HyFOM Reasoner: Hybrid Integration of Fuzzy Ontology

and Mamdani Reasoning

Cristiane A. Yaguinuma

, Walter C. P. Magalh

aes Jr.

, Marilde T. P. Santos

, Heloisa A. Camargo

and Marek Reformat

Department of Computer Science, Federal University of S

ao Carlos, S

ao Carlos, SP, Brazil

Embrapa Dairy Cattle, Juiz de Fora, MG, Brazil

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada

Keywords:

Knowledge Representation and Reasoning, Fuzzy Ontology, Mamdani Fuzzy Inference System, Hybrid

Reasoner.

Abstract:

Some real-world applications require representation and reasoning regarding imprecise or vague information.

In this context, the appropriate combination of fuzzy ontologies and Mamdani fuzzy inference systems can

provide meaningful inferences involving fuzzy rules and numerical property values. In general, this knowledge

is not obtained through typical fuzzy ontology reasoning and can be relevant for some ontology reasoning tasks

that depend on numerical property values. To address this issue, this paper proposes the HyFOM reasoner,

which provides a hybrid integration of fuzzy ontology and Mamdani reasoning. A real-world case study

involving the domain of food safety is presented, including comparative results with a state-of-the-art fuzzy

description logic reasoner.

1 INTRODUCTION

Many applications use ontologies to represent seman-

tic information that can be shared among people, soft-

ware agents and systems. In special, ontologies sup-

port not only representational primitives but also rea-

soning tasks that reveal meaningful knowledge.

However, there are some concepts whose mean-

ing cannot be fully captured using conventional on-

tologies. For instance, it is difﬁcult to model concepts

like creamy, dark, hot, large and thick, for which a

clear and precise deﬁnition is not possible, as they

involve so-called fuzzy or vague concepts (Straccia,

2006). Thus, there is a need for extending ontolo-

gies with concepts from the fuzzy set theory (Zadeh,

1965) to represent and reason over imprecise or vague

information.

In this sense, a number of fuzzy extensions of

ontologies have been developed, as pointed out by

(Lukasiewicz and Straccia, 2008). Some proposals

have incorporated concepts related to fuzzy variables

and linguistic terms, as they capture the vagueness

inherent in some real-world situations. These con-

cepts have been exploited by Mamdani Fuzzy Infer-

ence Systems (Mamdani FIS) (Mamdani and Assil-

ian, 1975) to infer numerical outputs based on fuzzy

variables and fuzzy rules. This well-known reasoning

approach could be also employed in fuzzy ontology-

based applications, providing numerical property val-

ues that are not inferred by typical fuzzy ontology rea-

soning. The inferred values could be considered in

other fuzzy ontology reasoning tasks, e.g. the fuzzy

instance check depending on speciﬁc property values.

Although some proposals have been developed to-

wards the combination of fuzzy ontology and fuzzy

rule reasoning, there are some issues to be consid-

ered. Fuzzy rule semantics and defuzziﬁcation meth-

ods should meet application requirements in order to

obtain meaningful results. The set of fuzzy infer-

ences should not be limited to fuzzy rule reasoning,

since fuzzy ontology reasoners also provide relevant

inferences regarding fuzzy concept knowledge. It is

important to incorporate fuzzy rule inferences to the

fuzzy ontology, as they may contribute to perform

other fuzzy ontology reasoning tasks.

Focusing on the mentioned issues, this paper de-

scribes the HyFOM reasoner (Hybrid Integration of

Fuzzy Ontology and Mamdani reasoning). It follows

a hybrid architecture to integrate fuzzy ontologies and

Mamdani rules aiming at providing meaningful infer-

ences to fuzzy ontology-based applications. The Hy-

FOM approach is explained as follows. Section 2 dis-

370

A. Yaguinuma C., C. P. Magalhães Jr. W., T. P. Santos M., A. Camargo H. and Reformat M..

HyFOM Reasoner: Hybrid Integration of Fuzzy Ontology and Mamdani Reasoning.

