=Paper=
{{Paper
|id=Vol-2663/paper-14
|storemode=property
|title=Using Higher-order Description Logics for Learning and Mining in Complex Domains
|pdfUrl=https://ceur-ws.org/Vol-2663/paper-14.pdf
|volume=Vol-2663
|authors=Francesca Alessandra Lisi
|dblpUrl=https://dblp.org/rec/conf/dlog/Lisi20
}}
==Using Higher-order Description Logics for Learning and Mining in Complex Domains==
https://ceur-ws.org/Vol-2663/paper-14.pdf
Higher-order Description Logics for
Learning and Mining in Complex Domains?
Francesca A. Lisi
Dipartimento di Informatica &
Centro Interdipartimentale di Logica e Applicazioni (CILA)
Università degli Studi di Bari “Aldo Moro”, Italy
FrancescaAlessandra.Lisi@uniba.it
Abstract. This short paper summarizes the work I have done over the
last years on the use of higher-order Description Logics (DLs) for learning
and mining in complex domains. In particular, the work proposes higher-
order DLs as a means for metamodeling and metaquerying in Concept
Learning and Knowledge Graph Mining, respectively.
Keywords: Higher-order Description Logics · Concept Learning · Knowl-
edge Graph Mining.
1 Introduction
Most learning and mining problems can be reformulated as Constraint Satisfac-
tion Problems (CSPs) or Optimization Problems (OPs). So, problem solving in
this context could in principle take advantage of generic solvers, by exclusively
using a description of the relevant domain knowledge and the conditions imposed
by the problem to be solved. However, in spite of focusing on problem specifica-
tion, research in this area has traditionally focused on designing effective specific
algorithms for solving the problem in hand. As stressed by De Raedt [7], there
is an increasing interest in providing the user with languages for learning and
mining. This change of perspective claims for a model+solver approach to learn-
ing and mining problems, in which the user specifies the problem by means of
a declarative modeling language and the system automatically transforms such
models into a format that can be used by a solver to efficiently generate a so-
lution. For instance, constraint programming has been successfully applied to
itemset mining problems (see, e.g., [12] for a comprehensive account). Another
notable example is the framework of Meta-Interpretive Learning (MIL) [26]. MIL
uses descriptions in the form of meta-rules (expressed in a higher-order dyadic
Datalog fragment) with procedural constraints incorporated within a meta-
interpreter, which could be eventually implemented by relying on Answer Set
Programming (ASP) solvers (see [10] for an updated overview).
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
F.A. Lisi
This short paper summarizes the work I have done over the last years on
the use of higher-order Description Logics (DLs) for learning and mining in
complex domains. In particular, the work proposes higher-order DLs as a means
for metamodeling and metaquerying in Concept Learning and Knowledge Graph
Mining, respectively.
2 Learning and Mining in Complex Domains
Machine Learning (ML) and Data Mining (DM) algorithms both look for regu-
larities in data, by means of some inductive reasoning mechanism such as gen-
eralization. However, it is conventional to distinguish between the two classes of
algorithms as for the scope of induction. In particular, learning algorithms usu-
ally aim at prediction on unseen data, whereas mining algorithms have typically
the scope of mere description of the given data.
Structure is inherent to data and knowledge in complex domains, and needs
appropriate means for representation. Among the many formalisms used for
representing structured knowledge, one of the most popular is the family of De-
scription Logics (DLs) [1], which has been the starting point for the definition of
the ontology language OWL.1 A DL knowledge base (or equivalently, an OWL
ontology) is a collection of logical axioms and assertions. RDF2 is another pop-
ular formalism for structured knowledge, which however is less expressive than
OWL. A knowledge graph (KG) is a huge collection of RDF triples. KGs can
be interlinked and overall they implement the so-called Web of Data, i.e., the
vision of the World Wide Web (WWW) as a distributed database system.
Structured knowledge poses several challenges to learning and mining al-
gorithms. In the following subsections I will briefly introduce the two cases of
interest for this work, namely Concept Learning and Knowledge Graph Mining.
