=Paper=
{{Paper
|id=Vol-3263/abstract-10
|storemode=property
|title=Modelling Multiple Perspectives by Standpoint-Enhanced DLs (Extended Abstract)
|pdfUrl=https://ceur-ws.org/Vol-3263/abstract-10.pdf
|volume=Vol-3263
|authors=Lucía Gómez Álvarez,Sebastian Rudolph,Hannes Strass
|dblpUrl=https://dblp.org/rec/conf/dlog/AlvarezRS22
}}
==Modelling Multiple Perspectives by Standpoint-Enhanced DLs (Extended Abstract)==
https://ceur-ws.org/Vol-3263/abstract-10.pdf
Modelling Multiple Perspectives by
Standpoint-Enhanced Description Logics
Extended Abstract
Lucía Gómez Álvarez, Sebastian Rudolph and Hannes Strass
Computational Logic Group, Institute of Artificial Intelligence, Faculty of Computer Science, TU Dresden
1. Introduction
Ontologies and knowledge bases reflect not only domain specific knowledge, but also the
individual points of view of their creators along with other contextual aspects. They may also
differ in modelling design decisions, such as the choice of conceptual granularity or specific
ways of axiomatising information. This semantic heterogeneity is bound to pose significant
challenges whenever the interoperability of independently developed knowledge specifications
is required.
Example 1. Consider the integration of two ontologies of forestry LC and LU, adopting a land
cover and a land use perspective respectively. LC characterises a forest as a “forest ecosystem”
with a minimum area (G1) where a “ forest ecosystem” is an ecosystem with a certain ratio of tree
canopy cover (G2). In contrast, LU defines a forest as a maximally connected area with “forest use”
(G3). Both sources LC and LU agree that forests subsume broadleaf, needleleaf and tropical forests
(G4), and they both adhere to the Basic Formal Ontology (BFO, Arp et al. 2015), an upper-level
ontology stipulating that “land” and “ecosystem” are disjoint categories (G5).
(G1) □LC [Forest ≡ ForestEcosystem ⊓ ∃hasLand.Area≥0.5ha ]
(G2) □LC [ForestEcosystem ≡ Ecosystem ⊓ TreeCanopy≥20% ]
(G3) □LU [Forest ≡ ForestlandUse ⊓ MCON] ∧ □* [ForestlandUse ⊑ Land]
(G4) □LC∪LU [(BroadleafForest ⊔ NeedleleafForest ⊔ TropicalForest) ⊑ Forest)]
(G5) (LC ∪ LU ⪯ BFO) ∧ □BFO [(Land ⊓ Ecosystem) ⊑ ⊥]
Ontology merging focuses on combining and merging different sources into a single conflict-
free conceptual model, a non-trivial task [2, 3] often involving a certain knowledge loss or
weakening in order to avoid incoherence and inconsistency [4, 5]. For instance, in Example 1,
forests are defined as ecosystems in LC (G1) and as lands in LU (G3), with ecosystem and land
being disjoint categories (G5). To merge LU and LC, one would typically (Opt-Weak) give up on
the disjointness axiom (G5), or (Opt-Dup) duplicate the conflicting predicates [6], here both
DL 2022: 35th International Workshop on Description Logics, August 7–10, 2022, Haifa, Israel
lucia.gomez_alvarez@tu-dresden.de (L. Gómez Álvarez); sebastian.rudolph@tu-dresden.de (S. Rudolph);
hannes.strass@tu-dresden.de (H. Strass)
0000-0002-2525-8839 (L. Gómez Álvarez); 0000-0002-1609-2080 (S. Rudolph)
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
Forest and all its subclasses (G4). As an alternative, we propose Standpoint Logic, a simple,
yet versatile generic approach to extend existing KR formalisms by the capability to express
domain knowledge relative to diverse, possibly conflicting standpoints.
