A Learning Method based on Bisimulation in Inconsistent KnowledgeSystems Thi Hong Khanh Nguyen, Quang-Thuy Ha and Trong Hieu Tran Abstract— Inconsistencies may naturally occur in the con
Trang 1A Learning Method based on Bisimulation in Inconsistent Knowledge
Systems
Thi Hong Khanh Nguyen, Quang-Thuy Ha and Trong Hieu Tran
Abstract— Inconsistencies may naturally occur in the
consid-ered application domains in Artificial Intelligence, for example
as a result of data mining works in distributed sources In
order to solve inconsistent knowledge, several paraconsistent
description logics have been proposed In this paper, we face the
problem of concept learning for an inconsistent knowledge base
system based on bisimulation This algorithm allows learning a
concept from a training information system in a paraconsistent
descriptive logic system with a set of positive items, negative
items, and inconsistent items Here, we present a system for
learning concept in an inconsistent knowledge base and discuss
preliminary experimental results obtained in the electronic
application domain
I INTRODUCTION Description logics (DLs) are a family of formal languages
which are well suited for representation and reasoning in a
domain of particular interest DLs is of particular importance
in providing theoretical models for semantic systems It is
the basis for building languages for modeling ontologies in
which OWL is a language that is recommended by the W3C
International Standard for use in Semantic Web systems
[14], [21] Description logics have usually been considered
as syntactic variants of restricted versions of classical
first-order logic [1] On the other hand, in Semantic Web and
multiagent applications, knowledge fusion frequently leads
to inconsistencies [13] A way to deal with inconsistencies
is to follow the area of paraconsistent reasoning [22], [21],
[15], [16], [4], [28], [8], [31]
Concept learning in DLs is similar to binary classification
in traditional machine learning The differences are that in
DLs objects are described not only by attributes but also
by the relationships between the objects As bisimulation
is the notion for characterizing indiscernibility of objects
in DLs It is very useful for concept learning in these
DLs [27], [30], [26], [9], [12] Consider a domain with
Thi Hong Khanh Nguyen, Quang-Thuy Ha and Trong Hieu Tran are
with Faculty of Information Technology, University of Engineering and
Technology, Vietnam National University, Hanoi, Vietnam
Thi Hong Khanh Nguyen is with Electricity Power University, Vietnam
khanhnth@epu.edu.vn,thuyhq@vnu.edu.vn,
hieutt@vnu.edu.vn
individuals represented by single and binary predicates In
DL language, predicates such as concept names and role names, respectively Domains can be described by different sources For instance, individuals may be some objects in an area on earth and sources of information may be computer systems of different satellites objects described by some boolean attributes and binary relationships between them Another example is the following: some banks cooperate to share information about their customers to a certain extent; the bank’s customers are individuals; atomic concepts can be credibility, wealth, and financial discipline; Atomic roles can
be some relationships based on the transformation Sources may provide consistent or inconsistent assertions Based on them, the paraconsistent interpretation can be generated as
an integrated information system
Concept learning in description logics has been studied by many researchers and is divided into three main approaches The first approach focuses on the ability to learn in descrip-tion logics and builds some simple algorithms [24], [18], [7], [11], [19] The second approach studies the learning concept in the description logic using refinement operators [6], [5], [17], [20], [2] Lehman and el [29] employed inte-grating refinement operators in terminological decision trees Third approach exploits bisimulation for concept learning problems in description logics [10], [27], [30], [9] In [19] Lambrix and Larochia proposed a simple concept learning algorithm based on the concept Lehmann and Hitzler [20], Badea and Nienhys [2], Iannone et al [17] studied concept learning in DLs by using refinement operators as in inductive logic programming Apart from refinement operators, scoring functions, and search strategies also play important roles in the algorithms proposed in those works
