Algebraic Structure of Recursively Constructed References and Its

Application to Knowledge Base

Susumu Yamasaki

1 a

and Mariko Sasakura

2 b

Okayama University, Tsushima-Naka, Okayama, Japan

IPDRE, Tottori University, Hamasaka, Tottori City, Japan

Keywords:

Algebraic Structure, Model Theory, Knowledge Base.

Abstract:

From management views on complex website page structures, we formulate an algebraic structure of re-

cursively constructed page references as presenting situations of them to the website with 3-valued domain.

Algebraic structure of references, abstracted from website page references, is here expressed as a ﬁnite or

countably inﬁnite set of rules, where each rule is deﬁned, by representing the recursive relations among web

page references. The situations of a reference with request to the website can be denoted as the acquisitive

positive, rejective negative and suspended negative, respectively. With respect to algebraic structure, a ﬁxed

point of the mapping associated with the rule set may be a model denoting consistent evaluations to assign

the situations of references to 3-valued domain. Model theory for representation of consistent evaluations of

references and the rule set (constructed with references) is newly settled if a ﬁxed point consistently exists. A

retrieval derivation to detect acquisitive positives and rejective negatives can be presented, to be sound with

respect to the model, based on the inference by negation as failure, which is related to the suspended negative.

As multiple knowledge base formed by a tuple of rule sets, this paper next presents algebraic structure of a

distributed knowledge system constrained by a state, and sequential applications of such systems, containing

state transitions. Model theory can be deﬁned with ﬁxed point of the mapping associated with the distributed

knowledge system, although the ﬁxed point may not be always applied to modeling. If consistent ﬁxed point

modeling is available, we may have a model of the distributed knowledge system, constrained by a state. Then

the application of such a distributed knowledge system may be considered as causing state transitions, follow-

ing modeling and designed state transitions.

1 INTRODUCTION

The well organized website contains web pages so

that complex information system may be constructed.

In this paper, we study the algebraic structure of web

page references, in more details, from the purpose of

analyzing the page references recursively constructed

among references, whose situations are in the cases

of being active, inactive or different, in close relation

of a reference request to the website. The active, inac-

tive and different situations are then interpreted as tak-

ing acquisitive, rejective and suspended values (which

are, as algebraic elements, assigned to references),

such that algebraic element may be knowledgeable by

itself. In this context, the algebraic structure of re-

cursively constructed references are formulated with

3-valued domain. The structure is complex enough,

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

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

however, references recursion may be compact to an-

alyze and to apply to knowledge base with processing.

The 3-domain is deﬁned by the set of truth, falsity and

the unknown, for the acquisitive, rejective and sus-

pended values, respectively. What values are to be

consistent assignments of references in a given alge-

braic structure is discussed, by the method of evalua-

tion of references and the whole structure, as a model

theory. We newly present a ﬁxed point model for the

algebraic structure, to make it clearer for the complex

structure of references to be captured as consistent in

the recursive construction of references. Based on the

evaluation of each reference in a 3-valued domain, the

model theory is regarded as relevant to consistency of

data integrity, when the complex structure of refer-

ences is applied to knowledge base.

For model theory, retrieval derivation to the struc-

ture with rejective negation is also presented, as

knowledge processing. With reference to knowledge

Yamasaki, S. and Sasakura, M.

Algebraic Structure of Recursively Constructed References and Its Application to Knowledge Base.

DOI: 10.5220/0012544500003708

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 9th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2024), pages 83-90

ISBN: 978-989-758-698-9; ISSN: 2184-5034

processing, we may see the algebraic structure, from

the view of knowledge base with logic.

About logic for knowledge and reasoning, we

have seen the backgrounds:

(a) The propositional system and algebra for inquis-

itive logic with negative variants are organized, by

means of intermediate logic in the class between

those classes of inuitionistic and propositional cal-

culi. (Bezhanishvili et al., 2022). Nonmonotonic

logic on reasoning and inference mechanism should

contain applicative aspects with respect to design and

programming (Hanks and McDermott, 1987; Reiter,

2001).

(b) Modal logic contains semantics based on state

space in Kleene-Kripke theory. Not only proposi-

tional logic but also ﬁrst-order modal logic is formally

dealt with (Fitting, 2002). With reference to knowl-

edge processing and computability, quantiﬁed modal

logic is discussed (Rin and Walsh, 2016). For step-

wise removal of objects, there is dynamic modality

(Bentham et al., 2022). Two-dimensional modality is

studied (Gessel, 2022).

dynamic logic for action was presented (Spalazzi and

Traverso, 2000). As other dynamic aspects of knowl-

edge processing, relevant works are made with the

concepts of justiﬁed belief truth (Egre et al., 2021),

reduction of decidability (Rasga et al., 2021) and in-

ference theory (Tennant, 2021).

