=Paper=
{{Paper
|id=None
|storemode=property
|title=The Limitations of Description Logic for Mathematical Ontologies: An Example on Neural Networks
|pdfUrl=https://ceur-ws.org/Vol-938/ontobras-most2012_paper7.pdf
|volume=Vol-938
|dblpUrl=https://dblp.org/rec/conf/ontobras/FreitasL12
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==The Limitations of Description Logic for Mathematical Ontologies: An Example on Neural Networks==
https://ceur-ws.org/Vol-938/ontobras-most2012_paper7.pdf
The Limitations of Description Logic for Mathematical
Ontologies: An Example on Neural Networks
Fred Freitas, Fernando Lins
Informatics Center - Federal University of Pernambuco (CIn - UFPE)
Av. Prof. Luis Freire, s/n, Cidade Universitária, 50740-540, Recife – PE – Brazil
{fred,fval}@cin.ufpe.br
Abstract. In this work, we discuss appropriate formalisms to account for the
representation of mathematical ontologies, and, particularly, the problems for
employing Description Logics (DL) for the task.DL has proven to be not
expressive enough to such tasks, because it cannot represent, for instance,
simple numerical constraints, except of cardinality constraints over instances
of individuals and relation instances. Therefore, for such representations, we
advocate the use of a more expressive formalism, based at least on first-order
logic using a top ontology as support vocabulary. We also provide a thorough
example on the representation of an Artificial Neural Network (ANN) defined
in KIF (Knowledge Interchange Format), along with a discussion on the
requirements needed for such a representation and the results achieved.
1 Introduction
All In this work, we introduce a discussion on the appropriate formalism to carry out the
task of representing mathematical knowledge, and particularly the problems for doing it
with Description Logics (BAADER ET AL, 2003). It departed from an attempt to
develop an Ontology of Multilayer Perceptron (MLP) Artificial Neural Networks
(ANN). Our initial intent was to define it using the Semantic Web standardized
description logic language OWL (Ontology Web Language) (WELTY ET AL, 2004),
which would offer us a lot of benefits, like mature development tools, availability of
reasoners, large community of users, etc.
On creating such ontology, our work was focused in developing a conceptual
model that would include the necessary mathematics to enable the future development
of knowledge-based applications that can reason over artificial neural networks, such as
an intelligent agent capable of interacting with users to answer questions about neural
networks that can even create and run them.
During the development of the ontology, we faced several representational
problems while trying to represent the mathematical expressions inherent to neural
networks in Description Logic (DL). Since this formalism is based on set theoretic
semantics, it fits properly to representing domains in which relationships among sets
suffice. For such domains, DL terminologies can accurately describe the domains’
classes. On the one hand, the choice of this formalism for the Semantic Web lies
actually on these grounds. On the other hand, it is not endowed with any apparatus to
deal with many types of mathematical operations and constraints that would turn it into
a more generic formalism, like what can be represented by Prolog, for instance.
Moreover, a core mathematical ontology is required as a vocabulary to make for the
needed complex well-defined mathematical meanings.
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Therefore, for such representations, we advocate the use of a more expressive
formalism, based at least on full first-order logic, and using a rich, well-defined support
vocabulary. In order to illustrate the discussion, we first introduce the required basic
mathematical definitions about MLP neural networks and the expressiveness
requirements that a language should fulfill to represent the ontology. Then, we discuss
the practical expressiveness limitations that hamper the use of Description Logic for
such representations. Finally, we present our MLP ANN ontology and some of its more
important definitions. We developed it in the Standard Upper Ontology Knowledge
Interchange Format (SUO-KIF) language (PEASE 2009) and used the top ontology
Suggested Upper Merged Ontology (SUMO) (NILES & PEASE, 2001) as support
vocabulary, which proved well-tailored for our mathematical representations. We
complete the article giving a very brief introduction of the SUO-KIF language and the
top ontology SUMO, discussing some related work, future work and conclusions.Full
papers must respect the page limits defined by the conference. Conferences that publish
just abstracts ask for one-page texts.
2 An Example: Representing the MLP ANN in an Ontology
Artificial Neural Networks (ANNs) were developed as a rough abstraction of biological
neural networks. An ANN consists of a number of nodes that perform simple numerical
processing. Its nodes are highly interlinked and grouped in layers. Mathematically
speaking, it implements complex function approximators (HAYKIN, 1994). ANNs are
used in the computer mainstream to perform pattern learning and recognition.
