Abstraction of Prevention Conceived in Distributed Knowledge Base

Susumu Yamasaki

1 a

and Mariko Sasakura

2 b

Okayama University, Tsushima-Naka, Okayama, Japan

Department of Computer Science, Okayama University, Tsushima-Naka, Okayama, Japan

Keywords:

Knowledge Acquisition, Algebraic Approach, Distributed System.

Abstract:

This paper is concerned with prevented information common in distributed knowledge base. The distributed

knowledge is a framework to reﬂect knowledge acquisition and retrieval in a seminar class with participants.

The presented accounts are listened to, by participants such that the participants may acquire knowledge.

To make the acquired knowledge formal, it is restricted to rule-type and assumed as logically causal or al-

gebraically structural. For such formalized knowledge, the meaning of knowledge may be considered by a

consistent ﬁxed point of the mapping associated with knowledge rules. The ﬁxed point approach has com-

plexity if the mapping is nonmonotonic (not monotonic), because the ﬁxed point is not always available for the

mapping. In this paper, we face such complexity caused by the nonmonotonic mapping which is reasonable

from interactions between positive and negative informations in acquiring knowledge. With the assumption of

some consistent ﬁxed point existence, retrieval derivation to the ruled knowledge is inductively designed such

that it may be sound with respect to the ﬁxed point meaning of knowledge. The derivation adopts negation by

failure, making adjustments between succeeding and failing cases, inherently involved in acquired knowledge.

As regards the whole system in accordance with the seminar class, distributed knowledge base is formalized

by taking a direct product of knowledge regarded as acquired by each participant. To make complexity of

mutually interactive communications simpler and to possibly formulate common prevention conceived in the

distributed knowledge base, we present a mapping associated with distributed knowledge base containing

common negatives. Then the meaning of the whole system with common prevention information may be the-

oretically deﬁned by a consistent ﬁxed point of a mapping. A consistent ﬁxed point of the mapping does not

always exist, however, if it is deﬁnable, retrieval derivation in distributed knowledge base may be designed

with an oracle of assumed prevention information in such a distributed knowledge base.

1 INTRODUCTION

A community may be regarded as containing dis-

tributed knowledge base which is formed with knowl-

edge of community members. Then some common

knowledge can be abstracted, although formulation

of knowledge arrangements is complex as a process.

This paper aims at the formation of common knowl-

edge, especially, preventive negative. This is moti-

vated as a problem in a seminar circle which is empir-

ical in the virtual reality by M. Sasakura, et al. (2021).

The treatment of common negative may involve ap-

plicative and theoretical interests, while positive in-

formation can be enjoyed individually.

As regards knowledge with logic, there have been

advanced methodologies.

https://orcid.org/0000-0001-7895-5040

https://orcid.org/0000-0002-8909-9072

(a) Nonmonotonic logic on reasoning and inference

mechanism is now rather classical but should be still

worthwhile examining (Hanks and McDermott, 1987;

Reiter, 2001).

(b) With respect to modal logic which may be tools

for epistemic aspects, ﬁrst-order modal logic is for-

mally dealt with (Fitting, 2002), and second-order

abstraction is considered for modality such that am-

biguity of knowledge may be discussed by B. Kooi

(2016). Quantiﬁers over epistemic agents are intro-

duced (Naumov and Tao, 2019), as well.

logic is discussed (Rin and S.Walsh, 2016). Topolog-

ical space may be taken for modality (Goldblatt and

Hodkinson, 2020).

(d) For stepwise removal of objects, dynamic modal-

ity is presented (Bentham et al., 2022). Dynamic logic

for action was presented by L. Spalazzi et al. (2000).

Justiﬁed belief truth is discussed (Egre et al., 2021).

Yamasaki, S. and Sasakura, M.

Abstraction of Prevention Conceived in Distributed Knowledge Base.

DOI: 10.5220/0011708600003485

In Proceedings of the 8th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2023), pages 39-46

ISBN: 978-989-758-644-6; ISSN: 2184-5034

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

(e) Reduction of decidability is compiled (Rasga

et al., 2021) and inference theory is formulated (Ten-

nant, 2021).