DOI: 10.5220/0004452803700378

In Proceedings of the 15th International Conference on Enterprise Information Systems (ICEIS-2013), pages 370-378

ISBN: 978-989-8565-59-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

cusses related work on the combination of fuzzy on-

tologies and fuzzy inference systems. Section 3 de-

scribes the main features of the HyFOM reasoner. A

case study about the domain of food safety, including

a comparison with a fuzzy description logic reasoner,

is presented in Section 4. Finally, Section 5 concludes

this paper and points out ongoing research.

2 RELATED WORK

In relation to the current approaches aiming to com-

bine fuzzy ontologies and fuzzy inference systems,

there are some issues that should be considered:

1. Does the semantics provided to represent and rea-

son over fuzzy rules meet the application needs?

2. Are fuzzy rule inferences integrated to the fuzzy

ontology so that they can contribute to other on-

tology reasoning tasks?

3. Does the application need to control the interac-

tion between fuzzy ontology and fuzzy rule infer-

ences?

4. Does the set of possible fuzzy inferences comprise

both fuzzy ontology and fuzzy rule reasoning?

Some fuzzy ontology languages based on ex-

pressive fuzzy description logics, e.g. (Bobillo and

Straccia, 2008; Bobillo and Straccia, 2009), pro-

vide implication operators that can be used to rep-

resent fuzzy rules. As described by (Guillaume and

Charnomordic, 2012), these rules are called implica-

tive rules, which are combined conjunctively. Never-

theless, such semantics can be too restrictive depend-

ing on the application, possibly resulting in knowl-

edge base inconsistency even for allowed property

values (question 1). Another approach provided by

the fuzzyDL reasoner (Bobillo and Straccia, 2008) is

representing fuzzy rules using fuzzy concept deﬁni-

tions along with a query interface to call defuzziﬁca-

tion methods. However, the defuzziﬁed output is not

incorporated into the fuzzy ontology, unless explicitly

done by application, affecting questions 2 and 3.

There are proposals that consider a crisp ontology

integrated with Mamdani rules, as in (Bobillo et al.,

2009; Wlodarczyk et al., 2010). In this context, an

ontology reasoner is used for consistency checking

regarding crisp deﬁnitions but the set of fuzzy infer-

ences is limited to fuzzy rule reasoning (question 4).

Some studies (Lee et al., 2010; Huang et al., 2011)

have adopted the Fuzzy Markup Language (FML)

(Acampora and Loia, 2005) to express FIS-related in-

formation such as fuzzy rules, linguistic variables and

fuzzy rule reasoning methods. OWL-FC (de Maio

et al., 2012) also represents these elements with a

high-level speciﬁcation for fuzzy control systems that

enables links to domain ontology concepts. In gen-

eral, these proposals focus on fuzzy rule reasoning,

mainly using the fuzzy ontology to represent a FIS

knowledge base. Thus, their set of fuzzy inferences

does not include fuzzy concept knowledge reasoning

(question 4), provided by fuzzy ontology reasoners.

(Bragaglia et al., 2010) propose a hybrid architec-

ture combining forward rules and fuzzy ontology rea-

soning. Although the set of fuzzy inferences covers

both fuzzy ontological and fuzzy rule reasoning, their

proposal does not exploit Mamdani reasoning neither

defuzziﬁcation methods. As mentioned earlier, Mam-

dani FIS provides useful inferences that can comple-

ment the set of fuzzy inferences demanded by some

applications, an issue related to question 1.

Aiming to deal with the discussed limitations, the

HyFOM reasoner is described in Section 3, combin-

ing fuzzy ontology and Mamdani reasoning based on

a hybrid architecture.

3 THE HyFOM REASONER

In this paper, the purpose of combining fuzzy ontol-

ogy and fuzzy rule reasoning is providing expressive

inferences that are not obtained through typical fuzzy

ontology reasoning. Speciﬁcally, a Mamdani FIS can

be used to infer numerical property values based on

fuzzy rules combining different properties and their

respective linguistic terms. In this sense, when ap-

plications require knowledge associated with a nu-

merical property value, it can be inferred based on a

Mamdani FIS by getting inputs from the fuzzy on-

tology. The inferred output is then returned to the

ontology, possibly contributing to other fuzzy ontol-

ogy reasoning tasks. Figure 1 presents a SADT di-

agram (Marca and McGowan, 1987) describing the

HyFOM reasoner approach to integrate fuzzy ontol-

ogy and Mamdani FIS reasoning.