Compared with the backgrounds, this paper is

concerned with knowledge base, which the algebraic

structure of references denotes, as well as knowledge

retrieval sound with respect to ﬁxed point model. The

primary aim of this paper is relevant to nonmono-

tonic logic. We next obtain the algebraic structure

of a distributed knowledge system containing a tuple

of knowledge bases, where the whole system is con-

strained by a state (environment). After knowledge

processing to each knowledge base, state transitions

may be supposed for further knowledge processing in

next states, where each state constrains a distributed

knowledge system.

Revising the work (S.Yamasaki et al., 2023), we

take knowledge processing with common negatives

in a state, constraining a distributed knowledge sys-

tem. On condition that only negatives may be com-

mon, consistency in the whole distributed knowledge

system is treated, without explicit communications

among knowledge bases. The state transitions are

characterized as sequential applications of state con-

strained systems.

We here note the literatures concerned with be-

haviours of distributed systems, paying attention to

the traverses of states in a system: Sequences travers-

ing states in a distributed system may be closely re-

lated to the method of automata (Droste et al., 2009).

As regards varying or moving processes, formulations

have been developed in mobility of ambients (Cardelli

and Gordon, 2000; Merro and Nardelli, 2005).

This paper captures the environment just by a state

name. About the traverse of states caused by applica-

tions of knowledge processing, we have semantics for

the effect of the sequential applications, on the basis

of algebraic semiring.

The paper is organized as follows. Section 2 for-

mally provides a rule set as algebraic structure of re-

cursively constructed references. In Section 3, the

meaning (model) of the algebraic structure may be

newly deﬁned by ﬁxed point of a mapping associ-

ated with the rule set. Section 4 presents the retrieval

derivation from the algebraic structure with respect to

its model. In Section 5, model theory of a distributed

knowledge system with common negatives is devel-

oped, as application of the rule sets to knowledge

bases and to a distributed knowledge system. Sec-

tion 5 is also concerned with the traverses of states,

with the model theory of the distributed systems con-

strained by states. Concluding remarks are given in

Section 6.

2 RECURSIVELY

CONSTRUCTED REFERENCES

As knowledge acquisition means, the website is es-

tablished. We note the static structure constructed by

organizing the website references of web pages. It is

concerned with recursive construction, as well as with

negatives owing to the situations of the website.

Algebraic Structure of References

As is seen, we observe web page reference with re-

quest to some website, as classiﬁed into several situa-

tion sets. In this paper, we suppose 3 situation sets of

the website collecting pages with references to them:

(a) The website is active so that the reference to the

site may be considered as in an acquisitive set

(for knowledge).

(b) The website is inactive so that the reference may

be regarded as in a rejective set.

is different from the one which the reference

should be settled to, such that the reference may

be interpreted as in a suspended set.

Figure 1 presents views on the situation sets.

We also observe the relation among references

where the referenced page may contain ﬁnitely many

COMPLEXIS 2024 - 9th International Conference on Complexity, Future Information Systems and Risk

Reference

Site active Site inactive Site different

Acquisitive Set Rejective Set Suspended Set



+

Figure 1: Web page reference is classiﬁed.

page references, as likely denoted by:

⟨Re f

, . . . , Re f

⟩ (contained in) Re f ,

where “Re f ” is a reference considered recursively

constructed with references Re f

, . . ., Re f

(n ≥ 0).

From application views, the reference is knowledge-

able, as well, in the sense that the reference may re-

cursively contain references.

We take Ac, Re and Sus as the acquisitive set,

the rejective set and the suspended set of references,

respectively, with Knowl as knowledge containing a

reference. Making use of a rule of the form

(Ac, Re, Sus)) ▷ Knowl,

and of a (ﬁnite or countably inﬁnite) set of rules, we

study the meaning of algebraic structure of rule set.

Its model theory is concerned with knowledge ac-

quirements.

Algebraic Rule

We ﬁrstly assume a domain A as a set of references

(as knowledgeable objects). We then make use of:

not

= {not a | a ∈ A},

∼

= {

∼

a | a ∈ A}.

where (i) not a is to be interpreted as rejection of a (or

its negation as in Heyting algebra), and (ii)

∼

a is as

suspension of a, exactly deﬁned later in model theory.

