Mapping Ontology with Probabilistic Relational Models

An Application to Transformation Processes

Cristina Manfredotti

, Cedric Baudrit

, Juliette Dibie-Barth´elemy

and Pierre-Henri Wuillemin

INRA AgroParisTech, 16, rue Claude Bernard, 75231 Paris Cedex 5, France

Institut National de la Recherche Agronomique, Institut de M´ecanique et d’Ing´enierie, Talence, France

Sorbonne Universites, UPMC, Univ Paris 06, CNRS UMR 7606, LIP6, Paris, France

Keywords:

Ontology, Probabilistic Graphical Models, Probabilistic Relational Models.

Abstract:

Motivated by the necessity of reasoning about transformation experiments and their results, we propose a map-

ping between an ontology representing transformation processes and probabilistic relational models. These

extend Bayesian networks with the notion of class and relation of relational data bases and, for this reason,

are well suited to represent concepts and ontologies’ properties. To easy the representation, we exemplify a

transformation process as a cooking recipe and present our approach for an ontology in the cooking domain

that extends the Suggested Upper level Merged Ontology (SUMO).

1 INTRODUCTION

A transformation process is a dynamic process com-

posed of a sequence of operations which allows in-

puts to be transformed in several different outputs. It

relies on data and knowledge coming from heteroge-

neous sources, often suffers from lack of information

and contains uncertain data, the observations being

acquired with seldom precise instruments, different

from a process to another. Reasoning on a transfor-

mation process supposes to be able, for instance, to

predict future outputs given certain inputs or given

that some inputs are missing, to diagnose how to ob-

tain the best output by determining the important in-

puts, to control the process and to suggest the best

sequence of operations. In this paper, we provide a

step forward toward reasoning on transformation pro-

cesses. To do that, we have to face two main locks:

(1) data and knowledge heterogeneity and (2) uncer-

tainty quantiﬁcation.

In order to face the ﬁrst lock, a relevant solution

is to use ontologies (Fridman Noy, 2004; Doan et al.,

2012). Many works propose solutions to manage un-

certainty in ontologies such as adapting the querying

process using fuzzy sets (Buche et al., 2005), rea-

soning using a possibilistic and probabilistic descrip-

tion logic reasoner (Qi et al., 2010; Lukasiewicz and

Straccia, 2008), reasoning in fuzzy ontology (Bobillo

et al., 2013) or using existing knowledge to predict

unﬁlled information (Sa¨ıs and Thomopoulos, 2014).

In this paper, we propose to quantify uncertainty in

reasoning with probability theory.

We propose to explore a novel way to reason

on transformation processes facing the two locks in-

troduced above: we combine the representation ex-

pression of ontologies with the reasoning possibili-

ties of probabilistic relational models which provides

a consistent framework to process uncertainty. Prob-

abilistic relational models add the notion of class to

Bayesian networks which allows to do ﬁltering, pre-

diction, classiﬁcation and smoothing. The notion of

‘class’, common to ontologies (concepts) and proba-

bilistic relational models, leads us to choose this prob-

abilistic model to be paired with the ontology’s repre-

sentation model. The ﬁrst step of this combination

consists in proposing a mapping between a transfor-

mation process ontology and a probabilistic relational

model. The next step, not presented in this paper, will

be to learn the parameters of the model from an onto-

logical database and then to implement methods able

to reason on the learned model.

We present all our ﬁndings in the domain of cook-

ing recipes becauseit well exempliﬁes a general trans-

formation process, being simple and easy to under-

stand. We ﬁrst present background on probabilistic

relational models. We detail, in Section 3, an ontol-

ogy of transformation processes and, in Sections 4,

its mapping with a probabilistic relational model. We

discuss our ﬁndings in Section 5 providing a compar-

ison with the state of the art.

Manfredotti, C., Baudrit, C., Dibie-Barthélemy, J. and Wuillemin, P..

Mapping Ontology with Probabilistic Relational Models - An Application to Transformation Processes.

In Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2015) - Volume 2: KEOD, pages 171-178

ISBN: 978-989-758-158-8

171

2 PRMs

A Bayesian network (BN) (Koller and Friedman,

2009) is the representation of a joint probability over

a set of random variables that uses a Directed Acyclic

Graph (DAG) to encode probabilistic relations be-

tween variables (Figure 1).

Figure 1: Two Bayesian networks.

Probabilistic Relational Models (PRMs) extend

the BN representation with a relational structure be-

tween (potentially repeated) fragments of BN called

classes (Torti et al., 2010). A class is deﬁned as a

DAG over a set of inner attributes and a set of outer

attributes from other classes referenced by so-called

reference slots (Figure 2).

Figure 2: A relationa schema formed by two classes X and

Y. ρ is a reference slot in Y which indicates that attributes

of class Y (D, E, F) can have parents in class X (A, B, C).

The probabilistic models are deﬁned at class level

over the set of inner attributes, conditionally to the set

of outer attributes and represent generic probabilistic

relations inside the classes that will be instantiated for

each speciﬁc situation. In this way, PRMs provide a

high-level, qualitative description of the structure of

the domain and the quantitative information provided

by the probabilitydistribution (Friedman et al., 1999).

In a PRM, the (relational) schema describes a set

of classesC, associated with attributes A(C) and refer-

ence slots R(C)

. A slot chain is deﬁned as a sequence

of reference slots that allows to put in relation at-

tributes of objects that are indirectlyrelated. A system

in the PRM provides a probability distribution over

a set of instances of a relational schema (Wuillemin

Using the standard object-oriented notation, we will

write C.X (respectively C.Y) to refer to a given attribute X

(respectively, reference slot Y) of a class C.

and Torti, 2012). In this paper we present a general

approach to deduce relational schemas from a given

ontology of transformation processes.

3 TRANSFORMATION

PROCESSES

A cooking recipe is a well known transformation pro-

cess. For this reason and its simplicity, we propose

to illustrate our ontology on the cooking domain. We

present, in the following, the ontology, its concepts

and relations and an example of recipe. Finally, we il-

lustrate examples of forms of uncertainty that can be

found in a transformation process.

3.1 Our Ontology

An ontology is designed to represent the knowledge

on a domain with concepts, relations between these

concepts and instances of these concepts (Guarino

et al., 2009). When deﬁning an ontology, it is impor-

tant to refer to an upper level ontology to guarantee

its genericity. Muljarto et. al. deﬁnes an ontology for

food transformation extending the upper level ontol-

ogy DOLCE (Muljarto et al., 2014). In this paper,

we propose, instead, to extend the Suggested Upper

level Merged Ontology (SUMO) because it separates

physical from abstract entities and gives a deﬁnition

of object, separated from the deﬁnition of process.

Despres presents an ontology of numeric cook-

ing (Despres, 2014). We keep four of the concepts

introduced in her work: ingr´edient called product,

mat´eriel called device (using the SUMO concept’s

name), technique de base called operation and ´etapes

de r´ealisation, realization step. To these, we add two

concepts, the concept attribute already deﬁned in the

SUMO ontology, and the concept observation that

records the values assumed by the attribute during the

process. Figure 3 presents the general relation schema

of these concepts that are detailed below.

Figure 3: The general relation schema of the concepts used

to describe the proposed ontology. Subconcepts are con-

nected with discontinuous lines.

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3.1.1 Concepts and their Relations

We deﬁne a recipe (a transformation process) as a se-

quence of realization steps. Each realisation step is

composed of one or more operation(s) applied either

to one or more product(s) using one or more device(s)

or to a device in order to change some of its prop-

erties. The product output(s) of one operation can

be the input of another following it in the sequence

given by the recipe. In Figure 4, we report part of

the SUMO ontology highlighting the concepts we use

and the ones we deﬁne.

Figure 4: Part of the SUMO ontology, highlighting in italic

the concepts we use and in bold the concepts we deﬁne. We

have omitted part of the concepts we do not use.