According to Figure 1, the main inputs and con-

trols of the HyFOM reasoner are Mamdani rules rep-

resenting a Mamdani rule base; a fuzzy ontology; and

an individual of the fuzzy ontology. The Mamdani

rule base contains a set of rules combining numerical

properties and linguistic terms in the antecedent (in-

put properties) to infer the value of a property in the

consequent (output property). An example of a Mam-

dani rule is: If property1 is high and property2

is medium then property3 is low, where prop-

erty1 and property2 are input properties and prop-

erty3 is an output property, all of them described by

linguistic terms (high, medium and low, respectively).

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371

Figure 1: Main steps for integrating fuzzy ontology and Mamdani FIS reasoning with HyFOM reasoner.

The fuzzy ontology models a speciﬁc domain in

terms of concepts, properties, relationships, instances

and linguistic terms associated with numerical prop-

erties. The properties and linguistic terms used in

Mamdani rules should be deﬁned in the fuzzy ontol-

ogy with equal names, for mapping purposes. Ap-

plications pass an individual of the fuzzy ontology to

check if it has any property value that can be inferred

based on Mamdani reasoning. If a class is passed as

input, the integration approach is done for all its indi-

viduals.

The mechanisms (arrows at the bottom of activity

boxes) are the resources required to complete a pro-

cess, which may include people with particular skills

and computational tools, according to the SADT

speciﬁcation. In the proposed approach, the mecha-

nisms are inference engines and domain experts who

supervise the outputs of the integration process. Fol-

lowing a hybrid architecture, inference engine imple-

mentations are reused, including a crisp ontology rea-

soner, a fuzzy ontology reasoner and a Mamdani FIS.

The crisp ontology reasoner performs efﬁcient query

answering and reasoning related to crisp deﬁnitions

and assertions in the ontology. The fuzzy ontology

reasoner provides reasoning tasks regarding fuzzy

deﬁnitions and assertions, such as fuzzy concept sub-

sumption and fuzzy instance check. The Mamdani

FIS is responsible to infer the value of a numerical

property based on fuzzy rules and fuzzy operations.

The activity boxes presented in Figure 1 com-

bine inputs, controls and mechanisms to produce out-

puts. In the activity A1, the HyFOM reasoner iden-

tiﬁes which are the input and output properties used

in Mamdani rules so that their values can be obtained

from the fuzzy ontology. In the activity A2, the Hy-

FOM reasoner ﬁrstly checks if the output property

value for a particular individual can be obtained from

the fuzzy ontology. If so, then the fuzzy ontology is

able to provide its value thus there is no need to in-

volve Mamdani rules. If not, the appropriate input

property values should be obtained from the fuzzy on-

tology so that the Mamdani FIS can be invoked to in-

fer the output property value.

Still in the activity A2, property values (either out-

put or input) are obtained from the fuzzy ontology

based on the following procedure. The crisp ontology

reasoner is invoked to check if the property value is

either asserted or inferred based on crisp deﬁnitions.

The main reason for using a crisp ontology reasoner is

due to its optimized access to assertions and conven-

tional ontology reasoning tasks. Still, if property val-

ues cannot be obtained, the fuzzy ontology reasoner is

invoked because fuzzy concept deﬁnitions and impli-

cations can support reasoning associated with prop-

erty values. For example, a fuzzy concept deﬁnition

such as C1 ≡ ∃property1.high along with an as-

sertion C1(ind1) indicates that individual ind1 has

value high for property1. Thus, even if property

values are not explicit, they may be inferred based on

fuzzy ontology axioms and deﬁnitions. The inferred

values can be crisp (a speciﬁc number) or fuzzy (a

fuzzy set), depending on application preferences.