The most commonly used ANN is the Multilayer Perceptron (MLP) (HAYKIN,
1994). It has three layers (entry, hidden and output) or more (when more than one
hidden layer is used). Each layer consists of a set of “artificial neurons” connected with
the neighbor layers, as displayed in the figure 1 below.
Figure 1− A MLP Artificial Neural Network, its layers and neurons.
A MLP ANN “learns” the implicit function it should approximate by computing,
correcting and propagating classification errors throughout all its neurons, as follows. In
the training phase, the MLP ANN is given a set of problem instances, usually
represented as numerical values, along with their solutions, and process the data many
times to assign correctly the neurons’ parameters. Synapses among artificial neurons are
modeled via activation functions, which trigger the delivery of numerical stimuli as
inputs to the next layer’s neurons. At the end of the learning phase, the network is tuned
with the parameters so as to reflect as closely as possible the classification function
implicit in the training set.
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The Backpropagation algorithm (HAYKIN, 1994) is used to train the MLP
ANN. It is divided into two phases. In the forward phase, the network is run and a
classification output is calculated. In the backward phase, it compares the output with
the correct result and, if wrong, it calculates and propagates the error through the
weights of the connections among all the neurons of the network. Therefore, the
learning ability of MLP ANNs lies on the neurons. The algorithm iterates between these
two phases until it reaches a stopping criteria. These two phases are explained in detail
in the next subsections.
2.1 Forward Phase
Here, the ANN basically computes summations of a product. At first, each neuron
receives its entry and propagates it to the first hidden layer’s neurons, if its activation
function “triggers”.
𝑝
For each neuron of the hidden layer, the potential of activation of
the neuron j (𝑛𝑒𝑡𝑗 , 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎) is calculated by summating all synapses weights
(𝑤𝑗𝑖 ) from neurons i that reach neuron j, multiplied by the outputs of neurons ii:
𝑛
𝑛𝑒𝑡𝑗𝑝 = � 𝑥𝑖𝑝 𝑤𝑗𝑖 (1)
𝑖= 1
Then, the activation potential is applied to an activation function that will return
the output of the neuron. One of the most used activation functions is the logistic
function (also called sigmoid), that can be seen below:
1
𝑃�𝑛𝑒𝑡𝑗𝑝 � = 𝑝 (2)
−𝑛𝑒𝑡𝑗
1+𝑒
The output of the neuron, given by the result of the activation function, typically
consists of a real value between -1 and 1. A synapse is defined as the propagation of the
output as an entry to the next layer, or, if it is the last layer, one of the outputs belonging
to the output vector of the network. Synapses in a neuron occur if its output is higher
than a given threshold, usually set to 0. Synapses are then propagated (or not) through
the network until they reach the output layer, finishing the forward phase. This phase is
executed when the network is in the execution mode too.
2.2 Backward Phase
The first step of this phase consists in calculating the error between the output of the
neuron j from the forward phase and the expected output from the instance of the
training set, as in the following formula
𝛿𝑗 = �𝑑𝑗 − 𝑦𝑗 �𝑓 ′ (𝑛𝑒𝑡𝑗 ) (3)
where 𝑑𝑗 is the desired output, 𝑦𝑗 the actual output, and 𝑓 ′(𝑛𝑒𝑡𝑗 ) the activation function
of the neuron’s derivate. Hidden nodes’ errors are computed as follows:
𝑛
′
𝛿𝑗 = 𝑓 (𝑛𝑒𝑡𝑗 ) � 𝛿𝑙 𝑤𝑙𝑗 (4)
𝑖= 1
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They are the product of the activation function derivate of the current neuron
and the summation of the products between the error (𝛿𝑗 ) and each of the last layers’
weights (𝑤𝑙𝑗 ). Finally, the synapse weights that connect current neuron j with its
antecessors (neurons i) are given by:
𝑤𝑗𝑖 (𝑡 + 1) = 𝑤𝑗𝑖 (𝑡) + 𝜂𝛿𝑗 (𝑡)𝑥𝑖 (𝑡) (5)
where 𝑤𝑗𝑖 (𝑡) stands for the old weight, 𝑤𝑗𝑖 (𝑡 + 1) means the new weight, 𝜂 is the
learning rate that regulates learning speed, 𝛿𝑗 (𝑡) is the error and 𝑥𝑖 (𝑡) is the input signal
of the neuron with the old weight. The rough idea is slowly (as tuned by 𝜂) distributing
the error issued by the network through all the layers and neurons.