Concerning knowledge representation, this paper

considers a rule set as knowledge, where each rule

can be given by some algebraic expression reﬂecting

logical implication or causal relation, based on alge-

bra. The rule set causes nonmonotonic acquiring of

knowledge. In this sense, the classical nonmonotonic

logic is most relevant to the present paper, based on

algebraic expressions. Some common prevention in

distributed knowledge base (each of which is formed

by algebraic expressions) causes complexity, to be de-

tected.

For the behaviours of distributed systems, com-

munication theories are embedded with the ideas of

rewriting process and process algebra, as well as of

sequence generations:

(a) Rewriting process with reductions for calculus

is formulated by functionalism as in the work (C.

Bertolissi et al., 2006).

(b) Mobile ambients are dealt with in process of

behaviours (Cardelli and Gordon, 2000; Merro and

Nardelli, 2005).

tem may be covered by the method of automata (M.

Droste et al., 2009).

Among the above processes, the presentation of

the seminar (referred to, in this paper) is regarded as

the sequence generated in the above third method.

The seminar presentation of this paper contains a

stream of photo images, which is related to the se-

quence generated by traversing states. On the other

hand, the distributed system of knowledge is captured

in a static sense. Just common, negative (as a meet

of reports from the participants) can be examined at

each site of participants.

This paper then makes theoretical analyses and

designs on:

(a) knowledge representation abstracted from the

seminar participants,

(b) meaning of acquired knowledge and retrieval, and

tributed knowledge system containing common nega-

tive.

The paper is organized as follows. Section 2

abstracts a theoretical method of sharing prevention

knowledge, from the seminar in a virtual reality space.

In Section 3, a rule set is formally given as algebraic

expressions whose meaning may be deﬁned by ﬁxed

point of a mapping associated with the rule set. A re-

trieval derivation for knowledge is detection and ex-

amination of the meaning which knowledge denotes.

In Section 4, a distributed knowledge with common

negative is dealt with, by expanding individual knowl-

edge base and derivation. Concluding remarks are

given in Section 5.

2 KNOWLEDGE ACQUISITION

AND PROCESSING

The seminar (which we have implemented in virtual

reality) conceives some knowledge processing prob-

lem. As in lecture and seminar, participants (who

listen to the presenter’s verbal accounts) may be re-

garded as acquiring knowledge. The class consists of

images through video camera, to be organized as the

streams to the seminar participants. After reviewing

the presentation, the participants deliver reports to be

gathered into a common virtual space.

Common Prevention in Distributed Knowledge

Base

There is a motivated problem of this paper: After

reporting, can the seminar participants hold a com-

mon knowledge? As common knowledge, preventive

notices (which the seminar presents and participants

would keep) are the subject of this paper. The dis-

tributed system is in general complex, owing to nega-

tives: The more positive and negative knowledge are

acquired, the more complicated the system is. We

here assume that:

(a) Each participant organizes his/her understanding

as knowledge base, with respect to the presentation

by verbal accounts in seminar.

(b) Each participant can manage knowledge consist-

ing of rules, where each rule is constructed in terms

of positive and negative informations deriving conclu-

sion.

ined, to be common in the distributed system, where

each knowledge may derive inferences, as the partici-

pant thinks of, after the seminar presentation.

(d) The knowledge acquisition (to be virtually imple-

mentable in the seminar) is schematized with human

interaction.

That is, the verbal accounts or explanations are

to be presented such that the participants can acquire

knowledge of rule sets, where the participants sup-

posedly acknowledge algebraic structure of knowl-

edge. Among acquired knowledge, common preven-

tion may be inferred. This scheme is arranged as in

Table 1. On the common prevention, algebraic analy-

sis is made for retrieval derivation to be designed.

Rule Set as Knowledge

The rule formation comes from logical or algebraic

COMPLEXIS 2023 - 8th International Conference on Complexity, Future Information Systems and Risk

Table 1: The acquired knowledge is formalized.

Presenter =⇒ Participants

Verbal accounts Acquired Rules

Examined ⇐= Negatives

Alert Communicated Prevention

property to understand knowledge. As regards pos-

itive and negative informations, theories on state

spaces are given (Hornicher, 2021). Merge of dis-

tributed knowledge is formulated (Christoff et al.,

2022). In this paper, common negative in a distributed

system is paid attention to, conceived from the semi-

nar problem so that the negative of logical or algebraic

basis may be shared.