After the input values are obtained, the activity A3

invokes the Mamdani FIS to infer the corresponding

output based on Mamdani rules. In the Mamdani FIS,

the fuzzy operations considered are min-max com-

position, minimum for rule semantics and maximum

for aggregation of outputs (Mamdani and Assilian,

1975). Several defuzziﬁcation methods are provided

to obtain a numerical value from the aggregated fuzzy

output, such as Center of Area (COA), Moment de-

fuzziﬁcation and Mean of Maxima (MOM). Finally,

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a new property assertion associating the output prop-

erty, the individual and the defuzziﬁed value is added

to the fuzzy ontology under domain expert supervi-

sion (activity A4). As a result, the output generated

by Mamdani FIS will be available for other fuzzy on-

tology reasoning tasks that may depend on it.

The HyFOM reasoner was implemented us-

ing reasoners and frameworks available for fuzzy

ontology-based applications and FIS. The crisp ontol-

ogy reasoner is based on the OWL API (Horridge and

Bechhofer, 2011) and Hermit (Motik et al., 2009), an

optimized reasoner providing efﬁcient access to crisp

ontology assertions and inferences. The fuzzy ontol-

ogy reasoner used is the fuzzyDL reasoner (Bobillo

and Straccia, 2008), which supports fuzzy concepts,

linguistic terms and fuzzy axioms, with a Java API to

access reasoning tasks. The Mamdani FIS is provided

by FuzzyJ Toolkit and Fuzzy Jess (Orchard, 2001),

including a Java API for handling fuzzy sets, fuzzy

rules, Mamdani inference and defuzziﬁcation meth-

ods.

Based on this platform, domain experts can model

the fuzzy ontology using the Prot

e ontology editor

with the FuzzyOWL2 plugin (Bobillo and Straccia,

2011). The resulted ontology can be processed by

OWL API, Hermit and fuzzyDL, provided that it is

parsed to the fuzzyDL syntax to allow fuzzy ontology

reasoning. The results inferred by the Mamdani FIS

are integrated to the fuzzy ontology using the OWL

API support for including new assertions.

Some contributions of the HyFOM reasoner are

demonstrated in a real-world case study described in

Section 4.

4 CASE STUDY ON FOOD

SAFETY

The HyFOM reasoner was applied in a case study

to support domain experts in evaluating the chemi-

cal risk of analytes (residues and contaminants) de-

tected in food samples. The experiments were spon-

sored by the Brazilian Ministry of Agriculture, Live-

stock and Supply (MAPA), focusing on the National

Plan for Control of Residues and Contaminants (PN-

CRC). PNCRC is responsible for monitoring the pres-

ence of residues of pesticides and veterinary drugs as

well as environmental contaminants in food products.

More details on MAPA and PNCRC are available in

(de Magalh

aes Junior et al., 2012).

According to the methodology explained by

(de Magalh

aes Junior et al., 2012), laboratory anal-

yses obtain the concentration of different analytes in

food samples. For each analyte, there is a maximum

level established by the Codex Alimentarius Commis-

sion, which is the maximum concentration of that sub-

stance ofﬁcially permitted in a speciﬁc food. Each an-

alyte concentration is confronted with its respective

maximum level (in percentage) to obtain the Concen-

tration Risk (CR) of the analyte detected in a food

sample. In this sense, if an analyte concentration is

lower or equal than its maximum level it is called a

compliant analyte in the sample; otherwise it is called

non-compliant. In addition to CR, there are other risk

factors determined by PNCRC experts:

• Trends associated with analyte concentration: in-

dicate whether the analyte concentration has a ten-

dency of decreasing, stabilizing or increasing in a

short, medium or long period of time, according

to its toxicological proﬁle;

• Adjustment period: estimated time for the

provider of the food sample to be brought into

compliance with MAPA’s requirements concern-

ing a speciﬁc analyte;

• Adjustment cost: estimated costs for the provider

of the food sample to be brought into compliance

with MAPA’s requirements concerning a speciﬁc

analyte.

The risk factors are combined to obtain the Aggre-

gate Risk (AR) associated with an analyte detected in

a food sample. Depending on the AR value, inter-

ventions should be applied to the providers of food

samples. Some types of intervention are: no inter-

vention for compliant analytes with negligible AR;

preventive intervention for analytes that have a short-

term increasing trend and medium AR; and maximum

intervention for non-compliant analytes that have in-

tolerable AR.