In the next section, we enlist the requirements for representing such knowledge.
3 Requirements of Candidate Formalisms and Languages
To address the problem of choosing the knowledge representation formalism and
language to represent the MLP ANN, we had to consider a number of representation
requirements. We needed enough expressive power to represent the mathematical
concepts on the ontology. Potential candidate formalisms and languages should, at the
same time, allow us to do it in an easy way, as well as give us the possibility of
computing results afterwards. Below, we list the most important requirements for that:
• Ability to represent numerical constraints - as seen above, MLP encompasses
plenty of calculations. The ontology must be capable of representing them in an
straight-forward way, as well as the constraints over them involving
mathematical operations like summations, exponentiations or function
derivations.
• Availability of a rich vocabulary of abstract definitions - the MLP ANN
Ontology makes use of a lot of abstract concepts that are fulfilled either by a
core mathematical ontology or by a top ontology which must include
mathematical expressions, set theory, graph theory, algorithms, sequences, etc.
• Ability to represent n-ary relations - the freedom to use unlimited arities in
relations leaves ontologists free to use some advanced constructs. For instance, a
synapse connects two neurons, thus possessing a ternary relation named
connects that has, as arguments, the synapse and the two neurons.
• Ability to represent functions - when dealing with mathematics concepts, we
often need to represent functions. In the MLP ANN, we had to represent the
activation function and the stopping criteria based on functions.
In the next section, we discuss the difficulties in fulfilling such criteria with DL.
4 Description Logics’ Representational Problems regarding Mathematical
Ontologies
DL and its Ontology Web Language OWL present in practice a number of limitations in
expressiveness that prevented us from using it for representing the knowledge presented
in the section 2. We discuss the main issues in the next subsections.
87
4.1 Description Logic Ontological Engagement
The first aspect to remark when embarking in such a discussion relates to the conceptual
and epistemological commitment (BRACHMAN 1978) of DL as a representation
formalism. Its representation purposes and consequent grounding on precise and
unambiguous set theoretic semantics (BAADER ET AL, 2003) allows the language
mainly to describe quite precisely domains that involve sets (as concepts or classes
according to the formalism’s jargon), elements (as class and relation instances), their
relations (also called properties or roles) and simple constraints that can be expressed in
axioms that use a limited set of constructs like relations’ cardinality, type checking of
relations’ domain and range, classes’ subsumption, complement, disjointness,
equivalence, instance membership to a class and instance equality/inequality.. This is
enough to make for the main reasoning task of concept classification, which means to
entail class subsumption, when this relation is not declared explicitly 1. Class
subsumption, equivalence, disjointness and inconsistency (standing for a class that
according to its definition cannot bear any instances) can be inferred by the reasoned,
also called classifier.
In the next subsection we discuss the use of numerical constraints in DL.
4.2 Numerical Representations and Constraints
Despite consisting of a powerful family of semantically well-founded formalisms’, DLs
are not endowed with the basic mathematical resources necessary for representing
neural networks, for instance the arithmetical operations. The only type of arithmetical
processing is counting and comparing the number of instances that participate in
relations for solving queries, like with the following axiom:
Worried-Woman ≡ Woman ⊓ (≥ 3 child.Man) (6)
It states that a worried woman is one who has three or more male children. The
classifier is able to solve queries that searches for the instances of the class Worried-
Woman by counting the instances of Woman which are related to instances of the class
Man via the relation child.
Although it is possible to define relations with numerical values as datatype
properties in OWL (types integer and float and special types date, time and dateTime),
we did not find in the literature DL extensions that address the general problem of
solving arithmetical constraints, ranging from the simplest ones (like the ones that use
the four basic arithmetical operations) to the most complex (using exponential,
differential, integral operators, differential equations, etc).
Description Logic consists indeed in a smart purely declarative formalism that
performs optimized classical reasoning. Sticking to this stance, no ultimate solution is
expected to the posed problem of including arbitrary arithmetical constraints, because
classifier will not cope with them, ought to the fact that elementary number theory has
been proved undecidable by Gödel in his classical incompleteness theorem (GÖDEL,
1 Classification based on subsumption is also used for solving equivalence and disjointness between classes and class
inconsistency. Even in assertional queries, i.e., the ones which include instances, subsumption checking is the basic
processing behind the reasoning (BAADER ET AL, 2003).