A rule is of the form r, in a formal way (which

would be clearer in Section 3), represented by posi-

tive and negative information to derive a conclusion,

which can be taken as knowledge acquisition:

r = (Pos, Neg, con) where

Pos, Neg ⊆ A (a domain)

Pos ::= {a}∪ Pos

Neg ::= {a

∗

} ∪ Neg

∗

stands for negative operation on A)

con ::= a or a

∗

(for a ∈ A)

The form contains recursion of representation in

Backus-Naur Form. If we see the structure as logi-

cal, we may interpret the knowledge as follows:

(a) If Pos = {a

, . . . , a

}, Pos denotes a

∧ . . . ∧ a

with conjunction ∧.

(b) If Neg = {b

∗

, . . . , b

∗

}, Neg denotes b

∗

∧ . . . ∧ b

∗

“⇒” (implication).

Then a set R of rules (where the rules are mutually

conjunctive) is assumed as knowledge (which each

participant acquires after the seminar). As auxiliary

means, a subset of the set {a

∗

| a ∈ A} may be re-

ported to the presenter. It is considerably common

negative information.

We analyze the meaning of knowledge and a dis-

tributed knowledge system with common negative.

Then we show some way to see the meaning and to

perform retrieval of the knowledge.

Although the following treatment on knowledge

as rule set is algebraic, the rule set is interpreted log-

ically as a conjunction of rules which are in accor-

dance to logical implications with premises and con-

clusion in 3-valued intuitionistic logic.

3 KNOWLEDGE

ORGANIZATION

In this section, a rule set is formally described. Each

rule reﬂects algebraic, logical or causal relation such

that a set of rules is regarded as knowledge.

3.1 A Rule Set

We ﬁrstly assume a domain A to construct a set of

knowledgeable objects. We then make use of:

∗

= {a

∗

| a ∈ A}

where a

∗

is to be interpreted as negative of a.

A rule is a triplet of the form (Pos, Neg, con)

where:

(a) Pos ⊆ A,

(b) Neg ⊆ A

∗

and

∗

for a ∈ A.

The expression (Pos, Neg, a) or (Pos

, Neg

, a

∗

)

may be adopted, if the form con (of the third item

in the rule triplet) is a ∈ A or a

∗

∈ A

∗

, respectively. A

name R is used to refer to a set of such triplets (a rule

set).

Example 1. We illustrate a set R of rules:

(i) ({revised}, { f ashioned

∗

}, evolved)

(ii) ({ f ashioned},

0, ruled)

(iii) (

0, {ruled

∗

}, revised)

Many-valued logic provability is presented

(Pawlowski and Urbaniak, 2018). From practical

views, answer set programming and problem solving

are compiled in 2-valued logic (Gebser and Schaub,

2016; Kaufmann et al., 2016).

Apart from answer set programming, 3-valued

evaluation is studied with the unknown (the unde-

ﬁned), for preventive negative to be paid attention

to. The 3-valued evaluation applicable to algebraic

expressions contains originality with respect to

complexity for both rule sets and retrieval. As a

3-valued evaluation, we adopt the following algebra:

Base Algebra of 3-Valued Domain

A bounded lattice K = ({ f , unk, t},

, ⊥, >)

equipped with the partial order v and an implication

⇒ may be taken:

(a) ⊥ and > are the least and the greatest elements f

and t, respectively, with respect to the partial order v

such that ⊥ = f v unk v t = >.

(b) The least upper bound ( join)

and the greatest

lower bound (meet)

exist for any two elements of

the set { f , unk, t}.

Abstraction of Prevention Conceived in Distributed Knowledge Base

in a way that z v (x ⇒ y) iff x

z v y.

t is the truth, f the falsehood and unk (the

unknown as t or f ) is the undeﬁned for the truth

value, where the partial order is deﬁned, regarding

the truth value.