With support of PNCRC experts, the main con-

cepts related to the chemical risk of analytes were

modeled in a fuzzy ontology. Initially, Prot

e and

FuzzyOWL2 plugin were used to model the main

concepts and properties, later parsed to the fuzzyDL

syntax to enable fuzzy ontology reasoning. List-

ing 1 shows how the risk factors were modeled in the

fuzzy ontology (fuzzyDL syntax), all of them asso-

ciated with linguistic terms deﬁned by the experts.

At the moment, the linguistic terms related to the

properties hasConcentrationTrend, hasAdjustmentPe-

riod and hasAdjustmentCost are nominal (crisp) val-

ues due to the available data, but there is an ongoing

work with PNCRC experts to fuzzify such deﬁnitions.

Listing 2 describes concept deﬁnitions representing

the types of intervention, which are based on the risk

factors and their linguistic terms. An instance of an

analyte-sample analysis is illustrated as well.

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373

(functional hasConcentrationRisk )

(range hasConcentrationRisk *real* 0.0 200.0 )

(define-fuzzy-concept negligibleCR

left-shoulder(0.0, 200.0, 20.0, 40.0) )

(define-fuzzy-concept acceptableCR

trapezoidal(0.0, 200.0, 20.0, 40.0, 80.0, 90.0) )

(define-fuzzy-concept nearCR

trapezoidal(0.0, 200.0, 80.0, 90, 100.0, 100.0) )

(define-fuzzy-concept equivalentCR

crisp(0.0, 200.0, 100.0, 100.0) )

(define-fuzzy-concept highCR

trapezoidal(0.0, 200.0, 100.0, 100, 120.0, 130.0) )

(define-fuzzy-concept intolerableCR

right-shoulder(0.0, 200.0, 120.0, 130.0) )

(functional hasConcentrationTrend )

(range hasConcentrationTrend *real* 0.0 3.0 )

(define-fuzzy-concept decreaseOrStabilize

crisp(0.0, 3.0, 0.0, 0.0) )

(define-fuzzy-concept longTermIncrease

crisp(0.0, 3.0, 1.0, 1.0) )

(define-fuzzy-concept mediumTermIncrease

crisp(0.0, 3.0, 2.0, 2.0) )

(define-fuzzy-concept shortTermIncrease

crisp(0.0, 3.0, 3.0, 3.0) )

(functional hasAdjustmentPeriod )

(range hasAdjustmentPeriod *real* 0.0 3.0)

(define-fuzzy-concept unnecessaryAP

crisp(0.0, 3.0, 0.0, 0.0))

(define-fuzzy-concept shortAP crisp(0.0, 3.0, 1.0, 1.0))

(define-fuzzy-concept mediumAP crisp(0.0, 3.0, 2.0, 2.0))

(define-fuzzy-concept longAP crisp(0.0, 3.0, 3.0, 3.0))

(functional hasAdjustmentCost )

(range hasAdjustmentCost *real* 0.0 3.0 )

(define-fuzzy-concept unnecessaryAC

crisp(0.0, 3.0, 0.0, 0.0))

(define-fuzzy-concept lowAC crisp(0.0, 3.0, 1.0, 1.0))

(define-fuzzy-concept mediumAC crisp(0.0, 3.0, 2.0, 2.0))

(define-fuzzy-concept highAC crisp(0.0, 3.0, 3.0, 3.0))

(functional hasAggregateRisk )

(range hasAggregateRisk *real* 1.0 15.0 )

(define-fuzzy-concept negligibleAR

crisp(1.0, 15.0, 1.0, 1.5) )

(define-fuzzy-concept veryLowAR

triangular(1.0, 15.0, 1.5, 1.5, 4.75) )

(define-fuzzy-concept lowAR

triangular(1.0, 15.0, 1.5, 4.75, 8.0) )

(define-fuzzy-concept mediumAR

triangular(1.0, 15.0, 4.75, 8.0, 11.25) )

(define-fuzzy-concept highAR

triangular(1.0, 15.0, 8.0, 11.25, 14.5) )

(define-fuzzy-concept veryHighAR

triangular(1.0, 15.0, 11.25, 14.5, 14.5) )

(define-fuzzy-concept intolerableAR

crisp(1.0, 15.0, 14.5, 15.0) )

Listing 1: Risk factors deﬁned in the fuzzy ontology.