.A rule is of the form r, in a formal way as represented

by acquisitive, rejective and suspended informations

contained in an acquired one. It is regarded as knowl-

edge (with the notations “

0” and “|” as the empty set

and “or”, respectively), from the application views as

in Section 5, denoted by the form as follows.

r = (Ac, Re, Sus) ▷ Knowl, where for a ∈ A:

Ac ⊆ A, Re ⊆ A

not

, Sus ⊆ A

∼

such that

Ac ::=

0 | {a} ∪ Ac

Re ::=

0 | {not a} ∪ Re

Sus ::=

0 | {

∼

a} ∪ Sus

Knowl ::= a | not a |

∼

That is, a rule is of the form (Ac, Re, Sus) ▷ Knowl

where: (a) Ac ⊆ A is a ﬁnite acquisitive set. (b) Re ⊆

not

is a ﬁnite rejective set. (c) Sus ⊆ A

∼

is a ﬁnite

suspended set. (d) Knowl = a, not a or

∼

a is acuired

knowledge (for a ∈ A).

The expression (Ac, Re, Sus) ▷ a, (Ac, Re, Sus) ▷

not a, or (Ac, Re, Sus) ▷

∼

a may be adopted, if the

form Knowl is a ∈ A or not a ∈ A

not

∼

a ∈ A

∼

respectively.

R is inductively deﬁned as a (ﬁnite or countably

inﬁnite) set of rules. It is regarded as knowledge base,

as in Section 5: R ::=

0 | r ∪ R.

As a denotation of the page reference, the rule

form (∪

(Ac

, Re

, Sus

)) ▷ Knowl may be consid-

erable, however, it can be replaced by a rule set

∪

{(Ac

, Re

, Sus

) ▷ Knowl}.

3 MODEL THEORY IN

ALGEBRAIC STRUCTURE

As a more general case than the one (Yamasaki

and Sasakura, 2023) and the conditional causal re-

lation (Yamasaki and Sasakura, 2021), we exam-

ine the meaning of the rule set deﬁned in Section

2. While many-valued logic provability is presented

(Pawlowski and Urbaniak, 2018), answer set pro-

gramming and problem solving are compiled in 2-

valued logic for practice (Gebser and Schaub, 2016;

Kaufmann et al., 2016).

The rejection negation is regarded as negation in

Heyting algebra, as well as in intuitionistic proposi-

tional logic. For the suspended negation, one more

negation is to be deﬁned, so that 3-valued evaluation

is of use, with the unknown (as a value). But we

do not deal with weak negation as in the case (Ya-

masaki, 2006). With a 3-valued domain, we present

ﬁxed point modeling, based on a mapping associated

with the rule set.

3.1 3-valued Domain

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

, ⊥, ⊤)

equipped with the partial order ⊑ 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 ⊑

such that ⊥ = f ⊑ unk ⊑ t = ⊤.

(b) The least upper bound ( join with

) and the great-

est lower bound (meet with

) exist for any two ele-

ments of the set { f , unk, t}.

{ f , unk, t}) is deﬁned in a way that z ⊑ (x ⇒ y) iff

z ⊑ y.

t is the truth, f is the falsity and unk (the unknown

as t or f ) is the undeﬁned for the truth value, where

Algebraic Structure of Recursively Constructed References and Its Application to Knowledge Base

the partial order is deﬁned, regarding the truth value.

Evaluation for Rule Set

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

(not a)

= if (eval

(a) = f ) then t else f

eval

(

∼

= if (eval

(a) = f ) then t

else if (eval

(a) = unk) then unk else f

eval

(Knowl)

= if (Knowl = a) then eval

(a)

else if (Knowl = not a) then eval

(not a)

else if (Knowl =

∼

a) then eval

(

∼

eval

(Ac) =

a∈Ac

V (a)

eval

(Re) =

not a∈Re

eval

(not a)

eval

(Sus) =

∼

a∈Sus

eval

(

∼

eval

((Ac, Re, Sus))

= eval

(Ac)

eval

(Re)

eval

(Sus)

eval

((Ac, Re, Sus) ▷ Knowl)

= if eval

((Ac, Re, Sus)) ⊑ eval

(Knowl)

then t else if (eval

(Knowl) = f ) then f else unk

eval

(R) =

((Ac,Re,Sus)▷Knowl)∈R

eval

((Ac, Re, Sus) ▷ Knowl)

(

operates on a countable set)

Note that eval

(

0) = t for Ac, Re, Sus and R to be the

empty set

Model of Rule set

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

such that eval

V (I,J)

(R) may be settled inductively as

above. 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 Model of Rule Set with Fixed Point

Theory

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.

Given a rule set R (based on the set A of ob-

jects), for each rule (Ac, Re, Sus) ▷ a (a ∈ A) to be

evaluated as t (following the deﬁnition of the evalua-

tion eval

with the valuation V ) such that eval

(R) =

t, we examine mutually exclusive cases. For each

(Ac

, Re

, Sus

) ▷ not b, or (Ac

, Re

, Sus

) ▷

∼

in R, we can exhaustively examine the cases for

(Ac

, Re

, Sus

) ▷ b or (Ac

, Re

, Sus

) ▷ c.