2.1 Concept Learning
Concept Learning deals with inferring the general definition of a category based
on members (positive examples) and nonmembers (negative examples) of this
category. In Concept Learning, the key inferential mechanism for induction is
generalization as search through a partially ordered space of inductive hypothe-
ses [23]. A popular form of Concept Learning is the one known under the name
of Inductive Logic Programming (ILP) [25] where the hypotheses are typically
expressed in the form of first-order Horn clauses (or other fragments of first-
order logic). A distinguishing feature of ILP with respect to other forms of Con-
cept Learning is the use of prior knowledge of the domain of interest, called
background knowledge (BK), during the search for hypotheses. In ILP it is also
common practice to exploit some declarative bias to, e.g., constrain the language
of hypotheses.
1
https://www.w3.org/TR/owl2-overview/
2
https://www.w3.org/RDF/
Higher-order DLs for Learning and Mining in Complex Domains
Fig. 1. Michalski’s example of eastbound (left) and westbound (right) trains (illustra-
tion taken from [22]).
Concept Learning in DLs has been paid increasing attention since the 90s.
Early work essentially focused on demonstrating the PAC-learnability for various
terminological languages derived from the Classic DL (see, e.g., [3]). Later
works such as [2,14] have followed the generalization as search approach by
extending the methodological apparatus of ILP to DL languages. More recently
there has been a renewed interest in more theoretical work (see, e.g., [13]).
There are several variants of the Concept Learning problem in the DL con-
text. The variant I consider as a showcase in this paper is the supervised one. In
the following, the set of all individuals occurring in A and the set of all individ-
uals occurring in A that are instances of a given concept C w.r.t. K are denoted
by Ind(A) and RetrK (C), respectively.
Definition 1 (Concept Induction - CSP version). Let K = (T , A) be a
DL KB. Given a (new) target concept name C, a set of positive and negative
−
examples Ind+C (A) ∪ IndC (A), and a concept description language DLH , the CSP
version of the Concept Induction problem (denoted by CI-CSP) is to find a concept
definition C ≡ D with D ∈ DLH such that: (i) K |= (a : D) ∀a ∈ Ind+ C (A),
and (ii) K |= (b : ¬D) ∀b ∈ Ind− C (A).
Example 1. For illustrative purposes of the CI-CSP problem, let us consider a
very popular classification problem proposed 40 years ago by Ryszard Michal-
ski [22] and illustrated in Figure 1. Here, 10 trains are described, out of which
5 are eastbound and 5 are westbound. The aim of this problem is to find the
discriminating features between these two classes referred to as EastTrain and
WestTrain (or, more briefly, as ET and WT) from now on.
For the purpose of this case study, let us consider an ALCO ontology, trains2,
encoding the original Trains data set. 3 With reference to trains2 (which there-
fore will play the role of K as in Def. 1), we might want to induce a SROIQ
concept definition for the target concept name ET (i.e., the language of hy-
potheses is some SROIQH based on SROIQ) from the following positive and
negative examples:
3
http://archive.ics.uci.edu/ml/datasets/Trains
F.A. Lisi
Fig. 2. Fragment of a knowledge graph (taken from [30]).
– Ind+
ET (A) = {et1, . . . , et5} ⊆ RetrK (ET)
– Ind−
ET (A) = {wt1, . . . , wt5} ⊆ RetrK (¬ET)
Note that the 5 positive examples for ET are negative examples for WT and vice
versa.
2.2 Knowledge Graph Mining
The analysis of data contained in a KG (referred to as KG Mining) is preliminary
to several crucial maintenance tasks, notably the automated completion of the
graph (aka link prediction), which pose several challenges due to the open and
distributed environment of the WWW infrastructure. In the KG community
approaches for link prediction are divided into statistics-based (see [27] for an
overview), and logic-based (e.g., [9,30]). The latter, which are the closest to the
work reported in this paper, basically extend and adapt previous work in ILP on
relational association rule mining. However, they differ in the expressiveness of
the mined rules. AMIE+ [9] can mine only Horn rules, whereas the methodology
presented in [30] can address the case of nonmonotonic rules.
Example 2. In the context of link prediction, the following rule
isM arriedT o(X, Y ), livesIn(X, Z) ⇒ livesIn(Y, Z) (1)
can be mined from the KG in Fig. 2 and applied to derive new facts such as
livesIn(alice, berlin), livesIn(dave, chicago) and livesIn(lucy, amsterdam) to
be used for completing the graph.