Evaluation of Rule Set and Model

A valuation V : A → { f , unk, t} is assumed with the

bounded lattice K. With respect to V , the value

eval

(E) of an expression E is given.

eval

(a) = V (a) (a ∈ A)

eval

∗

)

= if eval

(a) = f then t else f (a

∗

∈ A

∗

)

eval

(con)

= if con = a then eval

(a)

else if con = a

∗

then eval

∗

)

eval

(Pos) =

{V (a) | a ∈ Pos}

eval

(Neg) =

{V (a

∗

) | a

∗

∈ Neg}

eval

((Pos, Neg, con))

= if eval

(Pos) v eval

(con)

or eval

(Neg) v eval

(con) then t

else if eval

(con) = f then f else unk

eval

(R) =

(Pos,Neg,con)∈R

eval

((Pos, Neg, con))

For a pair (I, J) ∈ 2

× 2

such that I ∩ J =

0, i.e.,

I and J are disjoint, the valuation

V (I, J) : A → { f , unk, t}

is deﬁned to be:

V (I, J)(a)

= if a ∈ I then t else if a ∈ J then f

else unk

If the disjoint pair (I, J) (I ∩ J =

0) causes

eval

V (I,J)

(R) = t, the pair is called a model of the rule

set R.

3.2 Fixed Point Model

For a method to obtain a model of the given rule

set, a ﬁxed point of the mapping associated with the

rule set may be taken, although we cannot always

have such a ﬁxed point, because of nonmonotonic

functionality of the mapping.

Mapping for Modeling of Rule Set

Given a rule R, a mapping T

: 2

× 2

→ 2

× 2

deﬁned to be

(I, J) = (I

, J

)

such that the pair (I

, J

) is given for the pair (I, J) to

• if (∀(Pos, Neg, a) ∈ R. ((∃b ∈ Pos. b ∈ J) or

(∃c

∗

∈ Neg. c ∈ I))) then a ∈ J

• else if (∃(Pos, Neg, a) ∈ R. ((∀b ∈ Pos. b ∈ I) and

(∀c

∗

∈ Neg. c ∈ J))) and

(∀(Pos

, Neg

, a

∗

) ∈ R. ((∃b ∈ Pos

. b ∈ J) or

(∃c

∗

∈ Neg

. c ∈ I))) then a ∈ I

• else (∃(Pos, Neg, a) ∈ R. ((∀b ∈ Pos. b 6∈ J) and

(∃b

∈ Pos. b

6∈ I)) and (∀c

∗

∈ Neg. c ∈ J))) and

(∀(Pos

, Neg

, a

∗

) ∈ R. ((∃b ∈ Pos

. b ∈ J) or

(∃c ∈ Neg

∗

. c ∈ I))) entails a 6∈ I

∪ J

If T

(I, J) = (I, J) such that I ∩ J =

0, the pair is

called a consistent ﬁxed point of T

Proposition 1. If (I, J) is a consistent ﬁxed point of

then eval

V (I,J)

(R) = t.

Proof. All the rules of R are examined. For each a ∈

A, we check the evaluation of R with respect to the

valuation V (I, J).

(i) When a ∈ I,

(∀(Pos, Neg, a) ∈ R. eval

V (I,J)

((Pos, Neg, a)) = t).

Because a ∈ I,

(∀(Pos

, Neg

, a

∗

) ∈ R. ((∃b ∈ Pos

. b ∈ J) or

(∃c

∗

∈ Neg

. c ∈ I))).

It follows that

eval

V (I,J)

(Pos

) = f or eval

V (I,J)

(Neg

) = f .

Thus eval

V (I,J)

((Pos

, Neg

, a

∗

)) = t.

(ii) Assume that a ∈ J. Then

(∀(Pos, Neg, a) ∈ R. ((∃b ∈ Pos. b ∈ J) or

(∃c

∗

∈ Neg. c ∈ I))).

It follows that

eval

V (I,J)

(Pos) = f or eval

V (I,J)

(Neg) = f .

Thus eval

V (I,J)

((Pos, Neg, a)) = t. On the other hand,

eval

V (I,J)

∗

) = t

such that eval

V (I,J)

((Pos

, Neg

, a

∗

)) = t.

(iii) In case that a 6∈ I ∪ J:

(∃(Pos, Neg, a) ∈ R. ((∀b ∈ Pos. b 6∈ J) and

(∃b

∈ Pos. b

6∈ I)) and (∀c

∗

∈ Neg. c ∈ J))) and

(∀(Pos

, Neg

, a

∗

) ∈ R. ((∃b ∈ Pos

. b ∈ J) or

(∃c

∗

∈ Neg

. c ∈ I))).