From such an inspection, a pair (I, J) for the valu-

ation V (I, J) to model the rule set R is to be given as

a ﬁxed point of the formally deﬁned mapping Trans

(as below). The pair (I, J) = Trans

(I, J) (by Deﬁ-

nition 1) may be taken (to the valuation V (I, J)), as

satisfactory.

Deﬁnition 1. Given a rule R, a mapping Trans

: 2

→ 2

× 2

is deﬁned to be

Trans

(I, J) = (I

′

, J

′

)

such that the pair (I

′

, J

′

) may be given:

(a) if (∀(Ac, Re, Sus) ▷ a ∈ R. ((∃b ∈ Ac. b ∈ J) or

(∃not c ∈ Re. c ̸∈ J) or (∃

∼

d ∈ Sus. d ∈ I))),

then a ∈ J

′

(b) else if (∃(Ac, Re, Sus) ▷ a ∈ R.

((∀b ∈ Ac. b ∈ I) and (∀not c ∈ Re. c ∈ J) and

(∀

∼

d ∈ Sus. d ∈ J))) on condition that

(∀(Ac

, Re

, Sus

) ▷ not a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))) and

(∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))), then a ∈ I

′

ists in R and (∀(Ac, Re, Sus) ▷ a ∈ R.

((∃b ∈ Ac. b ̸∈ I) or (∃not c ∈ Re. c ̸∈ J) or

(∃

∼

d ∈ Sus. d ̸∈ J))) and

(∀(Ac

, Re

, Sus

) ▷ not a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))) and

(∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R.

((∃b ∈ Ac

. b ̸∈ I) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ̸∈ J)))), then a ̸∈ I

′

∪ J

′

else undeﬁned

If Trans

(I, J) = (I, J) such that I ∩J =

0, the pair

is called a consistent ﬁxed point of Trans

Proposition 2. If (I, J) is a consistent ﬁxed point of

Trans

then eval

V (I,J)

(R) = t.

Proof. For each a ∈ A, we examine the evaluation of

all the related rules in R with respect to the valuation

V (I, J).

COMPLEXIS 2024 - 9th International Conference on Complexity, Future Information Systems and Risk

(i) When a ∈ I,

(∀(Ac, Re, Sus) ▷ a ∈ R.

eval

V (I,J)

((Ac, Re, Suc) ▷ a) = t).

Because a ∈ I,

(∀(Ac

, Re

, Sus

) ▷ not a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))).

It follows that

eval

V (I,J)

(Ac

) = f or eval

V (I,J)

(Re

) = f or

eval

V (I,J)

(Sus

) = f .

Thus eval

V (I,J)

((Ac

, Re

, Sus

) ▷ not a) = t.

As is the similar case, for a ∈ I,

(∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))).

Then

eval

V (I,J)

(Ac

) = f or eval

V (I,J)

(Re

) = f or

eval

V (I,J)

(Sus

) = f .

Finally eval

V (I,J)

((Ac

, Re

, Sus

) ▷

∼

a) = t.

(ii) Assume that a ∈ J. Then

(∀(Ac, Re, Sus) ▷ a ∈ R.

((∃b ∈ Ac. b ∈ J) or (∃not c ∈ Re. c ̸∈ J) or

(∃

∼

d ∈ Sus. d ∈ I))).

Therefore

eval

V (I,J)

(Ac) = f or eval

V (I,J)

(Re) = f or

eval

V (I,J)

(Sus) = f .

It causes eval

V (I,J)

((Ac, Re, Sus) ▷ a) = t. On the

other hand, eval

V (I,J)

(not a) = t such that

eval

V (I,J)

((Ac

, Re

, Sus

) ▷ not a) = t

for any (Ac

, Re

, Sus

) ▷ not a in R.

Also we have eval

V (I,J)

(

∼

a) = t such that

eval

V (I,J)

((Ac

, Re

, Sus

) ▷

∼

a) = t

for any (Ac

, Re

, Sus

) ▷

∼

a in R.

(iii) In case that a ̸∈ I ∪ J: there is a rule of the form

(Ac, Re, Sus) ▷ a ∈ R such that eval

V (I,J)

(a) = unk,

where eval

V (I,J)

((Ac

′

, Re

′

, Sus

′

)) = unk or f for any

rule of the form (Ac

′

, Re

′

, Sus

′

) ▷ a. It follows that

eval

V (I,J)

((Ac

′

, Re

′

, Sus

′

) ▷ a) = t for any rule of the

form (Ac

′

, Re

′

, Sus

′

) ▷ a, on condition that

(∀(Ac

, Re

, Sus

) ▷ not a ∈ R.

