SEQUENTIAL KNOWLEDGE STRUCTURE IN DISTRIBUTED

SYSTEM WITH AWARENESS

Susumu Yamasaki

Department of Computer Science, Okayama University, 3-1-1 Tsushima-Naka, Okayama, Japan

Keywords:

Knowledge structure, Awareness, State transition.

Abstract:

This position paper deals with a formal system to manage sequential knowledge structure for Web site page

analyses in a distributed system, by means of the rule-based state transition. Agent technology in AI (of

modern approach), logic and database for relations between action and knowledge, process algebra and re-

lated logic with respect to distributed environments, and structural analyses (which may be static, interactive

or constrained) of referential knowledge for Web site pages may be relevant, however, the present paper is

concerned with the sequences like Web site page ones abstracted for a model of content retrievals through

communications among sites via the internet. Awareness depending of states on the communications between

sites may be adopted so that sequential knowledge acquisition and management would be possibly available.

1 INTRODUCTION

We make analyses on objective knowledge: As an ex-

ample of objective knowledge, the Web site page con-

tains varietiesof meanings. Exploring Web site pages,

the sequence makes sense which is often related to

state transitions caused by page contents. Under the

constraint of state transitions, we may be interested

in an automated implementation to form a sequence

composed of subsequences generated by distributed

calculi. For a model of forming sequence of objec-

tive knowledge, we assume a calculus as well as a

managing scheme based on awareness of reasonable

communications between calculi.

This paper is concerned with a model of forming

sequences of objective knowledge. It is a motive to

automate sequence formations, where:

(i) the sequence is composed of subsequences gener-

ated by distributed calculi, and

(ii) the constraint on state transitions for the formation

comes from awareness of a calculus.

To see sequential knowledge induced by Web

pages and to reach a formal system with some imple-

mentable procedures for content retrievals, we pay at-

tention to relevance to some established frameworks:

(1) Agent technologies as compiled (Russell, 1995)

may ﬁt descriptions of knowledge sequence for-

mation.

(2) Logic and database views are fundamental to an-

alyze knowledge structure (Minker, 1987; Shep-

herdson, 1987), to understand dynamic structure

with reference to knowledge (Mosses, 1992; Re-

iter, 2001), and to make use of distributed nega-

tives.

(3) Process algebra is concerned with sequence struc-

ture of communications (evaluations) (Hoare,

1985; Milner, 1989) even for distributed systems

(Bruns, 1996), while its logical formulation is dis-

cussed (Kucera and Esparza, 2003) such that logi-

cal frameworks may conceive modality and nom-

ination (Areces and Blackburn, 2003; Brauner,

2004).

(4) References of Web site pages may be examined

from static link view (Yamasaki, 2009) and in-

teractive mechanism with constraints (Yamasaki,

2007).

We can observe a way of knowledge acquisition,

in terms of Web site pages. Firstly, the Web site page

denotes a relation between states. Secondly, as re-

gards a visit sequence to Web site pages, we abstract

a sequential structure such as:

, P

, σ

, ..., σ

, P

, σ

n+1

(n > 0),

where:

(i) P

, ..., P

are pages, and

293

Yamasaki S..

SEQUENTIAL KNOWLEDGE STRUCTURE IN DISTRIBUTED SYSTEM WITH AWARENESS.

DOI: 10.5220/0003677402930298

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2011), pages 293-298

ISBN: 978-989-8425-80-5

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

(ii) σ

, ..., σ

are to denote states abstracted from

knowledge base.

Thirdly, we see structure of Web site pages such that:

• A page contains a list of references to pages.

As regards the description of recursive links, a se-

quence of pages included in a given page is taken in

this paper rather than a set of them, such that a se-

quential knowledge structure is studied. Such a re-

cursive structure forms constraints of visiting pages

as well as state transition sequences. We can, in

what follows, construct a calculus to realize sequen-

tial structures based on the above 3 points.

For a distributed system containing calculi (which

is deﬁned in this paper), we see that awareness

(Agotnes and Alechina, 2007) can manage the con-

nection of calculi with communication channels, to

compose subsequences generated by local calculi.

2 A MODEL FOR KNOWLEDGE

ACQUISITION

To model an acquisition scheme of keywords possibly

for Web usabiblity, we have considered the case that

the keywordscontain both positiveand negative infor-

mations to denote a content. In what follows, we have

some itemized aspects of behaviours of the model for

keywords acquisition.

