Rough Set - Advanced Artificial Intelligence - page 404

If θ 1 →Ψ, θ 2 →Ψ, ···, θ n →Ψ are all basic decision rule, θ 1 ∨θ 2 ∨···∨θ n →Ψ is
called as the composition of θ 1 →Ψ, θ 2 →Ψ,···, θ n →Ψ or compositional decision
rule for short.
To explore the value of PQ rule (consistent or inconsistent), we can use
following proposition.
Proposition 11.5 If and only if all {P ∨ Q}-basic formulas appear in {P ∨ Q}-

formal expressions of PQ rule's premise and consequence, the PQ-rule is true
(consistent) in S, otherwise false (inconsistent).

Any finite decision rule set in decision logic language is called as a decision
algorithm, and any finite decision rule is called as a basic decision algorithm.
If the decision rules in a basic decision algorithm are all PQ -decision rules,

the algorithm is called as PQ -decision algorithm or PQ -algorithm for short

(denoted as ( P , Q )). If and only if all decision rules in S are consistent, PQ

algorithm in S is consistent, otherwise inconsistent.

If for each x ∈U, there exists a PQ -rule θ→Ψ in an algorithm that x |=θ∧Ψ, the

PQ -algorithm is complete, otherwise incomplete.

Given a knowledge representation system, the non-empty subset of attribute P ,

Q can determine a PQ -decision algorithm uniquely. That is to say, PQ -algorithm

and PQ -decision table can be viewed as equivalent.

11.3.7 Inconsistent and Indiscernibility of Decision Rule

To test the consistency of a decision algorithm, we must consider the values of
all its decision rules. We can use proposition 11.5 to test it, while the following
proposition gives a simpler method.

For any PQ-decision rule θ 1 → ϕ 1 in a PQ-decision Proposition 11.6

algorithm, if and only if θ = θ 1 implicates ϕ = ϕ 1 , decision rule θ → ϕ in the

PQ decision algorithm is consistent in S.

Note that, to test the values of decision rule θ→Ψ, we must prove that the
premise of decision rule (formula θ) can distinguish the decision class Ψ with
other classes. That means the concept true can be replaced by the concept —