In multimodal Standpoint logic [7] propositions of the form □s 𝜑 and ♢s 𝜑 express information
relative to the standpoint s and read, respectively: “according to s, it is unequivocal/conceivable
that 𝜑”. In the semantics, inspired by the theory of supervaluationism [8], standpoints are
represented by sets of precisifications (akin to worlds in possible world semantics), such that
□s 𝜑 and ♢s 𝜑 hold if 𝜑 is true in all/some of the precisifications in s. This makes the single-
modal standpoint logic close to the modal logic K45 [9], where the usual Kripke relations can
be simplified into sets. Unlike in K45, in standpoint logic this simplification can be carried over
to the multi-modal case, and in addition combinations of standpoints are allowed, as in (G4).
In spite of its simple syntax and semantics, the language is remarkably versatile; it allows
for specifying knowledge relative (a) to a standpoint, e.g. (G1), (b) to the global standpoint,
denoted by *, e.g. (G3), and (c) to set-theoretic combinations of standpoints, e.g. (G4). Additional
definabe constructs include ℐLC∪LU [Forest ⊑ Land], meaning that, “according to LC ∪ LU, it is
indeterminate whether land subsumes forest” (with ℐs 𝜑 := ♢s ¬𝜑 ∧ ♢s 𝜑). The sharper operator
⪯, used to establish hierarchies of standpoints, can be defined via s1 ⪯ s2 := □s1 ∖s2 [⊤ ⊑ ⊥].
Intuitively, s1 ⪯ s2 expresses that standpoint s1 inherits the propositions of s2 , by virtue of
“s1 ⊆ s2 ” holding for the corresponding sets of precisifications. This is used in the example to
“import” background knowledge from the upper-level ontology BFO (G5) and it can be used in the
axiom * ⪯ (LC ∪ LU) to confine the interest to a given set of standpoints (each precisification
must be associated with at least one of those). Natural reasoning tasks over multi-standpoint
specifications include gathering unequivocal or undisputed knowledge, determining knowledge
that is relative to a standpoint or a set of them, and contrasting the knowledge that can be
inferred from different standpoints.
This extended abstract introduces the standpoint-enhanced version of the very expressive de-
scription logic 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 (cf. Section 2.2), which is tightly connected to the W3C-standardised
ontology language OWL 2 DL. Here, standpoint modalities are allowed only at the axiom level,
which – coupled with the semantics of the multi-modal standpoint logic framework – makes it
possible to establish a polytime translation from standpoint-enhanced 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 into plain
𝒮ℛ𝒪ℐ𝒬𝑏𝑠 (cf. Section 2.3). By virtue of this result, existing highly optimised OWL reasoners
can be used off the shelf to provide reasoning support for ontology languages from the OWL
family extended by standpoint modelling.
2. Sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠
Sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠 (S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] ) adds the standpoint modalities to 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 at
the sentence (axiom) level. After an introduction to 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 (Section 2.1), we provide the
syntax and semantics of S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] (Section 2.2), we state a small model property and finally we
present a satisfiability-preserving polynomial translation from S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] into plain 𝒮ℛ𝒪ℐ𝒬𝑏𝑠
(Section 2.3). This provides us with tight complexity results and it paves the way towards
practical reasoning in “Standpoint-OWL”, since it allows us to use highly optimised OWL 2 DL
reasoners.
2.1. 𝒮ℛ𝒪ℐ𝒬𝑏𝑠
𝒮ℛ𝒪ℐ𝒬𝑏𝑠 is a gentle extension of 𝒮ℛ𝒪ℐ𝒬 [10] allowing safe Boolean role expressions over
simple roles [11]. By focusing on the mildly stronger 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 instead of the more mainstream
𝒮ℛ𝒪ℐ𝒬, we can provide a more coherent and economic presentation, without giving up the
good computational properties and the availability of optimised algorithms and tools.
In what follows, let 𝑁I , 𝑁C , and 𝑁R be finite, mutually disjoint sets called individual names,
concept names and role names, respectively. 𝑁R is subdivided into simple role names 𝑁R s and
non-simple role names 𝑁R ns , the latter containing the universal role u and being strictly ordered
by ⋖. We briefly mention that unlike in the original definition of 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 , we fix simplicity
of roles and ⋖ explicitly upfront, simplifying the presentation without restricting expressivity.