In the SUMO ontology, cooking is a subconcept

of process. We deﬁne two subconcepts of the pro-

cess cooking: operation and realization step. An op-

eration can be applied to a device. For example, the

operation of pre-heating the oven at a certain temper-

ature has as input the device oven and operates chang-

ing its state. An operation can also be applied to one

or more product(s). The device mixer can be used

to whip eggs, whipping takes as input eggs and re-

turns eggs with changed properties. The operation

whipping uses the device mixer to modify some of the

properties of the object eggs given as input. Another

example of operation applied to one or more products

is the operation of mixing ﬂour and sugar. The device

spoon and bowl are used by the operation. The device

spoon is used to mix the two products in a bowl, to

return a product that is an intermediary mixture.

In the SUMO ontology, food and device are sub-

concepts of the concept object. We deﬁne a subcon-

cept of object that is superconceptof the concept food.

We call it product. This can be a food or an interme-

diary mixture with its own recipe. For instance, ﬂour

is an ingredient of a recipe of a cake, it is a food and

so a product; the mix made of ﬂour and sugar ready

to be added to eggs in the cake baking process is the

output of the mixing operation; the cream to be put on

top of a cake is an ingredient’s of the recipe which can

be separately prepared with its own recipe.

The SUMO concept attribute represents qualities

of objects or operations. The food ﬂour has attribute

type which can have value ‘whole grain’, the device

oven has attribute temperature which can have value

‘280°’and the operation mix has attribute speed with

value ‘quick’. To record the values of the attributes

we deﬁne the concept observation as a sub-concept

of the content bearing object SUMO concept

. While

making a cake, we can observe the mixture of ﬂour

and sugar and record its color and temperature (color

and temperature are attributes of the mixture, the ob-

servations about them are collected in the observa-

tion). While observing the mixture of butter and sugar

we will register also its granularity. Observations

cannot be modiﬁed by the transformation process.

In a recipe, there are operations that have a dura-

tion, we call them temporal operations and we dif-

ferentiate them from unitary operations. Temporal

properties can be described by the time ontology

the semantic web proposed in (Hobbs and Pan, 2004).

Temporal operation is a subconcept of the time on-

tology concept interval; unitary operation is a sub-

concept of the concept instant; those are both sub-

concepts of the time concept temporal entity (Fig-

ure 5). Thus, we can use properties of the time con-

cept temporal entity to represent temporal relations

between operations and so partially ordering the op-

erations of a recipe in realization steps.

3.1.2 A Recipe Example

The TAAABLE project

has the purpose of solving

A content bearing object is deﬁned as a self connected

object which expresses information.

http://www.w3.org/TR/owl-time/

http://intoweb.loria.fr/taaable3ccc/

Mapping Ontology with Probabilistic Relational Models - An Application to Transformation Processes

173

Figure 5: Operation’s subconcept hierarchy tree.

cooking problemson the basis of a recipe book (Badra

et al., 2008). They propose preparation graphs of a set

of recipes that their system has analysed with the on-

tology presented in (Despres, 2014). Being produced

automatically, the generated graphs may contain er-

rors. Consider, for instance, the following recipe for

the Aunt Lila’s cookies:

Aunt Lila’s cookies

/2 lb butter

2 c Nuts ground

2 c All-purposes ﬂour

4 tb Sugar

2 ts Vanilla

to roll Powdered sugar

Preheat oven to 180

◦

C. Cream sugar and butter until light and ﬂuffy.

Add vanilla and nuts. Add ﬂour gradually. Roll into small balls. Place

on baking sheet. Bake 15 to 20 minutes. Roll baked balls in powdered

sugar while still warm.

The graph for this recipe reported on the

TAAABLE Wiki presents some errors. In particular,

for the phrase ‘roll baked balls in powdered sugar’,

the automatic system recognizes as ingredient the

proposition‘in’ and as operation the term ‘powdered’.

Given the graph errors and the differences between

the two ontologies, we propose the graph of Figure 6.

Figure 6: The preparation graph for the Aunt Lila’s snow-

ball cookies based on our ontology.

The operation preheat the oven is a temporal oper-

ation which relates with an observation (the x°in the

rhombus in Figure 6). Representing the observation

of the temperature of the oven during time, could help

a decision process on when to put the cookies in the

oven, which can be an uncertain information.