3 Higher-order DLs for Learning and Mining
In several applications there is a need for modeling and reasoning about meta-
concepts, i.e., concepts whose instances are themselves concepts, and meta-
properties, i.e., relationships between meta-concepts. Metamodeling addresses
Higher-order DLs for Learning and Mining in Complex Domains
this need. Indeed, it allows one to treat concepts and properties as first-order
citizens, and to see them as individuals whose properties can be asserted and
reasoned upon. A common feature to metamodeling approaches is the use of
logical languages with higher-order constructs for a correct representation of
concepts and properties at the meta-level. Metaquerying is a special case of do-
main metamodeling. This is the case where the knowledge base does not contain
any axiom regarding meta-concepts or meta-properties, but the query language
allows for using meta-concepts and meta-properties, so that concepts and prop-
erties in the knowledge base can match the variables in the query, and may thus
be returned as answers to the query. This mechanism allows to express queries
that are beyond first-order logic.
Metamodeling (and metaquerying) has recently attracted an increasing in-
terest in the Knowledge Representation (KR) community, thus giving rise to a
stream of research aimed at extending DLs with higher-order features (see, e.g.,
[28,24,4,5,15]). In particular, Colucci et al. [4] introduce second-order features
in DLs under the Henkin semantics for modeling several forms of non-standard
reasoning. The Henkin style shows a desirable feature, i.e., the expressive power
of the language actually remains first-order.
In the following two subsections I briefly report the main achievements of my
research on metamodeling and metaquerying by means of higher-order DLs in
the context of Concept Learning and Knowledge Graph Mining.
3.1 Metamodeling in Concept Learning
In [16], I have extended Colucci et al.’s work on non-standard reasoning in DLs
[4] to several variants of Concept Learning, thus being the first to propose higher-
order DLs under Henkin semantics as a means for metamodeling in ML. The
idea is that each of these variants, besides being considered as non-standard
reasoning tasks, can be reformulated as a CSP or even as an OP. For the sake
of illustration I will focus on the case of CI-CSP.
−
Following Def. 1, let us assume that Ind+ C (A) = {a1 , . . . , am } and IndC (A) =
{b1 , . . . , bn }. A concept D ∈ DLH is a correct concept definition for the target
−
concept name C w.r.t. Ind+ C (A) and IndC (A) iff it is a solution for the following
second-order concept expression:
γCI-CSP := (a1 : X) ∧ . . . ∧ (am : X) ∧ (b1 : ¬X) ∧ . . . ∧ (bn : ¬X) (2)
that is, iff D can be a valid assignment for the concept variable X. The CI-CSP
problem can be modeled with the following second-order formula
φCI-CSP := ∃X.γCI-CSP (3)
The solvability of a CI-CSP problem is therefore based on the satisfiability of the
second-order formula being used for modeling the problem.
In [19,20], the proposed model+solver approach combines the efficacy of
higher-order DLs in metamodeling (as shown in [16]) with the efficiency of ASP
solvers in dealing with CSPs and OPs. The encoding into ASP is possible under
F.A. Lisi
the fixed-domain semantics [8], a non-standard model-theoretic semantics for
DLs which has been proposed in order to correctly address CSPs in OWL.
Example 3. According to (2), the intended CI-CSP problem of Example 1 corre-
sponds to the following second-order concept expression:
ET
γCI-CSP := (et1 : X) ∧ . . . ∧ (et5 : X) ∧ (wt1 : ¬X) ∧ . . . ∧ (wt5 : ¬X) (4)
The problem is then solvable if the following second-order formula:
φET ET
CI-CSP := ∃X.γCI-CSP (5)
ET
is true in SROIQH , i.e., if there exists a solution to γCI-CSP in SROIQH .
Let us now assume that SROIQH is the set of all SROIQ concept expres-
sions that can be generated starting from the atomic concept and role names
occurring in trains2 (except, of course, for the target concept name). Among the
ET
concepts belonging to SROIQH and satisfying γCI-CSP , there is
∃ hasCar.(ClosedCar u ShortCar) (6)
which describes the set of trains composed of at least one closed short car. It
provides a correct concept definition for ET w.r.t. the given examples, i.e., the
following concept equivalence axiom
ET ≡ ∃ hasCar.(ClosedCar u ShortCar) (7)
is a solution for the CI-CSP problem in hand.