Then, let ℛs be the set of simple role expressions such that 𝑟1 , 𝑟2 ::= s | s− | 𝑟1 ∪ 𝑟2 | 𝑟1 ∩ 𝑟2 |
𝑟1 ∖ 𝑟2 , with s ∈ 𝑁R
s , and let ℛ = ℛs ∪ 𝑁 ns be the set of (arbitrary) role expressions. The order ⋖
R
is extended to ℛ by making all elements of ℛs ⋖-minimal. The syntax and semantics of concept
and role expressions as well as the different types of 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 sentences (called axioms) are
defined as usual [10, 11]. Finally, any concept expression can be put in NNF, where negation
only occurs in front of concept names, nominals, or Self concepts.
2.2. Sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠 : Syntax and Semantics
Given a set 𝒮 of standpoint symbols containing the universal standpoint *, we define the set ℰ𝒮
of standpoint expressions via e1 , e2 ::= s | e1 ∪ e2 | e1 ∩ e2 | e1 ∖ e2 , where s ∈ 𝒮. Then, the set
S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] of Sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠 sentences is defined inductively over ℰ𝒮 and the
set of 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 axioms:
• if Ax is a 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 axiom then Ax ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] ,
• if 𝜑, 𝜓 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] then ¬𝜑 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] , 𝜑 ∧ 𝜓 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] , and 𝜑 ∨ 𝜓 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] ,
• if 𝜑 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] and e ∈ ℰ𝒮 then □e 𝜑 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] and ♢e 𝜑 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] .
In what follows, we assume that formulas are in SSNF, whereby their modal degree is at most 1.
The semantics of sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠 is defined by “plugging” the semantics of
𝒮ℛ𝒪ℐ𝒬𝑏𝑠 axioms into a multi-modal semantics where precisifications replace worlds and the
usual set of accessibility relations is replaced by the function 𝜎. Thus, a Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠
structure M is a tuple ⟨Δ, Π, 𝜎, 𝛾⟩ where (a) Δ is a non-empty set, the domain of M, (b) Π is
the non-empty set of precisifications, (c) 𝜎 is a function mapping each standpoint symbol to a
set of precisifications, with 𝜎(*) = Π, and (d) 𝛾 is a function mapping each precisification from
Π to an ordinary 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 interpretation ℐ = (Δ, ·ℐ ) composed of the (shared1 ) domain Δ
and a function ·ℐ mapping individual names to elements of Δ, concept names to subsets of Δ,
and role names to subsets of Δ × Δ.
Theorem 1. A sentential Standpoint-𝒮ℛ𝒪ℐ𝒬𝑏𝑠 formula 𝜑 is satisfiable iff it has a model with
at most |𝜑| precisifications.
Theorem 1 can be shown following a similar method to the propositional case [7] because
the use of modalities exclusively at the axiom level prevents the interaction between the modal
operators and implicit free variables. Proofs can be found at https://arxiv.org/abs/2206.06793.
1
We adopt the constant domain assumption, with which expanding and varying domains can be emulated [12].
2.3. Translation into Plain 𝒮ℛ𝒪ℐ𝒬𝑏𝑠
To conclude this short paper, we provide a polynomial translation mapping any S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠]
sentence 𝜑 (w.l.o.g. in NNF) to an equisatisfiable set of 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 axioms. With this, we
establish the decidability and complexity of S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] and we expose a strategy to support
standpoint extensions of languages of the OWL family with off-the-shelf reasoners.
Let Π|𝜑| be a set of size |𝜑| (Theorem 1) and our translation’s vocabulary V[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] (𝜑) consist
of all individual names inside 𝜑, plus, for each 𝜋 ∈ Π|𝜑| , the following symbols: a concept
name A𝜋 for each A ∈ 𝑁C ; a simple role name s𝜋 for each s ∈ 𝑁R s ; non-simple role names
r and r for each r ∈ 𝑁R ∖{u}; a simple role name s𝜌 for each unnegated RIA 𝜌 inside 𝜑;
𝜋 𝜋 ns 𝜋
a fresh constant name 𝑎𝜋𝜌 for each negated RIA 𝜌 inside 𝜑; and a concept name Ms𝜋 for each
s ∈ 𝒮. The non-simple role names inherit their ordering ⋖ from 𝑁R ns and we let r𝜋 ⋖ r𝜋 for
each r ∈ 𝑁R ∖{u}.