It follows that

eval

V (I,J)

(Pos) = unk and eval

V (I,J)

(Neg) = t.

COMPLEXIS 2023 - 8th International Conference on Complexity, Future Information Systems and Risk

On the assumption of a 6∈ I ∪ J, eval

V (I,J)

(a) = unk.

Thus eval

V (I,J)

((Pos, Neg, a) = t.

For any rule of the assumed form

(Pos

, Neg

, a

∗

) ∈ R,

eval

V (I,J)

(Pos

∪ Neg

) = f , while eval

V (I,J)

∗

) = f .

Thus eval

V (I,J)

(Pos

, Neg

, a

∗

) = t.

Example 2. Assume the rule set R as in Example 1.

Then, we have a model as ﬁxed points of T

({revised, evolved}, { f ashioned, ruled})

Thus, the consistent ﬁxed point (I, J) of T

is a

model of R.

Given a set R of rules, the following derivation is

an amended version of negation by failure.

Retrieval Derivation

suc and f ail are succeeding and failing derivations,

inductively deﬁned as follows.

(1)

0 : suc.

(2)

if ∃(Pos, Neg, a) ∈ R.

((∀(Pos

, Neg

, a

∗

) ∈ R. Pos

∪ Neg

: f ail) and

Pos ∪ Neg ∪ G : suc) then {a}∪ G : suc.

(3) If {a} : f ail and G : suc then {a

∗

} ∪ G : suc.

(4) If ∀(Pos, Neg, a) ∈ R. (Pos ∪ Neg : f ail) then

{a} : f ail.

(5) If {a} : f ail then {a} ∪ G : f ail.

(6) If {a} : suc then {a

∗

} ∪ G : f ail.

Example 3. Assume the rule set R as in Example 1.

Then we have a succeeding derivation.

1. For “{evolved} : suc” to hold,

“{revised} ∪{ f ashioned

∗

} : suc”

must hold, since there is

({revised}, { f ashioned

∗

}, evolved) ∈ R.

2. For “{revised} ∪ { f ashioned

∗

} : suc” to hold,

“{revised} : suc” must hold, since we can see

“{ f ashioned} : f ail”.

3. For “{revised} : suc” to hold, “{ruled

∗

} : suc”

must hold, since there is (

0, {ruled

∗

}, revised) ∈

4. We can see “{ruled} : f ail”. The check of

whether “{ruled

∗

} : suc” or not may be reduced

to the case for “

0 : suc” (initially assumed).

Thus we can display the following scheme:

{evolved}

with ({revised}, { f ashioned

∗

}, evolved) ∈ R

{revised} ∪{ f ashioned

∗

}

with { f ashioned} : f ail

{revised}

with (

0, {rules

∗

}, revised)

{ruled

∗

}

with {ruled} : f ail

0 : suc

The derivation is sound with respect to possibly

existing, consistent ﬁxed point model in the sense:

Proposition 2. Assume that (I, J) is a consistent ﬁxed

point of T

(1) If {a} : suc then a ∈ I.

(2) If {a} : f ail then a ∈ J.

Proof. (1) If {a} : suc then

(∀(Pos

, Neg

, a

∗

) ∈ R. Pos

∪ Neg

: f ail).

It follows that

(∃b ∈ Pos

.{b} : f ail), or

(∃c

∗

∈ Neg

: {c

∗

} : f ail)(i.e., ∃c

∗

∈ Neg

.{c} : suc).

By induction hypothesis, there is b ∈ J or c ∈ I, for the

assumed derivation {a} : suc. With respect to {a} :

suc, two cases are to be examined.

(a) If (

0, a) ∈ R, as a basis, it is evident that a ∈ I.

(b) If (Pos, Neg, a) ∈ R such that Pos ∪ Neg : suc, as

induction step, we can reason that

(∀b ∈ Pos. b ∈ I) and (∀c

∗

∈ Neg. c ∈ J),

from the condition that

(∀b ∈ Pos.{b} : suc) and (∀c

∗

∈ Neg.{c} : f ail).

This completes the induction step, such that a ∈ I.

(2) If {a} : f ail then

(∀(Pos, Neg, a) ∈ R. Pos ∪ Neg : f ail).