88
1931) as well as rational number theory, since the elementary number theory could be
defined in terms of the rationals, as proved by Julia Robinson (ROBINSON, 1949).
So, the hopes for proposing a general DL solution with arithmetical constraints
probably lies both on restricting the constructs for a limited set of operations and
solving them with built-in constructs that should be evaluated before classification (in
the flavor of Prolog, for instance). Such extension, on the other hand, would change the
formalism into a more procedural and less declarative one, and is not yet reported.
4.3 The Need for a Mathematical Background Knowledge in a Top Ontology
Due to the low expressiveness of OWL, any top-ontology developed using it would face
a lot of problems in representing complex abstract mathematical concepts. This
statement is corroborated by the fact that, despite the existence of several efforts for the
translation the top ontology Supper Upper-Merged Ontology (SUMO) to OWL, none of
them proved successful. Translations can be found in the SUMO site but they are in
OWL Full (which is known to be undecidable (BAADER ET AL, 2003)). Besides this
fact, any translations of full first order logic to DL will just prune the knowledge from
the original ontology that cannot be expressed in DL.
We indeed have tried to work with these translations, encountering many
practical problems. The two more relevant for us were the loss of a the rich set of
mathematical axioms provided by SUMO upper and the necessary mathematical
operations and constraints that could not be defined in DL, due to the lack of
expressiveness.
5 Solution: More Expressive Languages and Top Ontologies
Since DL does not fulfill our expressivity needs, the solution lies on relying on
employing a more expressive language, with first-order logic (FOL) expressivity at least
and, if possible, capable of defining basic arithmetic operations, and a top ontology that
includes some basic mathematical definitions. A Top Ontology defines a set of generic
concepts to be shared by various domain-specific ontologies. Complying with it assures
semantic interoperability among the ontologies and better ontological engagement
(BRACHMAN, 1978). Thus, based on the requirements enlisted above, we decided to
use the SUO-KIF (Standard Upper Ontology Knowledge Interchange Format) with
SUMO as the background mathematical knowledge required.
The SUO-KIF language has declarative semantics and is quite comprehensive in
terms of logical expressiveness, once it was intended primarily to implement the first-
order logic. It is an extension of KIF (GENESERETH, 1991), created with the objective
of supporting the development of SUMO. The goal was simplifying KIF, by
maintaining its syntax, logical operators and quantifiers, but leaving to the ontology
itself the declarations defining classes and instances, and thus eliminating the
dependency from the Frame-Ontology of KIF, what happens to Eng-Math (GRUBER &
OLSEN, 1994). Instead, one must create instances based on the concepts defined in
SUMO as an external resource. Another relevant capability of SUO-KIF when
compared to KIF’s Eng-Math is the deployment of the Sigma reasoner (PEASE, 2003).
Up to now, no reasoner can take KIF’s rich expressivity on. As for the top ontology, we
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opted for SUMO, since it endows us with the math we needed. In the next section, we
present the MLP ANN ontology, with some examples using SUO-KIF and SUMO.
2. The Resulting MLP ANN Ontology in SUO-KIF
In this section, we present the most important concepts of the Multi-Layer Perceptron
Artificial Neural Network (MLP ANN) Ontology. We deploy the basic taxonomy of the
ontology in Figure 2. Note that it uses plenty of concepts from SUMO, like the classes
Abstract, Physical, Relation, BinaryRelation, IrreflexiveRelation, etc. Next, we take a
deeper look at the ontology, describing some of the concepts and axioms that required a
higher level of expressiveness.
Figure 2. The ANN MLP Ontology basic taxonomy.
5.1 Activation function
The representation of the activation function constitutes a good example of a function
definition that takes advantage of the SUO-KIF expressiveness and the richness of the
mathematical vocabulary provided by SUMO. In the code below, we state the concept
of ActivationFunction as a function (irreflexive, intransitive, assymetric relation)
whose domain is the class Neuron.
(instance activationFunction BinaryPredicate)
(instance activationFunction IrreflexiveRelation)
(domain activationFunction 1 Neuron)
(domain activationFunction 2 UnaryFunction)
Then we represent the concept of a Logistic function, which is an unary function that
receives a real number as the only argument and returns an MLPRealNumber .