ns
The translation Tr(𝜑) of 𝜑 is then a set of 𝒮ℛ𝒪ℐ𝒬 axioms defined as follows: First, Tr(𝜑)
contains the RIA r𝜋 ⊑ r𝜋 for every r ∈ 𝑁R ns∖{u} and each 𝜋 ∈ Π . Second, for every unnegated
|𝜑|
RIA 𝜌 inside 𝜑 and each 𝜋 ∈ Π|𝜑| , Tr(𝜑) contains the RIA 𝐵𝐺𝜋 (𝜌), with 𝐵𝐺𝜋 defined by
𝑟1 ∘...∘ 𝑟𝑛 ⊑ r ↦→ s𝜋𝜌 ∘𝑟1𝜋 ∘...∘𝑟𝑛𝜋 ⊑ r𝜋 𝑟1 ∘...∘ 𝑟𝑛 ∘r ⊑ r ↦→ s𝜋𝜌 ∘ 𝑟1𝜋 ∘...∘ 𝑟𝑛𝜋 ∘ r𝜋 ⊑ r𝜋
𝜋
r ∘𝑟1 ∘...∘ 𝑟𝑛 ⊑ r ↦→ r ∘ 𝑟1𝜋 ∘...∘𝑟𝑛𝜋 ∘ s𝜋𝜌 ⊑ r𝜋 r∘ r ⊑ r ↦→ s𝜋𝜌 ∘ r𝜋 ∘ r𝜋 ⊑ r𝜋 ,
whereby the role expression 𝑟𝜋 is obtained from 𝑟 by substituting every role name s with s𝜋
(except u which remains unaltered). Third and last, Tr(𝜑) contains the GCI
⊤⊑
d d *
𝜋∈Π|𝜑| tr(𝜋, 𝜑) ⊓ 𝜋∈Π|𝜑| ∀u.M𝜋
where, by inductive definition,
tr(𝜋, Ax) = tr+ (𝜋, Ax) tr(𝜋, 𝜓1 ∨ 𝜓2 ) = tr(𝜋, 𝜓1 ) ⊔ tr(𝜋, 𝜓2 )
tr(𝜋, ¬Ax) = tr− (𝜋, Ax) tr(𝜋,′ □e 𝜓) = 𝜋∈Π|𝜑| (¬trℰ (𝜋, e) ⊔ tr(𝜋, 𝜓))
d
tr(𝜋,′ ♢e 𝜓) = 𝜋∈Π|𝜑| (trℰ (𝜋, e) ⊓ tr(𝜋, 𝜓))
⨆︀
tr(𝜋, 𝜓1 ∧ 𝜓2 ) = tr(𝜋, 𝜓1 ) ⊓ tr(𝜋, 𝜓2 )
The translation of unnegated and negated 𝒮ℛ𝒪ℐ𝒬 axioms
tr+ (𝜋, 𝜌) = ∀u.∃s𝜋𝜌 .Self tr− (𝜋, 𝜌) = ∃u. (∀r𝜋.¬{𝑎𝜋𝜌 }) ⊓ (∃𝑟𝜋1...∃𝑟𝜋𝑚 .{𝑎𝜋𝜌 })
(︀ )︀
tr+ (𝜋, 𝐶 ⊑ 𝐷) = ∀u.(¬𝐶 ⊔ 𝐷)𝜋 tr− (𝜋, 𝐶 ⊑ 𝐷) = ∃u.(𝐶 ⊓ ¬𝐷)𝜋
tr+ (𝜋, 𝐶(𝑎)) = ∃u. {𝑎} ⊓ 𝐶 𝜋 tr− (𝜋, 𝐶(𝑎)) = ∃u. {𝑎} ⊓ (¬𝐶)𝜋
(︀ )︀ (︀ )︀
tr+ (𝜋, 𝑟(𝑎, 𝑏)) = ∃u. {𝑎} ⊓ ∃𝑟𝜋 .{𝑏} tr− (𝜋, 𝑟(𝑎, 𝑏)) = ∃u. {𝑎} ⊓ ∀𝑟𝜋 .¬{𝑏}
(︀ )︀ (︀ )︀
. .
