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:
σ
1
, P
1
, σ
2
, ..., σ
n
, P
n
, σ
n+1
(n > 0),
where:
(i) P
1
, ..., P
n
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
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
(ii) σ
1
, ..., σ
n
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
= (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
1
. . . x
n
| n 0, x
1
, . . . , x
n
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
= (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
1
.. .Pr
n
to be connected by “and”)
versus Conclusion, like the inference rule.
(1)
move
R
(ε;σ;σ)
(2)
(x, G) R Sem[[x]]σ
1
= σ
2
move
R
(G;σ
2
;σ
2
)
move
R
(x;σ
1
;σ
2
)
(3)
move
R
(G
1
;σ
1
;σ
2
) move
R
(G
2
;σ
2
;σ
3
)
move
R
(G
1
G
2
;σ
1
;σ
3
)
The Meaning of the Relation move
R
Intuitively, the relation move
R
O
× Σ × Σ is de-
fined, such that by move
R
(γ;σ
1
;σ
2
), we mean that:
Given the sequence γ initiated, the state transition
from σ
1
to σ
2
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
1
= σ
2
and
the sequence G causes a state transition σ
2
σ
2
(from σ
2
to σ
2
), then the object x causes a transi-
tion σ
1
σ
2
(from σ
1
to σ
2
).
(iii) Forthe sequences G
1
and G
2
, respectivelycausing
the transitions from σ
1
to σ
2
and from σ
2
to σ
3
,
the sequence G
1
G
2
causes the transition from σ
1
to σ
3
.
We denote the derivation of the predicate
move
R
(G;σ
1
;σ
2
) with applications of the inference
rules (1)-(3) finitely many times, by the notation
move
R
(G;σ
1
;σ
2
)” 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
1
, σ
2
) stands for (σ
1
, σ
2
) E f fect[[β]].
The relation E f fect[[β]] means that if (σ
1
, σ
2
) is in-
cluded in, it denotes a possible transition from σ
1
to
σ
2
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
1
= σ
2
E f fect[[G]](σ
2
;σ
2
)
E f fect[[xG
]](σ
1
;σ
2
)
(3)
E f fect[[G
1
]](σ
1
;σ
2
) E f f ect[[G
2
]](σ
2
;σ
3
)
E f fect[[G
1
G
2
]](σ
1
;σ
3
)
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
R
and E f fect[[]].
Proposition 1.
G.[move
R
(G;σ
1
;σ
2
)] iff F.[E f fect[[F]](σ
1
;σ
2
)].
We have a procedure to extract a real sequence to
cause the state transition:
Object-sequence Formation for :
formation(G;σ
1
;σ
2
)
if G = ε
then if σ
1
= σ
2
then ε (empty sequence)
else
if G = x such that (x, G
) R and Sem[[x]]σ
1
= σ
2
then x followed by formation(G
;σ
2
;σ
2
)
else
if G = G
1
G
2
such that
formation(G
1
;σ
1
;σ
1
) and
formation(G
2
;σ
1
;σ
2
) are defined for some σ
1
then
formation(G
1
;σ
1
;σ
1
)
followed by f ormation(G
2
;σ
1
;σ
2
)
We can have the proposition between the effect of
procedure f ormation and E f fect[[]]:
Proposition 2. G.[formation(G;σ
1
;σ
2
) provides F]
iff E f fect[[F]](σ
1
;σ
2
).
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 = <
1
, . . . ,
n
;A > (n 1),
where each
i
is a system (O, Σ, Sem
i
, E f fect
i
, R
i
) for
knowledge structure, and awareness A is defined as
follows. We here assume a mapping
A : Σ {receive
i
, send
j
| 1 i, j n},
where if we have send
j
, receive
i
A (σ), then it is
described by:
j
σ
i,
which means that there may be a communication from
the calculus
j
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
σ
i for
any σ Σ and for any
i
. That is,
σ
is reflexive,
which is implicitly included in the following infer-
ence rules.
The inference rule of the provious section for
move
R
may be generalized to the one, move
R
i
(1 i n). The relation move
R
i
of the system
i
involves the usage of the function Sem
j
of the system
j
.
Inference Rules for
i
by Means of R
i
:
(1)
move
R
i
(ε;σ;σ)
(2)
(x, G) R
i
Sem
j
[[x]]σ
1
= σ
2
j
σ
2
i move
R
i
(G;σ
2
;σ
2
)
move
R
i
(x;σ
1
;σ
2
)
(3)
move
R
i
(G
1
;σ
1
;σ
2
) move
R
i
(G
2
;σ
2
;σ
3
)
move
R
i
(G
1
G
2
;σ
1
;σ
3
)
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
i
[[]]. Different from the relation E f fect[[]],
the relation E f fect
i
[[]] is concerned with the
sequence caused only by the follower relation of the
system
i
, but not any sequence caused by other
systems
j
(ı 6= j).