(instance Logistic UnaryFunction)
(instance Logistic TotalValuedRelation)
(domain Logistic 1 RealNumber)
(range Logistic 1 RealNumber)
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MLPRealNumber is defined as a real number ranging between -1 and 1:
(=>(instance ?NUMBER MLPRealNumber)
(and (or (greaterThan ?NUMBER -1)
(equal ?NUMBER -1))
(or (lessThan ?NUMBER 1)
(equal ?NUMBER 1))))
Finally we define the logistic function, as defined in subsection 2.1., def. (2):
(=>(equal (Logistic ?NUMBER1) ?NUMBER2)
(equal ?NUMBER2 (DivisionFn 1
(AdditionFn 1 (ExponentiationFn NumberE
(SubtractionFn 0 ?NUMBER1))))))
Next, we describe the ANN learning process.
5.2 Backpropagation algorithm process
The process is an iteration alternating the backward and forward phases, until the
halting criterion is met. We can represent it using four axioms. The first one states that
the backpropagation algorithms starts defining its net structure
We can represent this process using four axioms. The first one states that for all
backpropagation algorithms there exists a sub-process (the definition of the net
structure) starting together with the backpropagation algorithm:
(=>(and(instance ?B Backpropagation)
(instance ?NSD NetStructDefinition)
(subProcess ?NSD ?B))
(exists (?BL)
(and(instance ?BL BackpropLoop)
(subProcess ?BL ?B)
(exactlyEarlier (WhenFN ?NSD)
(WhenFn ?BL)))))
The second axiom states that right after the net structure definition, we start the
iteration over the backpropagation loop. The exactlyEarlier relation means that the
first argument finishes at the same instant that the second starts.
(=>(and(instance ?B Backpropagation)
(instance ?BL Backproploop)
(subProcess ?BL ?B)
(not (StopCriteriaFn ?B)))
(exists (?BL)
(and(instance ?BL2 BackpropLoop)
(subProcess ?BL2 ?B)
(exactlyEarlier (WhenFN ?BL)(WhenFn?BL2)))))
Finally, the last axiom states that, if the halting criterion is met, the algorithm
ends. The finishes relation means that both arguments finish at the same time:
(=>(and(instance ?B Backpropagation)
(instance ?BL BackpropLoop)
(subProcess ?BL ?B)
(StopCriteriaFn ?B))
(finishes (WhenFn ?B) (WhenFn ?BL)))
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The weights are updated (subsection 2.2., definition 5) according to the axiom
shown below. It states that if a weight is identified as one to be updated, then the old
weight holds only until the loop cycle lasts, and the new weight is calculated and set.
(=> (and
(instance ?UPDATEWEIGHT UpdateSynapseWeight)
(synapseToUpdate ?UPDATEWEIGHT ?SYNAPSE)
(connects ?SYNAPSE ?NODEA ?NODEB)
(hasOutput ?NODEB ?OUTPUT)
(holdsDuring (BeginFn ?UPDATEWEIGHT)
(hasWeight ?SYNAPSE ?OLDWEIGHT)))
(holdsDuring (EndFn ?UPDATEWEIGHT)
(and
(hasWeight ?SYNAPSE ?NEWWEIGHT)
(equal ?NEWWEIGHT (AdditionFn ?OLDWEIGHT
(MultiplicationFn ?LEARNRATE
(MultiplicationFn ?OUTPUT
(NeuronErrorFn ?SYNAPSE))))))))
5.3 An Example of Ternary Relation: The Synapse Definition
In the next encodings, we show an example of a ternary relation, the relation
connects, that links two neurons through a synapse. The first stretch of code shows
the basic characterizations of the relation in terms of inputs and outputs:
(instance connects TernaryPredicate)
(domain connects 1 Synapse)
(domain connects 2 Neuron)
(domain connects 3 Neuron)
Next, an axiom is defined, stating that all connections of a neuron are
linked with a neuron from its next layer. Note that a synapse takes part of the ternary
relation, thus sufficing for our representation needs.
(=>(and(instance ?N-A Neuron)
(connects ?SYNAPSE ?N-A ?N-B))
(exists(?LAYERA ?LAYERB)
(and(instance ?LAYERB Layer)
(instance ?LAYERA Layer)
(neuronLayer ?N-A ?LAYERA)
(neuronLayer ?N-B ?LAYERB)
(nextLayer ?LAYERA ?LAYERB))))
In the next section, we discuss related work regarding mathematical ontologies.