• A request containing keywords supposedly

searches the Web site pages involving keywords.

• The reliable response of a Web page to the request

causes an enumeration of the page and an inclu-

sion in a list of the request data.

• A managing process with a request is interactive

with multi-site (of a distributed system), where

each page contains a recursive link structure in a

site.

• How any page supposedly responds to the request

(as a program with knowledge content) is:

if the keywords of the page are consistent with

those of the request, then they are to be merged

with those of the request. If the keywords of the

page are inconsistent with those of the request, the

keywordskept in the request are revised to be con-

sistent with the page ones.

(The request searches a page in the sense that their

keywords are mutually consistent, and also acquire

consistent keywords from the page.)

We can design a whole system, consisting of a

managing program, a request, and sites with their own

pages:

(a) A managing program is interactive with a site

through a request of keywords. When there are

more than one interaction requirement of sites,

only one from a site is selected, and other require-

ments are kept until the interaction would be over.

(b) The request is a data structure with a function ac-

quires consistent keywords from pages in a site

and to integrates consistent keywords. The key-

words contained by it may be changed through

visits to site pages. In each site of the system,

there are pages under the site environment. Each

page of a site involves a program (to make the re-

quest data consistently revised) for keywords. If

the page contains consistent keywords with those

of the request, it is regarded as reliable. Other-

wise, the request may be consistently revised.

We can observe the state denoted by keywords of

the above data structure “request”, such that we now

have the structure of a formal system design and its

management, abstracting the Web site visit sequence

as well as knowledge acquisition. In this case, knowl-

edge acquisition may be made by state transitions,

changing situations of knowledge (which is realized

by keywords).

We will have a formal system involving knowl-

edge structure. It contains:

(i) a set of objects referring to pages.

(ii) a set of states.

(iii) a semantic function causing a state transition, as-

signed to each object.

(iv) a function to denote effects of an object sequence.

(v) a follower relation to represent an object sequence

succeeding an object, which is regarded as a rule

with constraints.

3 FORMAL SYSTEM FOR

KNOWLEDGE STRUCTURE

When the copying the page from each site (a spe-

ciﬁc local place) to the internet (the common space)

is allowed, the communication (the copying) between

any two sites is possible. On the assumption that the

copying of this direction is allowed, that is, the com-

munication of the page transfer, we have a system

in a distributed environments. Before the distributed

system description, we have a formal system as fol-

lows: A system for knowledge structure is a quintuple

ℑ = (O, Σ, Sem, E f fect, R), where:

(i) O is a set of objects.

(ii) Σ is a set of states.

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

294

(iii) Sem: O → (Σ → Σ) is a semantic function.

(iv) E f fect: O

∗

→ 2

Σ×Σ

is a function (to denote ef-

fects of a sequence of objects). Note that O

∗

. . . x

| n ≥ 0, x

, . . . , x

∈ O}. The empty se-

quence in O

∗

is denoted by ε.

(v) R ⊆ O× O

∗

is a follower relation, where (x, G) ∈

R means that x is followed by G.

Inference Rules by Means of the Follower Relation R:

We deﬁne the inference rules (1), (2) and (3), on the

assumption that a system

ℑ = (O, Σ, Sem, E f fect, R)

is given. Note that the inference rule is deﬁned by a

scheme of “assumptions vs. conclusion” as in proof

theory. That is, we have the notation: Assumptions

(of the form “Pr

.. .Pr

” to be connected by “and”)

versus Conclusion, like the inference rule.

(1)

move

(ε;σ;σ)

(2)

(x, G) ∈ R Sem[[x]]σ

= σ

′

move

(G;σ

′

;σ

)

move

(x;σ

;σ

)

(3)

move

;σ

) move

;σ

)

move

;σ

)

The Meaning of the Relation move

Intuitively, the relation move

⊆ O

∗

× Σ × Σ is de-

ﬁned, such that by move

(γ;σ

;σ

), we mean that:

• Given the sequence γ initiated, the state transition

from σ

to σ

is caused by rewriting (owing to

the follower relation) and reducing γ to the empty

sequence.

We have accounts for the above inference rules.

(i) The empty sequence ε causes the empty state tran-

sition.

