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Knowledge set and knowledge base

Let L be any language. We will make the notion of the totality of one's knowledge explicit by the following definitions:

Definition. tex2html_wrap_inline574 is a knowledge set for St if K satisfies the following conditions:

(KS1) K is consistent.

(KS2) tex2html_wrap_inline581, where tex2html_wrap_inline584.

(KS3) If tex2html_wrap_inline586 then tex2html_wrap_inline588 for some tex2html_wrap_inline608.

Definition. tex2html_wrap_inline592 is a knowledge base for St if B satisfies the following conditions:

(KB1) B is consistent.

(KB2) tex2html_wrap_inline600, where tex2html_wrap_inline602,

(KB3) If tex2html_wrap_inline604 then tex2html_wrap_inline606 for some tex2html_wrap_inline608.

By (KS2) (or (KB2)) we see that any element in K (or B, esp.) has the form tex2html_wrap_inline618. It is easy to see that if B is a knowledge base for St then tex2html_wrap_inline624 is a knowledge set for St.

Let tex2html_wrap_inline450 be consistent. We compare the following three conditions.

  1. If tex2html_wrap_inline630 then tex2html_wrap_inline632.
  2. If tex2html_wrap_inline634 then tex2html_wrap_inline636 for some tex2html_wrap_inline608.
  3. If tex2html_wrap_inline640 then tex2html_wrap_inline452 or tex2html_wrap_inline644.

Then we have the following

Lemma 5. (1), (2) and (3) are equivalent.

We now study the semantical characterization of knowledge sets. Let tex2html_wrap_inline646 be any KT5-model. For any tex2html_wrap_inline648 and tex2html_wrap_inline650, we define tex2html_wrap_inline651 by:
Since, as we will see below, tex2html_wrap_inline654 is a knowledge set for St, we call it the knowledge set for St at w.

Lemma 6. tex2html_wrap_inline654 is a knowledge set for St.

Let K be a knowledge set for St. We say tex2html_wrap_inline670 characterizes K if tex2html_wrap_inline674.

Theorem 7. Any knowledge set is characterizable.

Yasuko Kitajima
Fri Jun 20 13:39:43 PDT 1997