3.2 Metaquerying in Knowledge Graph Mining
In [21] it has been observed that an interesting alternative to language bias (i.e.,
the declarative bias used in, e.g., [30] to learn rules of a predefined form) is the
use of a meta-querying language that could take advantage of some useful meta-
information about the data to be analyzed, for instance, the schema of the KG
when available. In [17,18] I have proposed a new approach to KG Mining which
adapts the notion of metaquery introduced by [29] for DM in relational databases
to the novel context of KGs. In particular, a metaquery for KG Mining is a
second-order DL conjunctive query under the Henkin semantics. However, the
resulting metaquery language can be implemented with standard technologies of
the Web of Data such as SPARQL.4
Example 4. An example of a metaquery in this context is the following
M Q1 : mq(Q, Y, Z) ← P (X, Y ), Q(X, Z) (8)
which looks for the properties (Q) holding for the individuals Y . Note that P, Q
are higher-order variables whereas X, Y, Z are first-order variables.
4
https://www.w3.org/TR/sparql11-overview/
Higher-order DLs for Learning and Mining in Complex Domains
Metaqueries can be extended into implications, called metaquery extensions,
of the form
M Q1 → M Q2 (9)
which are actually a compact representation of two metaqueries, M Q1 and M Q2 ,
where M Q2 is longer than - we say extends - M Q1 . A shorter notation for (9)
is the following which stresses how M Q2 extends M Q1
M Q1 ⇒ (M Q2 \ M Q1 ) (10)
The left-hand side and the right-hand side of (10) are called the body and the
head of the metaquery extension, respectively. Note that in the case of query
extensions, the head does not correspond to the conclusion (as with clauses).
Following the standard terminology, one should rather bear in mind the un-
shortened notation, and call M Q2 the conclusion of the metaquery extension.
Metaquery extensions serve as a template for rules we are interested in when
applying rule mining algorithms to a given KG.
Example 5. Let us consider the following metaquery
M Q2 : mq(Q, Y, Z) ← P (X, Y ), Q(X, Z), Q(Y, Z) (11)
which looks for the properties (Q) holding for the individuals Y and shared with
the individuals X to which Y is related by some P . From (8) and (11) we can
build a metaquery extension as shown below
P (X, Y ), Q(X, Z) ⇒ Q(Y, Z) (12)
with reference to the KG depicted in Fig. 2, (1) is an instantiation of (12)
obtained by substituting the variables P and Q with the role names isM arriedT o
and livesIn, respectively.
4 Final remarks
The work summarized in this paper pursues an interesting direction of research
at the intersection of ML/DM and KR. For this research I have taken inspiration
from recent results in both areas, notably De Raedt et al.’s work on declarative
modeling for ML/DM [6], Colucci et al.’s work on non-standard reasoning in DLs
[4] and Gaggl et al.’s proposal of a fixed-domain semantics for DLs [8]. Inter-
estingly, the former two works pursue a unified view on the inferential problems
of interest to the respective fields of research. This match of research efforts in
the two fields has motivated the work presented in [16] with the aim of bridging
the gap between KR and ML/DM in areas such as the maintenance of knowl-
edge bases (or graphs) where the two fields have already produced promising
results though mostly independently from each other. New questions and chal-
lenges have then been raised by the cross-fertilization of these results. Notably,
the choice of a solver is a critical issue, which was more recently addressed
F.A. Lisi
in [19,20]. Finally, and from a broader perspective, the work here summarized
contributes to the current shift in AI from programming to solving as recently
argued by Geffner [11]. However, much work is still to be done.
As for the use of metamodeling in Concept Learning, I plan to implement
and test the approach by relying on available tools. Besides empirical evalua-
tion, I intend also to investigate how to express optimality criteria such as the
information gain function within the second-order concept expressions. Linking
the approach to existing work on ontologies for ML/DM problems is another
interesting direction of future research.
As for the use of metaquerying in Knowledge Graph Mining, several aspects
of the proposed approach should to be clarified before an implementation. First,
I plan to better define the semantics for the proposed metaquery language, also
concerning the link with SPARQL. Second, I intend to design algorithms for the
instantiation stage and choose the most appropriate evaluation measures for the
intended application.
Acknowledgements This work was partially funded by the INdAM - GNCS Project
2019 “Metodi per il trattamento di incertezza ed imprecisione nella rappresentazione
e revisione di conoscenza”.
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