When there is no rule of the form (Pos, Neg, a), then

a ∈ J. Otherwise, it follows that

(∃b ∈ Pos.{b} : f ail) or (∃c

∗

∈ Neg.{c

∗

} : f ail)

(the latter of which is caused by ∃c

∗

∈ Neg.{c} : suc).

By induction, b ∈ J or c ∈ I. This concludes that a ∈

Abstraction of Prevention Conceived in Distributed Knowledge Base

Negation by Failure

In the Retrieval Derivation, the item (2) contains the

point: For “{a} : suc” to hold, the following condition

is required:

(∀(Pos

, Neg

, a

∗

) ∈ R. Pos

∪ Neg

: f ail)

This condition contains complexity, which the

original negation by failure rule (K. Clark, 1978) does

not need. The condition comes from the evaluation

for the triplet with the implication in:

Pos ∪ Neg implies con.

The meaning of the implication is concerned with 3-

valued domain, as above mentioned for the deﬁnition

of eval

(•). Then:

(i) If {a} : f ail then {a

∗

} : suc.

(ii) In case that {a

∗

} : f ail, it may be possible that

{a} : suc. (Therefore, if {a} : suc then {a

∗

} :

f ail.)

This amended negation by failure is well enough to

escape from inﬁnite procedure for failure such that

succeeding derivation contains only ﬁnite steps of ex-

ecutions.

On the other hand, the rule set R

R = {(

0, {b

∗

}, a), (

0, {a

∗

}, b)}

induces 2 consistent ﬁxed points of T

({a}, {b}) and ({b}, {a})

Because of no ﬁnite failure for {b}, we cannot have

{a} : suc. At the same time, we cannot obtain {a} :

f ail such that it is no case of {b} : suc.

4 DISTRIBUTED KNOWLEDGE

BASE

In this section, distributed knowledge base is just a tu-

ple of knowledges each of which is described in Sec-

tion 3. But the negative from each knowledge must

be common. How we can have the meaning of dis-

tributed knowledge base with common negatives is

presented to apply to the theory for prevention alert

in distributed systems. Then we can have retrieval

derivation by means of oracle of assumed prevention,

with respect to the meaning of distributed knowledge

base.

4.1 Tuple of Rule Sets

As distributed knowledge base (which may be re-

garded as constructed by seminar participants), a tu-

ple of rule sets

τ = hR

, . . . , R

i (n ≥ 1)

is used. To represent a distributed knowledge base,

we extend the method of how each rule set denotes a

knowledge over a set A.

A valuation V

: A → { f , unk, t} (1 ≤ i ≤ n) is as-

sumed with the bounded lattice K, as in the case of

Section 3.

The evaluations of R

with the way as below are

to be executed for 1 ≤ k ≤ n.

Evaluation of Rule Sets Tuple and Model

With (V

, . . . , V

), the value Eval

,...,V

)

(τ) is given

by:

Eval

,...,V

)

(τ)

= Eval

,...,V

)

(hR

, . . . , R

= (eval

), . . . , eval

))

For a pair (I, J) ∈ 2

× 2

such that I ∩ J =

0, i.e.,

I and J are disjoint, the valuation

(I, J) : A → { f , unk, t} (1 ≤ k ≤ n)

is used as V (I, J) in Section 3.

If with the disjoint pairs (I

, J

) (1 ≤ k ≤ n) for a

tuple τ of rule sets,

Eval

),...,V

))

(τ) = (t, . . . , t)

then the pairs are called a model of τ.

Just taking a direct product of pairs each of which

may be deﬁned by the mapping of T

(1 ≤ k ≤ n),

we have a mapping of Tr

Mapping for Modeling of Rule Sets Tuple

Given a tuple of rule sets τ = hR

, . . . , R

i, a mapping

: (2

× 2

)

→ (2

× 2

)

is deﬁned to be

((I

, J

), . . . , (I

, J

))

= (T

, J

), . . . , T

, J

))

= ((I

, J

), . . . , (I

, J

))

such that each pair (I

, J

) is given for the pair (I

, J

)

with T

(as T

in Section 3):

