called normal
mode.
A component C possibly may not work correctly. In such a case, we have
a so–called failure mode. A component may have different failure modes.
The component’s behaviour in failure mode i can also be expressed by a
logical formula m i ∈ L P . For example, let C again produce the logical and
of its inputs. A possible failure mode of C, in which always O 1 = 0 is