[Subtitle: Non-deterministic information]

Dear Dominik,

You seem to know the background of NISs completely. Your comment is correct.

We need to define
(1) Consistency-based rules in NISs (An extension of rough sets concept)
(2) Criterion-based rules in NISs (NISapriori, an extension of Apriori related rules)
    (alpha,beta)-lower system, (alpha,beta)-upper system

Then, DEF-GC is defined by (0,1)-lower system, and possible rules are defined by (0,1)-upper system.
Therefore, NISapriori will be enough for handling two kinds of rules.

In Japan, we have spring holidays from 2nd to 6th in May. I am off now, so I will report the details of the above after the holidays.

Thank you.

Regards,
Hiroshi

[Subtitle: Non-deterministic information]
Background (No.4) A Discernibility Function and Consistency based Rules in NISs

Dear Dominik and everyone,

I would continue to show the background of NISs and rule generation. Consistency based rules defined below in NISs will be extensions of rules in DISs.

F is a certain rule:
(1) F is (DEF-DEF) type implication.
(2) F is consistent in all derived DISs.

F is a possible rule:
(1) F is an implication.
(2) F is consistent in some derived DISs.

In DISs, we often employ a discernibility matrix and a discernibility function proposed by Professor A.Skowron. We can adjust such a matrix and a function to NISs. In the attached file, we show the real execution of flu2.csv in the previous post. This program was implemented in prolog, and it takes the next steps.

(1) For the decision attribute value, a set of decision class is fixed.
(2) For this class, an extended discernibility function is internally generated.
(3) The minimal solutions of this function are obtained as the condition parts of the decision attribute value.If the solution is not unique, we add some attribute values, and we interactively obtain the condition parts.

In the next post, I will explain that NISapriori can also generate the same results.
Thank you.

Regards,
Hiroshi

[References]
(1) A.Skowron and C.Rauszer: The Discernibility Matrices and Functions in Information Systems, In Intelligent Decision Support - Handbook of Advances and Applications of the Rough Set Theory, Kluwer Academic Publishers, pp.331-362, 1992.
(2) Hiroshi Sakai, Michinori Nakata: Discernibility Functions and Minimal Rules in Non-deterministic Information Systems.  RSFDGrC (1) 2005: 254-264
(3) Hiroshi Sakai, Michinori Nakata: An Application of Discernibility Functions to Generating Minimal Rules in Non-Deterministic Information Systems. JACIII 10(5): 695-702