3.2 Uncertainty in Transformation

Processes

Data and knowledge in transformation processes are

widely tainted with uncertainty. Often the instruments

used to take measurements during a transformation

process are able to return only an estimation of the

quantity observed. Devices are generally calibrated

according to some environmental conditions that can

be difﬁcult to be repeated somewhere else. They also

have some built-in characteristics that are different

from device to device. Moreover, the problems of

missing data (e.g. the salt ingredient is not always

mentioned in a recipe) and missing values (e.g. ”to

roll” powdered sugar) are known problems in trans-

formation processes. Our aim is to provide a model

able to handle all these uncertainties.

Different languages model uncertainty in ontolo-

gies. BayesOWL (Pan et al., 2005), OntoBayes (Yang

and Calmet, 2005) and PR-OWL (da Costa et al.,

2008; Carvalho et al., 2013) are extensions of the

Web Ontology Languagecalled OWL to model uncer-

tainty in semantic web. PROWL provides a method to

write ontologies containing probabilistic information.

This information can be processed but it cannot be en-

riched as in the case of learning or updating from new

data. BayesOWL and OntoBayes add to the ontology

a BN that models the uncertainty on the domain, pro-

viding a pair ontology-BN. In (Helsper and van der

Gaag, 2002) BNs are built to integrate knowledge ex-

pressed by experts in an ontology. The BNs built with

these approaches cannot summarize the information

contained in the ontology because BNs cannot repre-

sent relational information. In this way, the two mod-

els need to be paired.

Different approaches have been presented that

map ontologies into BNs, see for instance (Devitt

et al., 2006) and (Fenz, 2012) where, with different

approaches, BNs are built starting from a knowledge

base modelled as an ontology. These approaches take

advantage of the information provided by the ontol-

ogy, simplifying the BN learning. Learning a BN,

they ﬂatten the information coming from the ontology

loosing its relational aspect.

The method proposed in (Truong et al., 2005)

brings together ontology and PRMs, merging them in

a new model on which different types of reasoning are

supported. To implement Bayesian reasoning on this

model, a BN is constructed from the uniﬁed model. In

this way, as in the works above, the reasoning is done

on a BN and not on probabilistic relational model.

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

174

In (Ishak et al., 2011) an approach for learning

probabilistic graphical models from ontology is pre-

sented. Their approach learns object-oriented BNs by

morphing a given ontology. Object-oriented BNs are

another extension of BNs using the object-oriented

paradigm. Differently from PRMs, object-oriented

BNs cannot manipulate reference slots but determine

a set of “interface”nodeswhich allow the communica-

tion between objects. Thus, object-oriented BNs are

less generic and, in our opinion, less suitable (because

less similar) to ontology morphing than PRMs.

With the aim of maintaining the structural and

relational information expressed in the ontology, we

present, in this paper our mapping of an ontology of

transformation processes into PRMs. Having a PRM

for the Aunt Lila’s cookies recipe would help us rea-

soning about different questions that are not possible

to be answered with an ontology. For instance, we

could compute the probability of having tasty Aunt

Lila’s cookies, given the fact that we have/haven’t

cream very well butter and sugar (this is the predic-

tion problem). We could also infer the probability of

having done a good job in creaming butter and sugar

having observed very tasty cookies (inference prob-

lem). The deﬁned PRM can be used to suggest a spe-

ciﬁc sequence of operations to obtain a certain output.

For instance, given the butter at a certain temperature,

we could suggest the best speed at which using the

mixer to cream it with sugar (process control). Fi-

nally, we could use the PRM to simulate experiences

under different condition.

4 MAPPING

Our approach maps a transformation processes ontol-

ogy into a PRM’s relational schema. We describe the

mapping for the ontology’s concepts: object, unitary

and temporal operation, attribute and observation.

The SUMO concept object and its subconcepts

product, device and observation (see Figure 3) is rep-

resented by a class (called class object).