(define-concept CompliantAnalysis

(and AnalyteSampleAnalysis

(<= hasConcentrationRisk 100.0 )))

(define-concept NonCompliantAnalysis

(and AnalyteSampleAnalysis

(> hasConcentrationRisk 100.0 )))

(define-concept NoIntervention (and CompliantAnalysis

(some hasAggregateRisk negligibleAR)))

(define-concept PreventiveIntervention

(and (some hasConcentrationTrend shortTermIncrease)

(some hasAggregateRisk mediumAR)))

(define-concept MaximumIntervention

(and NonCompliantAnalysis

(some hasAggregateRisk intolerableAR)))

(instance analysis1 AnalyteSampleAnalysis)

(related analysis1 metidation hasAnalyte)

(related analysis1 milkSample1 hasSample)

(instance analysis1 (= hasConcentrationRisk 121.2 ))

(instance analysis1 (= hasConcentrationTrend 3.0 ))

(instance analysis1 (= hasAdjustmentPeriod 3.0 ))

(instance analysis1 (= hasAdjustmentCost 2.0 ))

Listing 2: Concept deﬁnitions in the fuzzy ontology.

Instead of using the Chem-risk approach (de Ma-

galhes Junior, 2011) to compute AR, domain ex-

perts were requested to express their knowledge us-

ing Mamdani rules combining the risk factors to in-

fer AR. In this case study, fuzzy rules contribute to

make the process more transparent and interpretable

for PNCRC and MAPA decision makers, due to the

linguistic terms that are closer to human language.

Then, the results obtained with the hybrid reasoner

were compared with Chem-risk, which provides ap-

propriate results according to PNCRC experts. A to-

tal of 17 Mamdani rules were modeled, some of them

illustrated in Listing 3 using Fuzzy Jess.

Using the fuzzy ontology and Mamdani rules, the

HyFOM reasoner was applied to provide recommen-

dations on aggregate risk and intervention actions re-

lated to food samples. Following the approach de-

scribed in Section 3, individuals of the concept Ana-

lyteSampleAnalysis (see an example in Listing 2) are

passed to the HyFOM reasoner to obtain AR values

based on Mamdani rules. The corresponding input

property values are obtained from the fuzzy ontology,

as they are modeled as property assertions. Then, the

Mamdani FIS generates the output values, which are

returned to the fuzzy ontology under expert supervi-

sion. After that, the outputs can be considered in the

fuzzy instance check involving the concepts NoInter-

vention, PreventiveIntervention and MaximumInter-

vention to recommend the appropriate intervention.

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374

(defrule rule1

(hasAdjustmentCost ?c&:(fuzzy-match ?c "unnecessaryAC"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "negligibleCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "negligibleAR"))))

(defrule rule2

(hasAdjustmentCost ?c&:(fuzzy-match ?c "unnecessaryAC"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "acceptableCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "veryLowAR"))))

(defrule rule3

(hasAdjustmentCost ?c&:(fuzzy-match ?c "unnecessaryAC"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "nearCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "veryLowAR"))))

(defrule rule4

(hasAdjustmentCost ?c&:(fuzzy-match ?c "unnecessaryAC"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "equivalentCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "lowAR"))))

(defrule rule5

(hasAdjustmentCost ?c&:(fuzzy-match ?c "lowAC"))

(hasAdjustmentPeriod ?p&:(fuzzy-match ?p "unnecessaryAP"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "negligibleCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "veryLowAR"))))

(defrule rule6

(hasAdjustmentCost ?c&:(fuzzy-match ?c "lowAC"))

(hasAdjustmentPeriod ?p&:(fuzzy-match ?p "unnecessaryAP"))

(hasConcentrationRisk ?v&:(fuzzy-match ?v "acceptableCR"))

(assert (hasAggregateRisk

(new FuzzyValue ?*hasAggregateRiskFvar* "veryLowAR"))))

Listing 3: Mamdani rules to infer aggregate risk.