(ii) If there is (x, G) ∈ R such that Sem[[x]]σ

= σ

′

and

the sequence G causes a state transition σ

′

→ σ

(from σ

′

to σ

), then the object x causes a transi-

tion σ

→ σ

(from σ

to σ

(iii) Forthe sequences G

and G

, respectivelycausing

the transitions from σ

to σ

and from σ

to σ

the sequence G

causes the transition from σ

to σ

We denote the derivation of the predicate

move

(G;σ

;σ

) with applications of the inference

rules (1)-(3) ﬁnitely many times, by the notation

“move

(G;σ

;σ

)” itself.

The Meaning of the Expression E f fect

In accordance with the inferences, the effects of an

object sequence are deﬁned as follows. Note that

E f fect[[β]](σ

, σ

) stands for (σ

, σ

) ∈ E f fect[[β]].

The relation E f fect[[β]] means that if (σ

, σ

) is in-

cluded in, it denotes a possible transition from σ

by means of a sequence β. That is, β is an effective

sequence for the transition.

(1)

E f fect[[ε]](σ,σ)

(2)

(x, G) ∈ R Sem[[x]]σ

= σ

′

E f fect[[G]](σ

′

;σ

)

E f fect[[xG

′

]](σ

;σ

)

(3)

E f fect[[G

]](σ

;σ

) E f f ect[[G

]](σ

;σ

)

E f fect[[G

]](σ

;σ

)

The following propositions are regarded as the

special cases of corresponding Corollary 1 and Theo-

rem 3 in distributed systems.

We have the following proposition between the

transition expressed by move

and E f fect[[−]].

Proposition 1.

∃G.[move

(G;σ

;σ

)] iff ∃F.[E f fect[[F]](σ

;σ

)].

We have a procedure to extract a real sequence to

cause the state transition:

Object-sequence Formation for ℑ:

formation(G;σ

;σ

) ⇐

if G = ε

then if σ

= σ

then ε (empty sequence)

else

if G = x such that (x, G

′

) ∈ R and Sem[[x]]σ

= σ

′

then x followed by formation(G

′

;σ

′

;σ

)

else

if G = G

such that

formation(G

;σ

′

) and

formation(G

;σ

′

;σ

) are deﬁned for some σ

′

then

formation(G

;σ

′

)

followed by f ormation(G

;σ

′

;σ

)

We can have the proposition between the effect of

procedure f ormation and E f fect[[−]]:

Proposition 2. ∃G.[formation(G;σ

;σ

) provides F]

iff E f fect[[F]](σ

;σ

SEQUENTIAL KNOWLEDGE STRUCTURE IN DISTRIBUTED SYSTEM WITH AWARENESS

295

4 DISTRIBUTED SYSTEM FOR

KNOWLEDGE STRUCTURE

We now deal with the distributed system consisting

of calculi (as in the previous section) where the

communications between calculi are free so that

the object in the set O can be transferred from one

calculus to another.

A Distributed System

In what follows, we have a formal system to involve

an effective sequence for each calculus, to be ex-

tracted. Now a distributed system (for knowledge

structure) is an n-tuple

DS = < ℑ

, . . . , ℑ

;A > (n ≥ 1),

where each ℑ

is a system (O, Σ, Sem

, E f fect

, R

) for

knowledge structure, and awareness A is deﬁned as

follows. We here assume a mapping

A : Σ → {receive

, send

| 1 ≤ i, j ≤ n},

where if we have send

, receive

∈ A (σ), then it is

described by:

j →

which means that there may be a communication from

the calculus ℑ

to ℑ

through the state in Σ. Note

that we may see details of awareness (Agotnes and

Alechina, 2007). It can be supposed that i →

i for

any σ ∈ Σ and for any ℑ

. That is, →

is reﬂexive,

which is implicitly included in the following infer-

ence rules.

The inference rule of the provious section for

move

may be generalized to the one, move

(1 ≤ i ≤ n). The relation move

of the system ℑ

involves the usage of the function Sem

of the system

ℑ

Inference Rules for ℑ

by Means of R

(1)

move

(ε;σ;σ)

(2)

(x, G) ∈ R

Sem

[[x]]σ

= σ

′

j →

′

i move

(G;σ

′

;σ

)

move

(x;σ

;σ

)

(3)

move

;σ

) move

;σ

)

move

;σ

)

In accordance with the inference, we might have

the rules for effects of object sequences.

The relation E f fect[[−]] is extended to the one,

E f fect

[[−]]. Different from the relation E f fect[[−]],

the relation E f fect

[[−]] is concerned with the

sequence caused only by the follower relation of the

system ℑ

, but not any sequence caused by other

systems ℑ

(ı 6= j).