Nguyen and Szalas [26] applied bisimulation in DLs to model indiscernibility of objects Their work is pioneering
in using bisimulation for concept learning in DLs [27], [30], [9] The authors propose a general method for smoothing the domain of interpretation and through which to build the concept to be learned
Unlike all previous work handling consistent knowledge
2018 15th International Conference on
Control, Automation, Robotics and Vision (ICARCV)
Singapore, November 18-21, 2018
Trang 2bases In this paper, we develop a bisimulation based method
for inconsistent knowledge systems, for concept learning in
paraconsistent DLs
The rest of this paper is structured as follows In Section
2, we present notation and semantics of the paraconsistent
DLs considered in this paper We recall bisimilarity for
paraconsistent description logics in Section 3 A learning
algorithm based on bisimulation is presented in Section 4
In Section 5, we evaluate this algorithm by means of our
implementation Finally, in Section 6 we summarize our
work and draw conclusions
II PRELIMINARIES Following the recommendation of W3C for OWL,
like [25], [3], [23] we use the traditional syntax of DLs and
only change its semantics to cover paraconsistency
In this work, we consider a DL-signature is a set Σ =
I∪C∪R, where: a finite set C of concept names, a countable
set I of individual names, a countable set R of role names
We use uppercase letters like A, B to denote concept names,
lowercase letters like r, s to denote role names, and lowercase
letters like a, b to denote individual names
Let Φ be a set of features among I (inverse roles), O
(nominal), Q (qualified number restrictions), U (the universal
role) and Self (local reflexivity of a role)
Recall that, using the traditional semantics, every query is
a logical consequence of an inconsistent knowledge base A
knowledge base may be inconsistent, for instance, when it
contains both individual assertions A(a) and ¬A(a) for some
A ∈ C and a ∈ I Paraconsistent reasoning is
inconsistency-tolerant and aims to derive meaningful logical consequences
even when the knowledge base is inconsistent This problem
can be handled by three-valued logic (t: true, f: false and i:
inconsistent) The general approach is to define semantics s
such that, given a knowledge base KB , the set Conss(KB )
of logical consequences of KB w.r.t semantics s is a subset
of the set Cons(KB ) of logical consequences of KB w.r.t
the traditional semantics, with the property, that Conss(KB )
contains mainly only meaningful logical consequences of
KB and Conss(KB ) approximates Cons(KB ) as much as
possible That is, it is to weaken the traditional semantics in
an appropriate way
Paraconsistent semantics from the defined family, let’s say
s, is characterized by four parameters, denoted by sC, sR,
s∀∃Q, sGCI, with the following intuitive meanings:
• sC∈ {2, 3} specifies the number of possible truth values
of assertions of the form A(x), where A ∈ C
• sR∈ {2, 3} specifies the number of possible truth values
of assertions of the form r(x, y), where r ∈ R
• s∀∃Q ∈ {+, ±} specifies which from two considered semantics is used for concepts of the form ∀R.C, ∃R.C,
≤ n R.C or ≥ n R.C
• sGCI ∈ {w, m, s} specifies one of the three semantics for general concept inclusions: weak (w), moderate (m), strong (s)