Based on this case study, some experiments were

conducted using data provided by PNCRC. A total

of 114 beef sample analyses were available, involv-

ing 19 different analytes. The HyFOM reasoner was

executed with the 114 individuals of AnalyteSample-

Analysis deﬁned in the fuzzy ontology, using the Mo-

ment and COA defuzziﬁcation methods for generat-

ing the AR values. The results obtained with the Hy-

FOM reasoner were compared to the results provided

by Chem-risk approach and fuzzyDL reasoner.

The fuzzyDL reasoner was chosen for compari-

sion since it is one of the state-of-the-art fuzzy de-

scription logic reasoners. In addition, it supports

Mamdani FIS semantics with 3 defuzziﬁcation meth-

ods - Smallest of Maxima (SOM), Largest of Maxima

(LOM) and MOM. In general, MOM generates more

appropriate outputs compared with SOM and LOM,

as it takes the average between them. Note that the

HyFOM reasoner already uses fuzzyDL as a fuzzy

ontology reasoner, but only for inferences related to

fuzzy concept knowledge. Therefore, in the tests, a

”standalone” fuzzyDL reasoner is compared with the

HyFOM reasoner. As fuzzyDL does not provide spe-

ciﬁc constructors for handling Mamdani rules, two

fuzzyDL approaches for modeling fuzzy rules were

considered: (1) fuzzy implications and (2) fuzzy con-

cept constructors with MOM defuzziﬁcation. In these

two situations, the fuzzy ontology (Listings 1 and 2)

is reused but the rule set is replaced by the corre-

sponding fuzzy rules according to the two fuzzyDL

approaches.

Figure 2 presents the results obtained with Hy-

FOM reasoner, fuzzyDL and Chem-risk, which is the

reference for comparison. Some individuals of An-

alyteSampleAnalysis are omitted because the AR val-

ues remain unchanged for individuals with id ≤ 66. In

terms of mean squared error (mse), the HyFOM rea-

soner achieved a better overall performance with mo-

ment defuzziﬁcation (mse = 0.195) and COA (mse =

0.199) against fuzzyDL implications (mse = 0.295)

and fuzzyDL with MOM (mse = 0.342). The Fried-

man test was applied over the squared error values,

concluding that at least one of the means differs from

the rest. Dunn’s post test revealed that the fuzzyDL

implications results are signiﬁcantly different from

the rest, reﬂecting the distinct reasoning semantics

involved (Mamdani rules with defuzziﬁcation versus

implication rules). However, it is important to ana-

lyze the speciﬁc situations in which one approach per-

forms better than the others, to have a comprehensive

understanding of the results.

For analyses with AR = 1 according to Chem-risk,

the results from fuzzyDL implications are more pre-

cise than results provided by Mamdani rules with de-

fuzziﬁcation. Implicative rules in fuzzyDL generate

a numerical output corresponding to the minimum

value in the domain of discourse which belongs to the

conjunction of rule consequents. Thus, the numerical

output is not inﬂuenced by the shape of the fuzzy set,

as it happens with COA, Moment and MOM defuzzi-

ﬁcation methods, which generate a mse = 0.0625 in

this speciﬁc situation. On the other hand, such char-

acteristic favored the HyFOM reasoner results when

1 < AR ≤ 5 according to Chem-risk. In this case, the

defuzziﬁcation methods based on shape of the fuzzy

sets provided more precise results, differently from

the fuzzyDL implications and MOM that are inﬂu-

enced by the extremes of maximum degree.

HyFOMReasoner:HybridIntegrationofFuzzyOntologyandMamdaniReasoning

375

Figure 2: Aggregate risk obtained with fuzzyDL, HyFOM reasoner and Chem-risk.

For analyses with 5 < AR ≤ 12, there is no differ-

ence in the results provided by both HyFOM reasoner

and fuzzyDL. In such situation, the ﬁred rules involve

input properties with nominal values (hasConcentra-

tionTrend, hasAdjustmentPeriod and hasAdjustment-

Cost), thus fuzziness is not considered in the results.