((∃b ∈ Ac

. b ∈ J) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ∈ I))) and

(∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R.

((∃b ∈ Ac

. b ̸∈ I) or (∃not c ∈ Re

. c ̸∈ J) or

(∃

∼

d ∈ Sus

. d ̸∈ J))).

From the above condition: for any rule of the form

(Ac

, Re

, Sus

) ▷ not a ∈ R,

eval

V (I,J)

((Ac

, Re

, Sus

)) = f , while

eval

V (I,J)

(not a) = f . Thus

eval

V (I,J)

((Ac

, Re

, Sus

) ▷ not a) = t.

For any rule of the form

(Ac

, Re

, Sus

) ▷

∼

a ∈ R,

eval

V (I,J)

((Ac

, Re

, Sus

)) = unk or f , while

eval

V (I,J)

(

∼

a) = unk. Thus

eval

V (I,J)

((Ac

, Re

, Sus

) ▷

∼

a) = t.

4 RETRIEVAL DERIVATION IN

RULE SET

We conceive model theory in some case of replace-

ment of the rejective negation not by the suspended

negation

∼

. Negation as failure derivation (which

corresponds to the suspended negation

∼

) may be ex-

tended to be applicable to the model theory (in Sec-

tion 3). In the Japanese poem analysis (Yamasaki and

Yokono, 2008), this idea was adopted, while the anal-

ysis may be reﬁned by the following derivation.

Given a rule set R, we have the recursively de-

ﬁned derivation. Its implementation is applicable to

retrievals, when the algebraic rule set may be regarded

as knowledge base. Just a procedural aspect is now

presented, with a variable G ranging over the set A ∪

not

∪ A

∼

, such that G : suc and G : f ail may stand

for succeeding and failing derivations (of retrieval),

respectively.

(1)

0 : suc.

(2) When there is (Ac, Re, Sus) ▷ a ∈ R such that

(∀(Ac

, Re

, Sus

)▷ not a ∈ R. Ac

∪Re

∪Sus

f ail) and (∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R. Ac

∪

∪Sus

: f ail), with (Ac ∪Re∪Sus∪G : suc),

then {a} ∪ G : suc.

(3) If {a} : f ail and G : suc then {not a} ∪ G : suc.

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

∼

a} ∪ G : suc.

(5) If (∀(Ac, Re, Sus) ▷ a ∈ R. (Ac ∪ Re ∪ Sus :

f ail)), then {a} : f ail.

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

(7) If {a} : suc then {not a} ∪ G : f ail.

(8) If {a} : suc then {

∼

a} ∪ G : f ail.

The above retrieval derivation may contain sound-

ness with respect to some model (I, J) of R, in the

sense:

Algebraic Structure of Recursively Constructed References and Its Application to Knowledge Base

Proposition 3. Assume that for a rule set R, (I, J) is

a consistent ﬁxed point of Trans

. For the derivation

to R, we have:

(1) If {a} : suc, then a ∈ I.

(2) If {a} : f ail, then a ∈ J.

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

∀(Ac

, Re

, Sus

)▷ not a ∈ R. Ac

∪Re

∪Sus

: f ail,

where

(∃b

∈ Ac

.{b

} : f ail) or

(∃not c

∈ Re

.{not c

} : f ail)

(i.e., ∃not c

∈ Re

.{c

} : suc) or

(∃

∼

∈ Sus

. {

∼

} : f ail)

(i.e., ∃

∼

∈ Sus

. {d

} : suc).

By induction hypothesis, there is b

∈ J or c

∈ I or

∈ I, for the assumed derivation {a} : suc. Also

∀(Ac

, Re

, Sus

) ▷

∼

a ∈ R. Ac

∪ Re

∪ Sus

: f ail,

where

(∃b

∈ Ac

.{b

} : f ail) or

(∃not c

∈ Re

.{not c

} : f ail)

(i.e., ∃not c

∈ Re

.{c

} : suc) or

(∃

∼

∈ Sus

. {

∼

} : f ail)

(i.e., ∃

∼

∈ Sus

. {d

} : suc).

By induction hypothesis, there is b

∈ J or c

∈ I or

∈ 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 (Ac, Re, Sus) ▷ a ∈ R such that

Ac ∪ Re ∪ Sus : suc,

as induction step, we can reason that

(∀b ∈ Ac. b ∈ I) and (∀not c ∈ Re. c ∈ J)

and (∀

∼

d ∈ R. d ∈ J),

from the condition that (∀b ∈ Ac. {b} : suc)

and (∀not c ∈ Re. {not c} : suc)

and (∀

∼

d ∈ Sus.{

∼

d} : suc).