In the case sC = 2, the truth values of assertions of the form A(x) are t (true) and f (false) In the case sC = 3, the third truth value is i (inconsistent) In the case sC = 4, the additional truth value is u (unknown) When sC = 3, one can identify inconsistency with the lack of knowledge, and the third value i can be read either as inconsistent or as unknown
Similar explanations can be stated for sR In this paper, the problem is handled on three-valued logic Therefore, the set of considered paraconsistent semantics is as follows:
S = {2, 3} × {2, 3} × {+, ±} × {w, m, s}
For s ∈ S, an s-interpretation is a pair I = h∆I, ·Ii, where ∆I is a non-empty set, called the domain, ·I is the interpretation function, which maps every individual name
a to an element aI ∈ ∆I, every concept name A to a pair
AI= hAI+, AI−i of subsets of ∆I, and every role name r to
a pair rI = hrI+, r−Ii of binary relations on ∆I such that:
• if sC= 2 then AI+= ∆I\ AI
−
• if sC= 3 then AI+∪ AI
−= ∆I
• if sR= 2 then rI+= (∆I× ∆I) \ rI−
• if sR= 3 then rI+∪ rI
−= ∆I× ∆I The intuition behind AI = hAI+, AI−i is that AI
+ gathers positive items about A, while AI− gathers negative items about A Thus, AI can be treated as the function from ∆I
to {t, f, i} defined below:
AI(x) =
t for x ∈ AI+ and x /∈ AI
−
f for x ∈ AI− and x /∈ AI
+
i for x ∈ AI+ and x ∈ AI−
(1)
Informally, AI(x) can be thought of as the truth value of
x ∈ AI Note that AI(x) ∈ {t, f} if sC = 2, and AI(x) ∈ {t, f, i} if sC = 3 The intuition behind rI = hrI+, r−Ii is similar, and under which rI(x, y) ∈ {t, f} if sR = 2, and
rI(x, y) ∈ {t, f, i} if sR= 3
The interpretation function ·I maps a role R to a pair
RI = I+, RI− , defined for the case R /∈ R as follows:
(r−)I = I+)−1, (r−I)−1
UI = I× ∆I, ∅
Trang 3Function · maps a complex concept C to a pair C =
hCI
+, C−Ii of subsets of ∆I defined in [27] as follows:
>I = h∆I, ∅i
⊥I = h∅, ∆Ii
({a})I = I}, ∆I\ {aI}
(¬C)I = hCI
−, C+Ii (C u D)I = hCI
+∩ DI +, C−I∪ DI
−i (C t D)I = hCI
+∪ DI +, CI
−∩ DI
−i (∃R.Self)I = I| (x, x) ∈ R+I},
{x ∈ ∆I | (x, x) ∈ RI
−} ;
if s∀∃Q= + then
(∃R.C)I =
I| ∃y((x, y) ∈ RI
+∧ y ∈ CI
+)}, {x ∈ ∆I| ∀y((x, y) ∈ RI+→ y ∈ C−I)}
(∀R.C)I =
I| ∀y((x, y) ∈ RI
+→ y ∈ CI
+)}, {x ∈ ∆I| ∃y((x, y) ∈ RI
+∧ y ∈ CI
−)} ; (≥ n R.C)I=
I| #{y | (x, y) ∈ RI
+∧ y ∈ CI
+} ≥ n}, {x ∈ ∆I| #{y | (x, y) ∈ RI
+∧ y /∈ CI
−} < n}
(≤ n R.C)I=
I| #{y | (x, y) ∈ RI
+∧ y /∈ CI
−} ≤ n}, {x ∈ ∆I| #{y | (x, y) ∈ RI
+∧ y ∈ CI
+} > n} ;
if s∀∃Q= ± then
(∃R.C)I =
I| ∃y((x, y) ∈ R+I ∧ y ∈ C+I)}, {x ∈ ∆I| ∀y((x, y) /∈ RI
−→ y ∈ CI
−)}
(∀R.C)I =
I| ∀y((x, y) /∈ RI
− → y ∈ CI
+)}, {x ∈ ∆I| ∃y((x, y) ∈ RI
+∧ y ∈ CI
−)} ; (≥ n R.C)I=
I| #{y | (x, y) ∈ RI
+∧ y ∈ CI
+} ≥ n}, {x ∈ ∆I | #{y | (x, y) /∈ RI
−∧ y /∈ CI
−} < n}
(≤ n R.C)I=
I| #{y | (x, y) /∈ RI
−∧ y /∈ nCI
−} ≤ n}, {x ∈ ∆I | #{y | (x, y) ∈ RI
+∧ y ∈ CI
+} > n}
We denote Γ is a set of concepts, ΓI+ =T{CI
+ | C ∈ Γ},
ΓI
− =S{CI
− | C ∈ Γ} and ΓI = ΓI
+, ΓI
− Observe that, if
Γ is finite, then ΓI= (Γ)I
Example 1: An inconsistent knowledge system inL refers
to electronic devices:
Let I={Cellphone, Bluetooth, Laptop, Memory, Size, Weight}
C={Device, General}
R={hasGeneral}
A={General(Memory), General(Size), General(Weight), General(Bluetooth), Device(Cellphone), Device(Laptop), Device(Memory), hasGeneral(Cellphone,Size), hasGen-eral(Cellphone,Memory), hasGeneral(Cellphone,Bluetooth), hasGeneral(Laptop,Size), hasGeneral(Cellphone,Weight), hasGeneral(Cellphone,Memory)}
T ={Device = ∃hasGeneral.>}
This knowledge base is inconsistent because both concepts Device and General contain the object Memory
An interpretation of the inconsistent knowledge base is as follows:
∆I = {a, b, c, d, e, f }, CellphoneI = a, BluetoothI = b, LaptopI= c, M emoryI= d, SizeI = e, W eightI = f DeviceI= {a, c, d}, GeneralI= {b, d, e, f }
III BISIMULATION Bisimulation is as a binary relation between nodes of a labeled in a graph We will demonstrate how to modify and extend bisimulation to deal with richer logic languages Bisimulation is of interest to researchers and it is applied