• if (∀(Pos, Neg, a) ∈ R

. ((∃b ∈ Pos. b ∈ J

) or

(∃c

∗

∈ Neg. c ∈ I

))) then a ∈ J

• else if

(∃(Pos, Neg, a) ∈ R

. ((∀b ∈ Pos. b ∈ I

) and

(∀c

∗

∈ Neg. c ∈ J

))) and

(∀(Pos

, Neg

, a

∗

) ∈ R

. ((∃b ∈ Pos

. b ∈ J

) or

(∃c

∗

∈ Neg

. c ∈ I

))) then a ∈ I

• else (∃(Pos, Neg, a) ∈ R

. ((∀b ∈ Pos. b 6∈ J

) and

(∃b

∈ Pos. b

6∈ I

)) and (∀c

∗

∈ Neg. c ∈ J

))) and

(∀(Pos

, Neg

, a

∗

) ∈ R

. ((∃b ∈ Pos

. b ∈ J

) or

(∃c ∈ Neg

∗

. c ∈ I

))) entails a 6∈ I

∪ J

COMPLEXIS 2023 - 8th International Conference on Complexity, Future Information Systems and Risk

The ﬁxed point of the mapping Tr

may be ob-

tained such that each R

(1 ≤ k ≤ n) is modelled.

Proposition 3. If

((I

, J

), . . . , (I

, J

)) = ((I

, J

), . . . , (I

, J

))

such that I

∩ J

0 (1 ≤ k ≤ n) then

Eval

(V (I

),...,V (I

))

(τ) = (t, . . . , t),

i.e., eval

V (I

)

, J

) = t (1 ≤ k ≤ n).

Proof. For each R

, the proof of Proposition 1 can

be applied, where each T

is independent of another

(k 6= j). Tr

is deﬁned in terms of such mutually

independent T

. Therefore the above ﬁxed point of

is a tuple of the consistent ﬁxed points of T

(1 ≤

k ≤ n). Thus the above ﬁxed point of Tr

is associated

with

Eval

(V (I

),...,V (I

))

(τ)

which is a tuple of n values of t.

The ﬁxed points (I

, J

) (for 1 ≤ k ≤ n) are com-

ponents of the (tuple) ﬁxed point of τ.

4.2 Retrieval in Knowledge Base

Fixed point of

: (2

× 2

)

→ (2

× 2

)

((I

, J), . . . , (I

, J)) (I

∩ J =

0, 1 ≤ k ≤ n) can be

knowledge acquisitions of individuals, where preven-

tion information is common.

Since each (I

, J) is independent of another

, J), retrieval derivation to each R

is available,

which is deﬁned with an oracle O supposed to be a

common negative knowledge, following the deriva-

tion given in Section 3.

Retrieval Derivation to R

with Oracle O

(1)

0 : suc.

(2) If a 6∈ O where there is (Pos, Neg, a) ∈ R

such

that ((∀(Pos

, Neg

, a

∗

) ∈ R

. Pos

∪ Neg

: f ail)

and Pos ∪ Neg ∪ G : suc) then {a}∪ G : suc.

(3) If {a} : f ail and G : suc then {a

∗

} ∪ G : suc.

(4) If a ∈ O such that ∀(Pos, Neg, a) ∈ R

.(Pos∪Neg :

f ail) then {a} : f ail.

(5) If {a} : f ail then {a} ∪ G : f ail.

(6) If {a} : suc then {a

∗

} ∪ G : f ail.

The above retrieval derivation may contain sound-

ness with respect to (I

, J), in the sense:

Proposition 4. Assume that for

τ = hR

, . . . , R

((I

, J), . . . , (I

, J)) is a ﬁxed point of Tr

, where

∩ J =

0 (1 ≤ k ≤ n).

For the derivation to R

with the oracle Q ⊆ J, we

have:

(1) If {a} : suc then a ∈ I

(2) If {a} : f ail then a ∈ J.

Proof. Because O is supposedly a subset J, the rea-

son why soundness is held is almost the same as the

proof of Proposition 2 in Section 3. But we describe

formally in the similar way.

(1) If {a} : suc with a 6∈ O then

∀(Pos

, Neg

, a

∗

) ∈ R. Pos

∪ Neg

: f ail.

Thus (∃b ∈ Pos

.{b} : f ail) or (∃c

∗

∈ Neg.{c

∗

} : f ail)

(i.e., ∃c

∗

∈ Neg

.{c} : suc).