Deﬁnition. A class object in a PRM is a mapping be-

tween properties of the ontology concepts object and

PRM attributes.

In Figure 7, the concept input1 with proper-

ties att1 and att2 is mapped into the class object

Obj.input1 with attributes the variables att1 and att2.

We propose to represent the concept unitary oper-

ation by a speciﬁc class: the class operation.

Deﬁnition. A class operation in a PRM is deﬁned

by (1) a DAG over

• the reference slots giving access to the properties

of the classes mapping the input object(s) and the

device object(s) of the operation,

• an attribute for each property of the operation and

• the attributes representing the properties of the

output object(s) of the operation;

and (2) a probability distribution over the attributes

representing the properties of the results objects of

the operation given the values of the attributes repre-

senting the input and the device objects properties.

Figure 7 shows (at the top) the relational schema

and (at the bottom) the PRM for two classes oper-

ation: operation1 and operation2. The output of

the ﬁrst operation is input for the other, so a refer-

ence slot (ρ

) exists between the two classes. Each

class object representing the inputs and the device

(Obj.input1, Obj.input2, Obj.Device1, Obj.input3 and

Obj.Device2) are referred to by a reference slot in the

class operation (ρ1, ρ2, ρ3, ρ5 and ρ6). The attributes

representing the properties of the output object of the

operation (att4, att5) deﬁne a class to which other

classes operation can refer (see ρ

in Figure 7)

Figure 7: (top) The relational schema and (bottom) the

PRM for two operation classes. A ρ

in a class represents

the reference slot giving access to the properties of the class

it refers to. Each square represents an object.

A temporal operation is mapped with a concate-

nation of (unitary) operation. Following the standard

deﬁnition of dynamic BNs (Murphy, 2002) we can

deﬁne a PRM mapping a temporal operation.

Deﬁnition. A temporal operation class maps a tem-

poral operation as a pair of classes operation with a

reference slot among them:

With respect to the literature on PRMs, we should rep-

resent the attributes representing the properties of the object

output of the operation as a class outside the class operation.

Here, we represent it inside, to mean that the output is, in-

deed, a superclass of the operation itself.

Mapping Ontology with Probabilistic Relational Models - An Application to Transformation Processes

175

• one (operation

) representing the dependencies

between variables at the beginning of the opera-

tion and

• another (operation

→

) representing the dependen-

cies from the generic instant of time i to the next

instant i+ 1, with a reference slot to itself.

The second class operation (operation

→

) refers to it-

self, creating a (possibly inﬁnite) loop. To avoid the

loop to run forever, we ﬁx the number of times this

class can refer to itself. In this way, we ensure the

overall model to describe a probability distribution.

Figure 8 shows the relational schema of the PRM for

a temporal and a unitary operation classes. As before,

the output of the temporal operation is input for the

unitary one, so a reference slot exists between the two

classes.The output of the class operation operation

is input of the class operation

→

. A reference slot ex-

ists, also, betweenoperation

→

and itself. The number

of time the temporal operation class can refer to itself

is ﬁxed (reported in the triangle).

Figure 8: The relational schema of the PRM for a temporal

operation class linked to a unitary operation class.

An ontology of transformation processes is

mapped into a relational schema of a PRM that is

a concatenation of classes representing realization

steps chained by reference slots. In our ontology, at-

tributes are abstract entities representing properties of

object or processes. We map ontology’s attributes, in

the PRM, as attributes of the classes mapping the ob-

jects of which they represent the property. Finally, ob-

servations are ontology concepts that record a partic-

ular measurement done over an object or process. In

a PRM, an observation is mapped to a class to which

an attribute can refer to.

4.1 A PRM for the Example

Reasoning about mapping an ontology for transfor-

mation processes in a PRM leads us to better deﬁne

the ontology. In a BN, the conditional probability dis-

tribution of a node depends upon the number of its

parents. Referring to the Aunt Lila’s cookies exam-

ple, the ontology of the operation add in Figure 6 is

the same no matter the number of products we have

to add together. For a PRM, instead, changing the

number of parents of an attribute changes its condi-

tional probability distribution. Following this obser-

vation, we enrich our ontology with concepts specify-

ing the number of inputs each operation can have. We

replace the operation add with two subclasses add2

(Figure 9). Then we map the new ontology into a

PRM following the approach presented in the previ-

ous subsection. In the following, we report the map-

ping for only three operations.