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

(2) If {a} : f ail, then

(∀(Ac, Re, Sus) ▷ a ∈ R. (Ac ∪ Re ∪ Sus : f ail)).

When there is no rule of the form (Ac, Re, Sus) ▷ a,

then a ∈ J. Otherwise, it follows that

(∃b ∈ Ac.{b} : f ail) or

(∃not c ∈ Re. {not c} : f ail)

(i.e., ∃not c ∈ Re. {c} : suc) or

(∃

∼

d ∈ Sus. {

∼

d} : f ail)

(i.e., ∃

∼

d ∈ Sus. {d} : suc),

which are respectively caused by (∃b ∈ Ac.{b} : f ail)

or (∃not c ∈ Re. {c} : suc) or (∃

∼

d ∈ Sus. {d} : suc).

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

that a ∈ J.

5 MULTIPLE KNOWLEDGE

BASE CONSTRAINED BY

STATE

A distributed knowledge system, constrained by a

state, is formulated as multiple knowledge base (a tu-

ple of the algebraic rule sets as above described). We

suppose that the negatives (rejective negations) from

each rule set is common. We then have the meaning of

multiple knowledge base with common negatives, by

presenting a treatment of ﬁxed point theory. The state

transitions may be caused by sequential applications

of state constrained, distributed knowledge systems,

where nondeterministic choices of the next state tran-

sition are admissible.

Distributed Knowledge System

With an assumed state set S, and a variable s over

the set S, we have got a formal deﬁnition of the state

constraint, distributed knowledge system (of multiple

knowledge base) DK containing the rule sets.

DK ::= (s)Σ

Σ ::= null | R[s];Σ

where R is deﬁned in Section 2 and “;” (semicolon)

means the concatenation, with the empty list null.

The list (sequence) Σ may be alternatively denoted

by the form of tuple Σ = ⟨R

], . . . , R

]⟩ (n ≥ 0),

where (s)Σ may be dealt with, as follows.

(a) Σ is a distributed knowledge system consisting of

rule sets R

, . . . , R

. If n = 0 then Σ = null (as the

empty tuple). s is a constraint state of (s)Σ.

(b) A ﬁxed point method with the models of R

, . . . ,

may be deﬁnable, on condition that the negatives

(rejective negations) should be common among the

models.

as in R

] indicates the next state, with some

model as an operation on the rule set R

and after the

operation.

For a distributed knowledge system (with a con-

straint state s), Σ = ⟨R

], . . . , R

]⟩, we may have

models of R

, . . . , R

, such that negatives are com-

mon. A tuple of rule sets σ = ⟨R

, . . . , R

⟩ (n ≥ 1) is

adopted, in correspondence to Σ. To get models of a

distributed knowledge system, we extend the method

(in Section 3) of how each rule set is denoted.

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

are to be executed

for 1 ≤ k ≤ n.

Evaluation and Model for Rule Sets

With (V

, . . . , V

), the value Eval

,...,V

)

(σ) is given

COMPLEXIS 2024 - 9th International Conference on Complexity, Future Information Systems and Risk

by:

Eval

,...,V

)

(σ)

= Eval

,...,V

)

(⟨R

, . . . , 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 σ of rule sets,

Eval

),...,V

))

(σ) = (t, . . . , t)

then the pairs are called a model of σ. Just taking a

direct product of pairs each of which may be deﬁned

by the mapping of Trans

(1 ≤ k ≤ n), we have a

mapping of Tr

Mapping Associated with Tuple of Rule Sets

Given a tuple of rule sets σ = ⟨R

, . . . , R

⟩, a mapping

: (2

× 2

)

→ (2

× 2

)

is deﬁned to be

((I

, J

), . . . , (I

, J

))

= (Trans

, J

), . . . , Trans

, J

))

= ((I

′

, J

′

), . . . , (I

′

, J

′

))

such that each pair (I

′

, J

′

) is given for the pair (I

, J

)

with Trans

, where Trans

follows Deﬁnition 1.

Following Proposition 2, the ﬁxed point of the

mapping Tr

may be possibly obtained such that each

(1 ≤ k ≤ n) is modeled, with the condition that

negatives are common:

Proposition 4. If

((I

, J), . . . , (I

, J)) = ((I

, J), . . . , (I

, J))

such that I

∩ J =

0 (1 ≤ k ≤ n) then

Eval

(V (I

,J),...,V (I

,J))

(σ) = (t, . . . , t),

i.e., eval

V (I

,J)

) = t (1 ≤ k ≤ n).