Constructive Deﬁnition of E f fect

(1)

E f fect

[[ε]](σ, σ)

(2)

(x, G) ∈ R

Sem

[[x]]σ

= σ

′

j →

′

i E f fect

[[F]](σ

′

;σ

)

E f fect

[[xF]](σ

;σ

)

(i = j)

(x, G) ∈ R

Sem

[[x]]σ

= σ

′

j →

′

i E f fect

[[F]](σ

′

;σ

)

E f fect

[[F]](σ

;σ

)

(i 6= j)

(3)

E f fect

[[F

]](σ

;σ

) E f f ect

[[F

]](σ

;σ

)

E f fect

[[F

]](σ

;σ

)

Effective Sequence from the Relation move

The following theorem suggests that we can have an

effective sequence F with E f fect

[[F]] on the basis of

the relation move

. The proof is presented in Ap-

pendix.

Theorem 1. On the assumption of a distributed system

(for knowledge structure) is an n-tuple

DS =< ℑ

, . . . , ℑ

;A > (n ≥ 1),

where each ℑ

is a system (O, Σ, Sem

, E f fect

, R

) for

knowledge structure, suppose that move

(G;σ

;σ

Then there is a sequence F such that

E f fect

[[F]](σ

, σ

The Relation Move

caused by E f fect

[[−]]

The following theorem suggests that we can havea re-

lation move

on the basis of the relation E f fect

[[−]].

The proof is in Appendix.

Theorem 2. If E f fect

[[F]](σ

, σ

) then there is a se-

quence G such that move

(G;σ

;σ

Equivalence between move

and E f f ect

[[−]]

By Theorems 1 and 2, we have:

Corollary 1. There is a sequence G such that

move

(G;σ

;σ

) iff there is a sequnce F such that

E f fect

[[F]](σ

, σ

KEOD 2011 - International Conference on Knowledge Engineering and Ontology Development

296

We next have a distributed procedure:

The procedure can demonstrate a sequence to

cause a transition between two given states.

Object-sequence Formation for ℑ

in DS:

d-formation

(G;σ

;σ

) ⇐

if G = ε

then

if σ

= σ

then ε (empty sequence)

else

if G = x such that (x, G

′

) ∈ R

, j →

′

and Sem

[[x]]σ

= σ

′

then

if (i = j)

then x followed by d-formation

′

;σ

′

;σ

)

else d-formation

′

;σ

′

, σ

)

else

if G = G

such that

d-formation

;σ

′

) and

d-formation

;σ

′

;σ

) are deﬁned

for some σ

′

then

d-formation

;σ

′

)

followed by d-formation

;σ

′

;σ

)

The following theorem, whose proof is given

in Appendix, states the effect(s) by the procedure

d-formation

as well as in terms of E f fect

[[−]].

Theorem 3. d-formation

(G;σ

;σ

) provides F for

some G iff E f fect

[[F]](σ

;σ

We can have concluding remarks on the dis-

tributed system where:

(i) Which calculi are a pair of a sender and a receiver

is controlled by awarenes of states. This notion is

not only a software method but an AI tool, if this

formal system is applicable to a model of a part of

brain works for sequence knowledge.

(ii) The abstract notions whcih we have presented,

move

(G;σ

;σ

), E f fect

[[F]](σ;σ

) and d-

formation

′

;σ

), are concerned with the

state transition from σ

to σ

, the extraction of

an effective sequence only from the calculus ℑ

and a sequence construction in the calculus ℑ

, re-

spectively .

The system is an abstract scheme to contribute to a

formation of sequences composed of distributed sub-

sequences, free from more speciﬁc e-Learning mech-

anism (Sasakura and Yamasaki, 2008) ane event-

formation (Yamasaki and Sasakura, 2008).