As it was mentioned previously, there is an ongoing

work to fuzzify these properties with support of PN-

CRC experts, so that more precise results can be ob-

tained in comparison with Chem-risk approach.

Finally, the HyFOM reasoner presents better re-

sults for analyses with AR > 12. In this situation,

fuzzyDL implications do not infer AR values due

to knowledge base inconsistency, since different rule

consequents not having an intersection are combined

conjunctively. This problem does not happen with

Mamdani rules, which are able to infer pertinent out-

puts. In addition, the defuzziﬁcation methods pro-

vided by the HyFOM reasoner generate a better ap-

proximation than MOM, one of the methods available

in fuzzyDL. Therefore, in relation to question 1 (Sec-

tion 2), the HyFOM reasoner can be considered more

appropriate for this case study as it provides rule se-

mantics and defuzziﬁcation methods that better meet

the application needs.

After fuzzy rule reasoning was performed, the AR

values should be available for other fuzzy ontology

reasoning tasks that depend on them. For example,

the fuzzy ontology reasoner should consider the AR

values to obtain the membership degree of an individ-

ual of AnalyteSampleAnalysis to the concept Preven-

tiveIntervention.

As the HyFOM reasoner incorporates the AR val-

ues to the fuzzy ontology as new property assertions,

the fuzzy instance check task is performed as ex-

pected. The AR inferred by the implicative rules

is also considered by the fuzzy ontology reasoning

task. However, when using fuzzyDL with Mamdani

FIS semantics and MOM, the generated outputs are

not taken into consideration by the fuzzy instance

check. In this case, the application should be aware

that the defuzziﬁcation results are not integrated and

should take appropriate actions regarding fuzzy con-

cept reasoning. This issue is related to the questions 2

and 3 (Section 2), which point out that fuzzy rule in-

ferences should be automatically integrated with the

fuzzy ontology. Thus, both the HyFOM reasoner and

fuzzyDL implications meet these integration require-

ments, while the other fuzzyDL approach does not.

5 CONCLUSIONS AND FUTURE

WORK

The proposed hybrid reasoning system was designed

focusing on addressing issues discussed in Section 2,

which are not fully accomplished by related work.

Regarding rule semantics, the HyFOM reasoner is

based on Mamdani rules while some fuzzy descrip-

tion logic reasoners support implicative rules that can

be too restrictive for some applications. Moreover,

the defuzziﬁcation methods provided are based on the

shape of the fuzzy set, generating outputs that repre-

sent a suitable balance among multiple ﬁred rules.

The case study involving the domain of food

safety demonstrated that the HyFOM reasoner pro-

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duces appropriate results comparable with the Chem-

risk approach already used in this domain. Fuzzy on-

tologies and Mamdani rules are interesting for this

context as they provide advantages regarding inter-

pretability and treatment of imprecision, inherent in

expert knowledge.

In terms of integration approach, the HyFOM rea-

soner automatically provides Mamdani FIS outputs to

other fuzzy ontology reasoning tasks. As the Hy-

FOM reasoner includes a fuzzy ontology reasoner

(fuzzyDL), the set of possible fuzzy inferences com-

prise both fuzzy ontology and fuzzy rule reasoning.

As illustrated by the case study, the integration of the

outputs from Mamdani FIS is important for the fuzzy

instance check task involving intervention actions that

depend on aggregate risk values.

As for future work, more real-world applications

of the HyFOM reasoner are being developed. In addi-

tion, there is an ongoing research about an integration

architecture involving a fuzzy tableau-based reasoner

and a fuzzy inference system. Other types of fuzzy

inference systems can be considered as well, such as

the non-parametric fuzzy system model proposed by

(Angelov and Yager, 2012), which is a new type of

simpliﬁed fuzzy rule-based system as an alternative

to the Mamdani and Takagi-Sugeno models.

ACKNOWLEDGEMENTS

The authors would like to thank the Brazilian research

agency CAPES for supporting this research. Special

thanks to the Embrapa Dairy Cattle and the Brazil-

ian Ministry of Agriculture, Livestock and Supply for

providing real-world data and domain expert support

for the case study.

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