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known. For example, it was feasible to diagnose a patient in spite of the fact
that some test results were not taken (here attributes correspond to tests,
so attribute values are test results). Since such missing attribute values do
not matter for the final outcome, we will call them “do not care” conditions.
The third possibility is a partial “do not care” condition: we assume that the
missing attribute value belongs to the set of typical attribute values for all
cases from the same concept. Such a missing attribute value will be called an
attribute-concept value. Calling it concept “do not care” condition would be
perhaps better, but this name is too long.
The main objective of this chapter is to study incomplete decision tables,
assuming that in the same decision table some attribute values may be lost,
some may be “do not care” conditions, and some may be attribute-concept
values. Decision tables with lost values and “do not care” conditions were
studied in [7-9, 12].
For such incomplete decision tables there are three special cases: in the
first case all missing attribute values are lost, in the second case all missing
attribute values are “do not care” conditions, and in the third case all miss-
ing attribute vales are attribute-concept values. Incomplete decision tables
in which all attribute values are lost, from the viewpoint of rough set theory,
were studied for the first time in [13], where two algorithms for rule induction,
modified to handle lost attribute values, were presented. This approach was
studied later in [23-25], where the indiscernibility relation was generalized to
describe such incomplete decision tables.
On the other hand, incomplete decision tables in which all missing at-
tribute values are “do not care” conditions, again from the view point of rough
set theory, were studied for the first time in [4], where a method for rule induc-
tion was introduced in which each missing attribute value was replaced by all
values from the domain of the attribute. Originally such values were replaced
by all values from the entire domain of the attribute, later by attribute values
restricted to the same concept to which a case with a missing attribute value
belongs. Such incomplete decision tables, with all missing attribute values be-
ing “do not care conditions”, were extensively studied in [14, 15], including
extending the idea of the indiscernibility relation to describe such incomplete
decision tables.
Rough set methodology for incomplete decision tables with missing at-
tribute values of the type attribute-concept values is presented in this chapter
for the first time, though it was briefly mentioned in [9].
In general, incomplete decision tables are described by characteristic rela-
tions, in a similar way as complete decision tables are described by indiscerni-
bility relations [7].
For complete decision tables, once the indiscernibility relation is fixed and
the concept (a set of cases) is given, the lower and upper approximations are
unique.
For incomplete decision tables, for a given characteristic relation and the
concept, there are three different possible ways to define lower and upper