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APPENDIX

Proof of Theorem 1:

It is proved by structural induction on the construction

of the predicate move

(G;σ

;σ

(a) In case that G = ε, it follows for the assumed

predicate move

(G;σ

;σ

) that σ

= σ

. Then

E f fect

[[ε]](σ

, σ

) where σ

= σ

SEQUENTIAL KNOWLEDGE STRUCTURE IN DISTRIBUTED SYSTEM WITH AWARENESS

297

(b) In case that G = x ∈ O, the predicate

move

(x;σ

;σ

) is derived by means of

the inference rule (2). Assume for the

predicate move

(x;σ

, σ

) that (x, G

′

) ∈ R,

Sem

[[x]]σ

= σ

′

, j →

′

i and move

′

;σ

′

, σ

)

for G

′

. For the predicate move

′

;σ

′

, σ

), by

induction hypothesis, there is a sequence F such

that E f fect

[[F]](σ

′

, σ

). By means of the rules

(2) for effects: If i = j, then E f fect

[[xF]](σ

, σ

If i 6= j, then E f fect

[[F]](σ

, σ

, we assume the predicate

move

;σ

). For the predicate

move

;σ

assume that move

;σ

′

) and move

;

′

;σ

) for some σ

′

. By induction hypothesis, we

can see that for some F

and F

E f fect

[[F

]](σ

, σ

′

), and

E f fect

[[F

]](σ

′

, σ

It follows from the rule (3) for effects: with F

E f fect

[[F

]](σ

, σ

This completes the induction step.

Q.E.D.

Proof of Theorem 2:

It is proved by structural induction on a sequence

G in the premise of the theorem. For the premise:

(a) While E f fect

[[ε]](σ

, σ

) holds, we indepen-

dently have the predicate move

(ε;σ

;σ

) for

= σ

, which is sufﬁcient for the proof.

(b) Assume the case that E f fect

[[F]](σ

, σ

) by

(x, G

′

) ∈ R

, j →

′

i, Sem

[[x]]σ

= σ

′

and

E f fect

[[F

′

]](σ

′

, σ

) (F = xF

′

). By induction hy-

pothesis, move

′

;σ

′

;σ

) for some G

′

. Then

we have the predicate move

(G;σ

;σ

) by the

move

deﬁnition.

[[F

]](σ

, σ

). It is sup-

ported that, for some F

and F

(i) E f fect

[[F

]](σ

, σ

′

), and

(ii) E f fect

[[F

]](σ

′

, σ

By induction hypothesis, we have both

move

;σ

′

)

for some G

and

move

;σ

′

;σ

)

for some G

. It follows that move

;

;σ

). This completes the induction.

Q.E.D.

Proof of Theorem 3:

It is proved by structural induction on the procedure

d-formation with reference to existing G.

(a) (Basis)

We see that d- formation

(ε;σ

;σ

) provides ε iff

and E f f ect

[[ε]](σ

;σ

(b) (Induction 1) Assume that G = x ∈ O. By the deﬁ-

nitions of d- formation

and E f fect

, we see that:

(i) d-formation

(x;σ

;σ

) provides F iff there is

some (x, G

′

) ∈ R

such that Sem

[[x]]σ

= σ

′

j →

′

i and

F = x. d-formation

′

;σ

′

;σ

) (i = j)

or F = d-formation

′

;σ

′

;σ

) (i 6= j).

(ii) E f fect

[[F]](σ

;σ

) iff there is some (x, G

′

) ∈

such that Sem

[[x]]σ

= σ

′

, j →

′

i and

E f fect

[[F

′

]](σ

′

, σ

) for F = xF

′

(i = j),

or E f fect

[[F

′

]](σ

′

, σ

) for F = F

′

(i 6= j).

By induction hypothesis, for some G

′

d-formation

′

;σ

′

;σ

) provides F

′

iff

E f fect

[[F

′

]](σ

′

;σ

It follows that d-formation

(x;σ

;σ

) provides F

iff

E f fect

[[F]](σ

;σ

This completes the induction 1.

6= ε. By the

deﬁnitions of d-formation

and E f f ect

, we see

that:

(i) d-formation

;σ

) provides F iff

F = d-formation

;σ

′

)

followed by d-formation

;σ

′

;σ

)

for some σ

′

(ii) E f fect

[[F]](σ

;σ

) iff F = F

for some F

and F

with some σ

′

, where

E f fect

[[F

]](σ

, σ

′

), and

E f fect

[[F

]](σ

′

, σ

By induction hypothesis, we see that:

d-formation

;σ

′

) provides F

iff E f fect

[[F

]](σ

, σ

′

), and

d-formation

;σ

′

;σ

) provides F

iff E f fect

[[F

]](σ

′

, σ

It follows that d-formation

;σ

)

provides F

iff E f fect

[[F

]](σ

, σ

). This

completes the induction 2.

Q.E.D.

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