Constructive Definition of E f fect
i
(1)
E f fect
i
[[ε]](σ, σ)
(2)
(x, G) R
i
Sem
j
[[x]]σ
1
= σ
2
j
σ
2
i E f fect
i
[[F]](σ
2
;σ
2
)
E f fect
i
[[xF]](σ
1
;σ
2
)
(i = j)
(x, G) R
i
Sem
j
[[x]]σ
1
= σ
2
j
σ
2
i E f fect
i
[[F]](σ
2
;σ
2
)
E f fect
i
[[F]](σ
1
;σ
2
)
(i 6= j)
(3)
E f fect
i
[[F
1
]](σ
1
;σ
2
) E f f ect
i
[[F
2
]](σ
2
;σ
3
)
E f fect
i
[[F
1
F
2
]](σ
1
;σ
3
)
Effective Sequence from the Relation move
R
i
The following theorem suggests that we can have an
effective sequence F with E f fect
i
[[F]] on the basis of
the relation move
R
i
. The proof is presented in Ap-
pendix.
Theorem 1. On the assumption of a distributed system
(for knowledge structure) is an n-tuple
DS =<
1
, . . . ,
n
;A > (n 1),
where each
i
is a system (O, Σ, Sem
i
, E f fect
i
, R
i
) for
knowledge structure, suppose that move
R
i
(G;σ
1
;σ
2
).
Then there is a sequence F such that
E f fect
i
[[F]](σ
1
, σ
2
).
The Relation Move
R
i
caused by E f fect
i
[[]]
The following theorem suggests that we can havea re-
lation move
R
i
on the basis of the relation E f fect
i
[[]].
The proof is in Appendix.
Theorem 2. If E f fect
i
[[F]](σ
1
, σ
2
) then there is a se-
quence G such that move
R
i
(G;σ
1
;σ
2
).
Equivalence between move
R
i
and E f f ect
i
[[]]
By Theorems 1 and 2, we have:
Corollary 1. There is a sequence G such that
move
R
i
(G;σ
1
;σ
2
) iff there is a sequnce F such that
E f fect
i
[[F]](σ
1
, σ
2
).
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
i
(G;σ
1
;σ
2
)
if G = ε
then
if σ
1
= σ
2
then ε (empty sequence)
else
if G = x such that (x, G
) R
i
, j
σ
2
i,
and Sem
j
[[x]]σ
1
= σ
2
then
if (i = j)
then x followed by d-formation
i
(G
;σ
2
;σ
2
)
else d-formation
i
(G
;σ
2
, σ
2
)
else
if G = G
1
G
2
such that
d-formation
i
(G
1
;σ
1
;σ
1
) and
d-formation
i
(G
2
;σ
1
;σ
2
) are defined
for some σ
1
then
d-formation
i
(G
1
;σ
1
;σ
1
)
followed by d-formation
i
(G
2
;σ
1
;σ
2
)
The following theorem, whose proof is given
in Appendix, states the effect(s) by the procedure
d-formation
i
as well as in terms of E f fect
i
[[]].
Theorem 3. d-formation
i
(G;σ
1
;σ
2
) provides F for
some G iff E f fect
i
[[F]](σ
1
;σ
2
).
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
R
i
(G;σ
1
;σ
2
), E f fect
i
[[F]](σ;σ
2
) and d-
formation
i
(G
;σ
1
;σ
2
), are concerned with the
state transition from σ
1
to σ
2
, 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
R
i
(G;σ
1
;σ
2
).
(a) In case that G = ε, it follows for the assumed
predicate move
R
i
(G;σ
1
;σ
2
) that σ
1
= σ
2
. Then
E f fect
i
[[ε]](σ
1
, σ
2
) where σ
1
= σ
2
.
SEQUENTIAL KNOWLEDGE STRUCTURE IN DISTRIBUTED SYSTEM WITH AWARENESS
297
(b) In case that G = x O, the predicate
move
R
i
(x;σ
1
;σ
2
) is derived by means of
the inference rule (2). Assume for the
predicate move
R
i
(x;σ
1
, σ
2
) that (x, G
) R,
Sem
j
[[x]]σ
1
= σ
2
, j
σ
2
i and move
R
i
(G
;σ
2
, σ
2
)
for G
. For the predicate move
R
i
(G
;σ
2
, σ
2
), by
induction hypothesis, there is a sequence F such
that E f fect
i
[[F]](σ
2
, σ
2
). By means of the rules
(2) for effects: If i = j, then E f fect
i
[[xF]](σ
1
, σ
2
).
If i 6= j, then E f fect
i
[[F]](σ
1
, σ
2
).