Proof. For each R

, the proof of Proposition 2 can

be applied, where each Trans

is independent of an-

other Trans

(k ̸= j), except that the set J is com-

mon. Tr

is deﬁned in terms of such mutually inde-

pendent Trans

. Therefore the above ﬁxed point of

is a tuple of the consistent ﬁxed points of Trans

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

is asso-

ciated with Eval

(V (I

,J),...,V (I

,J))

(σ), which is a tuple

of n values of t.

The ﬁxed points (I

, J) (1 ≤ k ≤ n) are components

of the (tuple) ﬁxed point of σ, in:

Σ = ⟨R

], . . . , R

]⟩ (n ≥ 1)

With (I

, J) (1 ≤ k ≤ n) as models in Σ, we may design

question answering on the model (I

, J) of the rule set

of Σ, and the state transition to s

for 1 ≤ k ≤ n. We

have nondeterministic choices to s

by (I

, J), (I

, J) :

s 7→ s

for 1 ≤ k ≤ n. With respect to models (I

, J)

(1 ≤ k ≤ n), the distributed derivation (Yamasaki and

Sasakura, 2023) and its soundness can be corrected

and revised to the derivations working for rule sets R

(1 ≤ k ≤ n):

(1)

0 : k-suc.

(2) When there is (Ac, Re, Sus) ▷ a ∈ R

such that

(∀(Ac

, Re

, Sus

) ▷ not a ∈ R

. Ac

∪ Re

∪

Sus

: f ail) and (∀(Ac

, Re

, Sus

) ▷

∼

a ∈

. Ac

∪ Re

∪ Sus

: f ail), with (Ac ∪ Re ∪

Sus ∪ G : k-suc),

then {a} ∪ G : k-suc.

(3) If {a} : f ail and G : k-suc then {not a} ∪ G :

k-suc.

(4) If {a} : f ail and G : k-suc then {

∼

a}∪G : k-suc.

(5) If, for any 1 ≤ j ≤ n, (∀(Ac, Re, Sus) ▷ a ∈ R

(Ac ∪ Re ∪ Sus : f ail)), then {a} : f ail.

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

(7) If {a} : k-suc then {not a} ∪ G : f ail.

(8) If {a} : k-suc then {

∼

a} ∪ G : f ail.

Meaning of Distributed Knowledge Systems

The distributed knowledge system (with a constraint

state) DK is to denote the set of objects (of the form)

(s)Σ. DK

∗

stands for the set of ﬁnite sequences from

DK including the empty sequence λ. The sequence

of DK

∗

may denote a ﬁnite number of sequential ap-

plications of the objects (possibly considered as pro-

cesses) in DK, or a ﬁnite number of transition se-

quences of objects in DK. The semantics for the se-

quence in DK

∗

is deﬁned inductively as follows, for a

state set S and its power set 2

Sem : DK

∗

→ (S → 2

such that for a state s ∈ S,

Sem[[(s

′

)null]](s) =

0 (with any s

′

∈ S)

Sem[[λ]](s) = {s}

Sem[[(s

′

)Σ]](s)











, . . . , s

} if (s = s

′

) and (∃(I

, J).

, J) is a model of R

)

for any k (1 ≤ k ≤ n) in

Σ = ⟨R

], . . . , R

]⟩

0 otherwise

Sem[[Xy]](s)

= ∪

t∈Sem[[X]](s)

Sem[[y]](t) (X ∈ DK, y ∈ DK

∗

)

Algebraic Structure of Recursively Constructed References and Its Application to Knowledge Base

We may have an intuition of what Sem[[X]] is for

X ∈ DK. The state transitions are abstracted into

the meaning of the object (named X) containing the

distributed knowledge system Σ, on condition that

there is some model (I

, J) for each R

within the dis-

tributed system Σ. Now let

Sem[[(s)null]] = Ω (s ∈ S)

Sem[[λ]] = Λ (λ ∈ DK

∗

)

Sem[[x]] = [[x]] (x ∈ DK

∗

− {λ})

With these notations, let A be inductively deﬁned in

Backus Normal Form:

A ::= Ω | Λ | [[x]] | A + A | A ◦ A

Deﬁnition 5. For α, β ∈ A , we deﬁne α = β, if α(s)

= β(s) for any s ∈ S.

(1) The addition + is deﬁned on A : For s ∈ S,

(α + β)(s) = α(s) ∪ β(s)

(2) The multiplication ◦ is deﬁned on A : For s ∈ S,

(α ◦ β)(s) = ∪

t∈α(s)

β(t)

We have seen that Ω(s) =

0 and Λ(s) = {s} for

any s ∈ S. It is seen by induction that [[x]] ◦ [[y]] = [[xy]]

for x, y ∈ DK

∗

. Without such reduction to [[xy]] from

[[x]] ◦ [[y]], we pay attention to the algebraic structure

of A. Then we can have a semiring (A, +, ◦, Ω, Λ).