The operation add2 is mapped in a PRM with

three reference slots, two for the inputs of the oper-

ation (nuts and vanilla) and one for the device used

by the operation (bowl). The PRM deﬁnes a class

mixture1 output of the operation. In Figure 9 the rela-

tional schema of this PRM with arrows representing

possible dependencies between the attributes of the

classes are reported.

Figure 9: The PRM for the operation add2.

The operation bake is a temporal operation. It is

represented by a pair of classes: one representing how

the operation bake starts, the other representing the

probability distribution of the process of baking. The

PRM for the operation bake reported in Figure 10 is

equivalent to a PRM consisting of the ﬁrst class in the

pair and 20 copies (if the duration of a time step is

equivalent to 1 minute) of the second. Being mixture4

an output of the making balls operation, it is formed

by small balls to be put in the oven. The concept

mixture4 has property the diameter of the balls that

is mapped as an attribute of the PRM class mixture4.

The diameter attribute of mixture4 inﬂuences the con-

sistency of the output of the baking operation mix-

ture5, as expressed by the probabilistic dependency

that exists between these two attributes.

The operation add gradually is a special temporal

operation because the ontology does not give us the

number of times the probabilistic model has to loop

over the second class in the pair before passing to the

operation that is next to it (Figure 11). We are cur-

KEOD 2015 - 7th International Conference on Knowledge Engineering and Ontology Development

176

Figure 10: The PRM for the operation bake.

Figure 11: The PRM for the operation add gradually.

rently reasoning about two possible solutions to treat

this problem. The ﬁrst one being to rely on structure

uncertainty. If a probabilistic distribution p on the

number of times the loop has to be done is given, we

can make the structure uncertain. We add a parameter

θ parent of the operation following the temporal one.

The probability of the operation given θ is given by

p. The second being to deﬁne a simulation process

on top of the PRM ruled by the conditions underlin-

ing the exit of the loop (e.g. cook till brown). We

condition the loop exit to the truth of this condition.

5 CONCLUSIONS

We presented how to map an ontology of transforma-

tion processes to a PRMs’s relational schema. The

probabilisticmodel deﬁned starting from the ontology

is a powerful reasoning tool. It integrates data infor-

mation into the relational schema obtained from the

ontology. Incorporating this information, we could

deal with common data mining problems such as

missing data and data integration. We propose to

combine the two models while maintaining them sep-

arate: each formalism can beneﬁt from the strength of

the other and be, at the same time, a standing-alone

model. We illustrate our mapping on an ontology of

transformation processes in the cooking domain, re-

lying on the SUMO upper level ontology.

We propose a methodology able to automati-

cally map SUMO physical concepts (objects and pro-

cesses) into PRM classes and the SUMO abstract en-

tity attribute into PRM attributes. We propose a map-

ping for the ontology concepts operation and tempo-

ral operation. To map the former into a PRM we ex-

tend the standard deﬁnition of PRMs with ideas used

in dynamic Bayesian networks. To map temporal op-

erations that have an uncertain stop criterion, we pro-

pose the use of structure uncertainty or the deﬁnition

of a simulation process over the sequence of opera-

tions. These have drawbacks that we are studying.

Learning PRMs is an NP hard problem that can

be compared to learning Bayesian networks. Acquir-

ing the parameters of a PRM knowing its relational

schema is much easier. Even if we do not have exper-

imental result on that, we think that we can say that

learning the PRM of a transformation process whose

relational schema has been obtained mapping the on-

tology of that transformation process is much easier

than learning it from scratch.

We plan to pair the proposed approach with an

algorithm for learning PRM’s parameters. This will

provide the possibility to experiment the proposed ap-

proach. Finally, we would like to apply our mapping

to other transformation processes such as microorgan-

ism production and stabilization processes.

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