By induction hypothesis, there is b ∈ J or c ∈ I,

for the assumed derivation {a} : suc.

2 cases are also to be examined:

(a) If (

0, a) ∈ R, as a basis, it is evident that a ∈ I.

(b) If (Pos, Neg, a) ∈ R such that Pos ∪ Neg : suc, as

induction step, we can reason that

(∀b ∈ Pos. b ∈ I) and (∀c

∗

∈ Neg. c ∈ J),

from the condition that

(∀b ∈ Pos. {b} : suc) and (∀c

∗

∈ Neg. {c} : f ail).

This completes the induction step, such that a ∈ I.

(2) If {a} : f ail with a ∈ O then

∀(Pos, Neg, a) ∈ R. Pos ∪ Neg : f ail.

When there is no rule of the form (Pos, Neg, a), then

a ∈ J. Otherwise, it follows that

∃b ∈ Pos.{b} : f ail or ∃c

∗

∈ Neg. {c

∗

} : f ail

(the latter of which is caused by ∃c

∗

∈ Neg.{c} : suc).

By induction, b ∈ J or c ∈ I. This concludes that a ∈

Proposition 4 shows that the derivations to R

with

oracle O is sound with respect to the consistent ﬁxed

point (I

, J) of T

, where the oracle O, an assumed

subset of the set J can be regarded as common pre-

vention in the distributed knowledge base

τ = hR

, . . . , R

Example 4. Assume the rule set

= { (

0, c), ({c}, {d

∗

}, a), ({b},

0, {a

∗

}) }

The set is modeled by ({a, c}, {b, d}).

Assume the rule set

= { (

0, c), ({b, c},

0, a), (

0, a

∗

) }.

The set R

is modeled by ({c}, {a, b}). For the tuple

, R

i, {b} is a common prevention.

Abstraction of Prevention Conceived in Distributed Knowledge Base

5 CONCLUSION

Motivated by a seminar relation between the presen-

ter and participants, we settle knowledge acquisitions

and distributed knowledge base.

We established theoretical results:

(a) A triplet of a positive set, a negative set and con-

clusion as a rule, is adopted, such that a rule set may

be regarded as acquired knowledge whose evaluation

is made in 3-valued domain: The rule-set evaluation

is based on 3-valued bounded lattice structure.

(b) This evaluation involves complex difﬁculty caused

by strong negatives. We conclude that a ﬁxed point of

a mapping associated with a rule set may be a model

(which evaluates the rule set as true in the 3-valued

domain). A ﬁxed point does not always exist, how-

ever, it may be a model if it must be consistent (that

is, no case is evaluated as both positive and negative),

leaving the unknown (undeﬁned) as it is.

tion by failure, to be sound with respect to the model

which is a consistent ﬁxed point.

This is another theoretical dealing with the alge-

braic/logical complex, compared with the papers (Ya-

masaki and Sasakura, 2021a; Yamasaki and Sasakura,

2022). Then we deal with distributed knowledge base

containing common prevention, abstractly reﬂecting

the seminar system with the reports on negatives from

participants.

(d) We think of a tuple of triplets (for rule sets) as dis-

tributed knowledge base, which is regarded as a theo-

retical representation of a seminar system.

(e) The evaluation of distributed knowledge base is

made such that we may have some model by a con-

sistent ﬁxed point of a mapping associated with dis-

tributed knowledge base, although such a ﬁxed point

does not always exist. This mapping contains the

mappings associated with acquired knowledges (vir-

tually of participants).

(f) With some assumed prevention, distributed re-

trieval derivations are available by means of strate-

gies containing individual derivation for knowledge

as brieﬂy pointed out in (c).

As in the treatment of communication (Yamasaki

and Sasakura, 2021b), if both positive and negative

are common, then the whole distributed knowledge

base should be treated as single knowledge base. In

this case, Section 3 gives a theoretical base. Ow-

ing to implementation of communication among R

(1 ≤ k ≤ n), the retrieval derivation must be essen-

tially complex. If we could have access to another R

from R

only by reference to, a possible ﬁxed point

and the given procedural aspects of retrieval deriva-

tion may be abstractly simpler.

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