(c) In case that G = G
1
G
2
, we assume the predicate
move
R
i
(G
1
G
2
;σ
1
;σ
2
). For the predicate
move
R
i
(G
1
G
2
;σ
1
;σ
2
),
assume that move
R
i
(G
1
;σ
1
;σ
2
) and move
R
i
(G
2
;
σ
2
;σ
2
) for some σ
2
. By induction hypothesis, we
can see that for some F
1
and F
2
,
E f fect
i
[[F
1
]](σ
1
, σ
2
), and
E f fect
i
[[F
2
]](σ
2
, σ
2
).
It follows from the rule (3) for effects: with F
1
F
2
,
E f fect
i
[[F
1
F
2
]](σ
1
, σ
2
).
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
i
[[ε]](σ
1
, σ
1
) holds, we indepen-
dently have the predicate move
R
i
(ε;σ
1
;σ
2
) for
σ
2
= σ
1
, which is sufficient for the proof.
(b) Assume the case that E f fect
i
[[F]](σ
1
, σ
2
) by
(x, G
) R
i
, j
σ
2
i, Sem
j
[[x]]σ
1
= σ
2
and
E f fect
i
[[F
]](σ
2
, σ
2
) (F = xF
). By induction hy-
pothesis, move
R
i
(G
;σ
2
;σ
2
) for some G
. Then
we have the predicate move
R
i
(G;σ
1
;σ
2
) by the
move
R
i
definition.
(c) In case that E f fect
i
[[F
1
F
2
]](σ
1
, σ
2
). It is sup-
ported that, for some F
1
and F
2
,
(i) E f fect
i
[[F
1
]](σ
1
, σ
1
), and
(ii) E f fect
i
[[F
2
]](σ
1
, σ
2
).
By induction hypothesis, we have both
move
R
i
(G
1
;σ
1
;σ
1
)
for some G
1
and
move
R
i
(G
2
;σ
1
;σ
2
)
for some G
2
. It follows that move
R
i
(G
1
G
2
;
σ
1
;σ
2
). 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
i
(ε;σ
1
;σ
1
) provides ε iff
and E f f ect
i
[[ε]](σ
1
;σ
1
).
(b) (Induction 1) Assume that G = x O. By the defi-
nitions of d- formation
i
and E f fect
i
, we see that:
(i) d-formation
i
(x;σ
1
;σ
2
) provides F iff there is
some (x, G
) R
i
such that Sem
j
[[x]]σ
1
= σ
2
,
j
σ
2
i and
F = x. d-formation
i
(G
;σ
2
;σ
2
) (i = j)
or F = d-formation
i
(G
;σ
2
;σ
2
) (i 6= j).
(ii) E f fect
i
[[F]](σ
1
;σ
2
) iff there is some (x, G
)
R
i
such that Sem
j
[[x]]σ
1
= σ
2
, j
σ
2
i and
E f fect
i
[[F
]](σ
2
, σ
2
) for F = xF
(i = j),
or E f fect
i
[[F
]](σ
2
, σ
2
) for F = F
(i 6= j).
By induction hypothesis, for some G
,
d-formation
i
(G
;σ
2
;σ
2
) provides F
iff
E f fect
i
[[F
]](σ
2
;σ
2
).
It follows that d-formation
i
(x;σ
1
;σ
2
) provides F
iff
E f fect
i
[[F]](σ
1
;σ
2
).
This completes the induction 1.
(c) (Induction 2) Assume that G = G
1
G
2
6= ε. By the
definitions of d-formation
i
and E f f ect
i
, we see
that:
(i) d-formation
i
(G
1
G
2
;σ
1
;σ
2
) provides F iff
F = d-formation
i
(G
1
;σ
1
;σ
1
)
followed by d-formation
i
(G
2
;σ
1
;σ
2
)
for some σ
1
.
(ii) E f fect
i
[[F]](σ
1
;σ
2
) iff F = F
1
F
2
for some F
1
and F
2
with some σ
1
, where
E f fect
i
[[F
1
]](σ
1
, σ
1
), and
E f fect
i
[[F
2
]](σ
1
, σ
2
).
By induction hypothesis, we see that:
d-formation
i
(G
1
;σ
1
;σ
1
) provides F
1
iff E f fect
i
[[F
1
]](σ
1
, σ
1
), and
d-formation
i
(G
2
;σ
1
;σ
2
) provides F
2
iff E f fect
i
[[F
2
]](σ
1
, σ
2
).
It follows that d-formation
i
(G
1
G
2
;σ
1
;σ
2
)
provides F
1
F
2
iff E f fect
i
[[F
1
F
2
]](σ
1
, σ
2
). This
completes the induction 2.
Q.E.D.
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