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5.4 Discussion
The ontology defines the exact constraints that hold among the many elements of a
MLP ANN, as well as the calculations and sequential operations needed to update it
during the training phase. Once SUO-KIF is endowed with a reasoned able to deal with
the mathematical constraints, an application that employs it – an intelligent agent, expert
system, computer algebra system or theorem prover - is capable of creating a virtual
MLP ANN, run it (including all the calculations and updates), and - more important in
terms of declarativity – answer queries about each aspect of it, like stating how many
layers at least a MLP ANN should possess, etc. With that features, even an intelligent
tutor system can make a good use from its knowledge.
Note that, when an expressive formalism is used with a supportive top ontology,
most, if not all, mathematical knowledge can be represented by a similar solution.
Simple examples are other types of neural networks (even recurrent, constructive)
6 Related Work
In the field of Artificial Intelligence, the first successful systems to deal with
mathematical contents that go beyond simple numerical calculi came from the branch
known as Computer Algebra Systems (CAS) (BERTOLI ET AL 1998). They are
capable of performing algebraic and symbolic computation in an abstract way A
representative example of this trend was the REDUCE system (HEARN 2004), which is
still in use nowadays. It is able to calculating algebraic derivates and integrals of
complex functions among many other functionalities. Many of those systems employ
declarative solutions for algebraic problems and some of them are able of using and
presenting proofs indeed. A detailed comparison among computer algebra systems and
their features can be found in [13]. Nevertheless, the integration between this type of
system and the growing field of ontology is still lacking tough. The consequence is that
some valuable mathematical knowledge available in mathematical ontologies, like the
characterization of distinct relation types used here, are not being fully exploited in CAS
systems.
As for the problem of mathematical knowledge in ontologies, we have searched
the literature for possible solutions to our problems, before deciding for the SUO-KIF
language. We found solutions ranging from simple syntactic agreements, like MathML
(CARLISLE 2009) - a markup language without axioms -, to full-fledged languages and
top ontologies like SUO-KIF and SUMO. Among them, we found a proposal to mix
OWL with OpenMath to encode math contents (BUSWELL 2004). We indeed made
attempts to employ this approach, by to representing the math needed in the MLP ANN
ontology in OWL and processes using the Semantic Web Rule language (SWRL) built-
ins. However, none of these solutions fitted our needs. All of them were limited, in the
sense of depending upon external features (like the rules in SWRL), instead of using a
mathematical theory. They mostly represent the expressions in a structured way and
even the solutions using rules will produce fragmented pieces of knowledge that could
pose problems of maintenance in the future.
Furthermore, we tried out a more consistent approach, the use of EngMath
(GRUBER & OLSEN, 1994), a mathematical ontology for engineering. EngMath could
indeed constitute a sound alternative, as it takes advantage of the Kif-Numbers ontology
93
(GRUBER & OLSEN, 1994), a KIF vocabulary focused on numbers, arithmetic
operations and related definitions. We chose SUMO and SUO-KIF because, we
conclude that SUMO covers all definitions comprised in Kif-Numbers and EngMath.
7 Future Work and Conclusions
Despite the popularity and usefulness of DL languages, we claim that this formalism
suffers from a lack of expressiveness for the representation of mathematical knowledge.
In our practical experience portrayed here, the most relevant lesson learned was that we
could only overcome DL representational problems by using at least a full first-order
logic formalism with a supportive top ontology like SUMO, that contains the
mathematical background knowledge required to qualify and define properly the math
definitions for the new ontologies. We presented a thorough case study in that direction,
on the field of automatic learning, the MLP ANN ontology.
We envisage two types of future work. As for the use of the developed ontology,
we are heading for practical applications of the ontology. The creation of these
applications, such as an intelligent agent capable of interacting with users answering
questions about artificial neural networks (making use of the ontology) are in our
agenda. We also consider the translation of the ontology to other languages, such as
Prolog, so as to enable us to reason with it and run concrete applications.
Another more general future work lies on the investigation of ways to embed
ontologies into computer algebra systems. Devising a solution for this general problem
will certainly endow the latter with more powerful reasoning techniques while solving
mathematical problems. For instance, CAS systems could take advantage during
inference of the applicable mathematical constraints defined as relations qualifiers in the
ontology. These constraints need not be hardcoded in the systems, thus increasing
knowledge reuse.
Acknowledgments. The authors would like to thank prof. Richard Banach, from the
University of Manchester, England, who provided us with valuable input on the
limitations of Description Logic for Mathematical representations, and prof. Jacques
Calmet, from the University of Karlsruhe, Germany, who, as scientific ancestor,
indirectly introduced us to the field of Computer Algebra.
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