6 CONCLUSION

The algebraic structure of references to website with

situations is abstractly formulated by means of re-

cursion of reference constructions, to make complex

structures of references clearer. The algebraic struc-

ture is equipped with ﬁxed point model, when the

mapping, associated with the recursively constructed

references, may have a consistent ﬁxed point such that

rejective and suspended negations cannot be contra-

dictory to the acquisitive positive..

About algebraic structure of a distributed knowl-

edge system, organized by a tuple of algebraic struc-

tures as multiple knowledge bases, and sequential

applications of such systems, we have the results,

in knowledge management technologies: (a) A state

constraint structure is formulated such that it may rep-

resent a distributed knowledge system and be mod-

eled by ﬁxed point theory. (b) The structure con-

tains common negatives among knowledge bases con-

stituting the whole knowledge system. (c) As re-

gards sequential application of the distributed knowl-

edge systems, the traverses of states contain algebraic

semiring, with multiplication for the concatenation

(to form sequences) and with addition for nondeter-

ministic choices of state transitions, which are caused

by consecutive applications of distributed knowledge

systems.

REFERENCES

Bentham, J. V., Mierzewski, K., and Blando, F. (2022).

The modal logic of stepwise removal. Rev.Symb.Log.,

15(1):36–63.

Bezhanishvili, N., G.Grilletti, and Quadrellaro, D. (2022).

An algebraic approach to inquisitive and dna-logics.

Rev.Symb.Log., 15(4):950–990.

Cardelli, L. and Gordon, A. (2000). Mobile ambients. The-

oret.Comput.Sci., 240(1):177–213.

Droste, M., Kuich, W., and Vogler, H. (2009). Handbook of

Weighted Automata. Springer.

Egre, P., Marty, P., and Renne, B. (2021). Knowledge,

justiﬁcation and adequate reasoning. Rev.Symb.Log.,

14(3):681–727.

Fitting, M. (2002). Modal logics between propositional and

ﬁrst-order. J.Log.comput., 12(6):1017–1026.

Gebser, M. and Schaub, T. (2016). Modeling and language

extensions. AI Magazine, 3(3):33–44.

Gessel, T. V. (2022). Questions in two-dimensional logic.

Rev.Log.Comput., 15(4):859–879.

Hanks, S. and McDermott, D. (1987). Nonmonotonic logic

and temporal projection. Artiﬁ.Intelli., 33(3):379–

412.

Kaufmann, B., Leone, N., Perri, S., and Schaub, T. (2016).

Grounding and solving in answer set programming. AI

Magazine, 3(3):25–32.

Merro, M. and Nardelli, F. (2005). Behavioural theory for

mobile ambients. J.ACM., 52(6):961–1023.

Pawlowski, P. and Urbaniak, R. (2018). Many-valued logic

of informal provability: A non-deterministic strategy.

Review.Symb.Log., 11(2):207–223.

Rasga, J., Sernadas, C., and Carnielli, W. (2021). Reduction

techniques for proving decidability in logics and their

meet-combination. Bull.Symb.Log., 27(1):39–66.

Reiter, R. (2001). Knowledge in Action. MIT Press.

Rin, B. and Walsh, S. (2016). Realizability semantics for

quantiﬁed modal logic: generalizing ﬂagg’s 1985 con-

struction. Rev.Symb.Log., 9(4):752–809.

Spalazzi, L. and Traverso, P. (2000). A dynamic logic

for acting, sensing and planning. J.Log.Comput.,

10(6):787–821.

Tennant, N. (2021). What is a rule of inference.

Rev.Symb.Log., 14(2):307–346.

Yamasaki, S. (2006). Logic programming with default,

weak and strict negatinns. Theory Pract.Log.Program,

6(6):737–749.

Yamasaki, S. and Sasakura, M. (2021). Algebraic expres-

sions with state constraints for causal relations and

data semantics. In Communications in Computer and

Information Science 1446, pages 245–266.

Yamasaki, S. and Sasakura, M. (2023). Abstraction of

prevention conceived in distributed knowledge base.

In Proceedings of the 6th International Conference

on Complexity, Future Information Systems and Risk,

pages 39–46.

Yamasaki, S. and Yokono, H. (2008). An analytic method

of seasonal reference in japanese haiku-poem. IPSI

Trans. on AR, 4(1):27–33.

COMPLEXIS 2024 - 9th International Conference on Complexity, Future Information Systems and Risk