in practice in which three main applications are mentioned: (i) Separating the expressive powers of logic languages; (ii) Minimizing interpretations and labeled state transition systems; (iii) Concept learning in description logics
In this section, we consider an implementation of bisimilarity for concept learning in description logics when inconsisten-cies occur Bisimilation is applied for Description Logics in concept learning problems with consistent knowledge [10], [30], [12], [27] The idea is to use models of KB and bisimularity in this model to guide the search for concept C
Let Φ ⊆ {I, O, Q, U, Self} be a set of features, s ∈ S
a paraconsistent semantics, and I, I0 s-interpretations A non-empty binary relation Z ⊆ ∆I× ∆I 0
is called a (Φ, s)-bisimulation between I and I0 if the following conditions hold for every a ∈ ΣI, x, y ∈ ∆I, x0, y0 ∈ ∆I 0
, A ∈ C,
r ∈ R and every role R of ALCΦdifferent from U : (1) Z(aI, aI0)
(2) Z(x, x0) ⇒ [AI+(x) ⇒ AI+0(x0)]
(3) Z(x, x0) ⇒ [AI−(x) ⇒ AI−0(x0)]
(4) [Z(x, x0) ∧ RI+(x, y)] ⇒ ∃y0 ∈ ∆I0[Z(y, y0) ∧
RI+0(x0, y0)], (5) if s∀∃Q= + then [Z(x, x0) ∧ R+I0(x0, y0)] ⇒ ∃y ∈ ∆I[Z(y, y0) ∧
Trang 4R+(x, y)]
(6) if s∀∃Q= ± then
[Z(x, x0) ∧ ¬RI−0(x0, y0)] ⇒ ∃y ∈ ∆I[Z(y, y0) ∧
¬RI
−(x, y)],
(7) if O ∈ Φ then
Z(x, x0) ⇒ (x = aI ⇔ x0 = aI0),
(8) if Q ∈ Φ then
if Z(x, x0) holds and y1, , yn (n ≥ 1) are
pair-wise different elements of ∆I such that RI+(x, yi)
holds for every 1 ≤ i ≤ n, then there exist pairwise
different elements y10, , y0n of ∆I0 such that
R+I0(x0, yi0) and Z(yi, y0i) hold for every 1 ≤ i ≤ n,
(9) if Q ∈ Φ and s∀∃Q= + then
if Z(x, x0) holds and y0
1, , y0
n (n ≥ 1) are pair-wise different elements of ∆I0such that RI+0(x0, yi0)
holds for every 1 ≤ i ≤ n, then there exist
pairwise different elements y1, , yn of ∆I such
that RI+(x, yi) and Z(yi, y0i) hold for every 1 ≤
i ≤ n,
(10) if Q ∈ Φ and s∀∃Q = ± then if Z(x, x0) holds
and y10, , yn0 (n ≥ 1) are pairwise different
elements of ∆I0 such that ¬RI−0(x0, y0i) holds
for every 1 ≤ i ≤ n, then there exist pairwise
different elements y1, , yn of ∆I such that
¬RI
−(x, yi) and Z(yi, y0i) hold for every 1 ≤ i ≤ n,
if U ∈ Φ then
(11) ∀x ∈ ∆I∃x0 ∈ ∆I 0
Z(x, x0)
(12) ∀x0∈ ∆I 0
∃x ∈ ∆IZ(x, x0),
if Self ∈ Φ then
(13) Z(x, x0) ⇒ [rI+(x, x) ⇒ r+I0(x0, x0)]
(14) Z(x, x0) ⇒ [rI−(x, x) ⇒ r−I0(x0, x0)]
As a consequence, if one of the above conditions holds and
I, I0are s-interpretations (Φ, s)-bisimilar to each other, then
I |=sA iff I0|=sA [27]
IV CONCEPTLEARNING FORPARACONSISTENT
DESCRIPTIONLOGICS Concept learning problem is similar to binary classification
in traditional machine learning The difference is that in
paraconsistent description logics, objects are described not
only by attributes but also by binary relationships between
objects As bisimulation is the notion for characterizing
indiscernibility of objects in paraconsistent description log-ics It is very useful for concept learning in inconsistent knowledge base systems
Definition 2: (Learning problem in paraconsistent de-scription logics) Let I be a finite interpretation (given as
a training information system), a knowledge base KB in a
DL L and sets E+, E− of individuals, learn a concept C in
L such that:
(1) KB |= C(a) for all a ∈ E+ and a /∈ E−, (2) KB |= C(a) for all a ∈ E− and a /∈ E+, (3) KB |= C(a) for all a ∈ E+ and a ∈ E− The goal of learning is to find a correct concept with respect
to the examples This can be seen as a search process in the space of concepts A natural idea is imposing an ordering on this search space and use models of KB and bisimulation
in those models to guide the search for C
The main idea of this method is to smooth the ∆ domain
of the information system I using the selectors Based on that idea, the concept learning approach is broadly described
as follows:
Let Ap∈ C be a concept name standing for the decision attributeand suppose that Apcan be expressed by a concept
C in LΣ + ,Φ +, where Σ+ ⊆ Σ\Ap and Φ+ ⊆ Φ Let I be
a training information system over How can we learn that concept C on the basis of I
The trace of the learning algorithm in this case is as follows:
1) Starting from ∆I partition, we smooth this partition sequentially until we reach the partition corresponding
to Ap This smoothing process can be stopped sooner when the current partition is consistent with E or satisfies certain conditions
2) In the process of smoothing ∆I partition, the blocks created at all steps are Y1, Y2, , Yn Each generated block is denoted by a new index by increasing the value of n For each 1 ≤ i ≤ n, we set the following information:
• Yiis characterized by a concept Cisuch that CiI=
Yi,
• Record information about Yi is split by E,
• Saving the index of the largest block Yj such that
Yi⊆ Yj and Yj is not split by E
3) The current partition is denoted Y = {Yi 1, Yi2, , Yik} ⊆ {Y1, Y2, , Yn}
4) When the current partition becomes consistent with
Ap, return Ci 1t tCi j, where i1 ij are indices such that Yi1 Yij are all the blocks of the current partition
Trang 5that are subsets of Ap.
V PRELIMINARY EVALUATION
A The datasets
We applied the proposed model to the set of electric
devices We build several datasets including concepts, roles,
and individuals We labeled and tested the datasets with
different numbers of inconsistent concepts
The device dataset contains electronic device information
and attributes including information on 941 types of
configu-rations (concepts), 32 links between objects (roles), and 521
objects (individuals) Each object in the dataset is expressed
by a concept We use data from 7 out of 627 subjects for
training and validation Data for the other types of equipment
is used for testing
After some preprocessing steps on these datasets, i.e tested
the inconsistent dataset on Protege Protege error when
testing Reasoner with HermiT as shown in Fig 1
Fig 1 Test the inconsistent data set on Protege
The reason it’s inconsistent data here is that of the plastic
cover that the plastic is also on the screen Meanwhile,
the cover and the screen are disjointed (the two layers are
totally unrelated) to each other
B Experimental results
We took several experiments with different rates of
incon-sistent data to evaluate the effect of the proposed algorithm
In order to analyze the contribution of the labeled datasets,
we also generated some subsets with the size of 25, 30, 35,
40, 45, 50, 55, 60 inconsistent data rates
To illustrate the influence of the inconsistent parameter,
we additionally measured ten-fold cross-validation accuracy,
Recall, Precision, and F1-Measure The results are shown in
TABLE I
SYSTEM
Inconsistent Accuracy Precision Recall F1-Measure
Table 1 Since the inconsistent parameter acts as a termina-tion criterion, we observe, as expected that lower inconsistent values lead to significant increases in accuracy
Overall, the presented approach is able to learn accurate and inconsistent concepts with a reasonably low number of expensive reasoner requests Note that all the approaches are able to learn in a very expressive language with arbitrarily nested structures, as can be seen in the concept above Learning many levels of the structure has recently been identified as a key issue for structured Machine Learning and our work provides a clear advance on this front The results show that our approach is competitive with state-of-the-art Semantic Web systems when appearing in-conisistent in a system
VI CONCLUSIONS AND FUTURE WORKS
In this paper, a concept learning model for paraconsistent knowledge base system is introduced and discussed The key idea in this work is to use models of KB and bisimulation in those models to guide the search for C This mathematical technique, along with the partitioning strategies used, has been tested on two theoretical and experimental aspects This work can be extended along various directions We can extend the comparison with other methods like as refinement operators Through empirical evaluation of experiments, we examine the correlation between learning algorithms
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