tr+ (𝜋, 𝑎 = 𝑏) = ∃u. {𝑎} ⊓ {𝑏} tr− (𝜋, 𝑎 = 𝑏) = ∃u. {𝑎} ⊓ ¬{𝑏}
(︀ )︀ (︀ )︀
Therein, for any role expression 𝑟, we let 𝑟 denote r if 𝑟 = r is a non-simple role name, and
otherwise 𝑟 = 𝑟. Moreover, 𝐶 𝜋 denotes the concept expression that is obtained from 𝐶 by
first transforming it into negation normal form, second replacing concept names A with A𝜋 and
role expressions r by r𝜋 , and third replacing every ∃r for non-simple r with ∃r. Finally, trℰ
implements the semantics of standpoint expressions: Each {︀ expression e ∈ }︀ℰ𝒮 is transformed
into a concept expression trℰ (𝜋, e) over the vocabulary Ms𝜋 |𝑠 ∈ 𝒮, 𝜋 ∈ Π|𝜑| thus:
trℰ (𝜋, s) = ∀u.Ms𝜋 trℰ (𝜋, e1 ∪ e2 ) = trℰ (𝜋, e1 ) ⊔ trℰ (𝜋, e2 )
trℰ (𝜋, e1 ∩ e2 ) = trℰ (𝜋, e1 ) ⊓ trℰ (𝜋, e2 ) trℰ (𝜋, e1 ∖ e2 ) = trℰ (𝜋, e1 ) ⊓ ¬trℰ (𝜋, e2 )
With all definitions in place, we obtain the desired result, which concludes the paper.
Theorem 2. Given 𝜑 ∈ S[𝒮ℛ𝒪ℐ𝒬𝑏𝑠] , the set Tr(𝜑) is a valid 𝒮ℛ𝒪ℐ𝒬𝑏𝑠 knowledge base, is equi-
satisfiable with 𝜑, is of polynomial size wrt. 𝜑, and can be computed in polynomial time.
Acknowledgments
Lucía Gómez Álvarez was supported by the Bundesministerium für Bildung und Forschung
(BMBF, Federal Ministry of Education and Research) in the Center for Scalable Data Analytics and
Artificial Intelligence (ScaDS.AI). Sebastian Rudolph has received funding from the European
Research Council (Grant Agreement no. 771779, DeciGUT).
References
[1] R. Arp, B. Smith, A. D. Spear, Building ontologies with basic formal ontology, MIT Press,
2015.
[2] J. Euzenat, A. Mocan, F. Scharffe, Ontology alignments, in: M. Hepp, P. D. Leenheer, A. D.
Moor, Y. Sure (Eds.), Ontology Management. Computing for Human Experience, volume 7,
Springer, 2008, pp. 177–206. doi:10.1007/978-0-387-69900-4_6.
[3] L. Otero-Cerdeira, F. J. Rodríguez-Martínez, A. Gómez-Rodríguez, Ontology matching:
A literature review, Expert Systems with Applications 42 (2015). doi:10.1016/j.eswa.
2014.08.032.
[4] C. Pesquita, D. Faria, E. Santos, F. M. Couto, To repair or not to repair: Reconciling
correctness and coherence in ontology reference alignments, in: Procs. of the 8th Int. Conf.
on Ontology Matching, volume 1111, CEUR, 2013, pp. 13–24.
[5] A. Solimando, E. Jiménez-Ruiz, G. Guerrini, Minimizing conservativity violations in
ontology alignments, Knowledge and Information Systems 51 (2017).
[6] I. Osman, S. Ben Yahia, G. Diallo, Ontology Integration: Approaches and Challenging
Issues, Information Fusion 71 (2021) 38–63. doi:10.1016/j.inffus.2021.01.007.
[7] L. Gómez Álvarez, S. Rudolph, Standpoint logic: Multi-perspective knowledge representa-
tion, in: F. Neuhaus, B. Brodaric (Eds.), Procs. of the 12th Int. Conf. on Formal Ontology in
Information Systems, volume 344 of FAIA, IOS Press, 2021, pp. 3–17.
[8] K. Fine, Vagueness, truth and logic, Synthese 30 (1975) 265–300. doi:10.1007/
BF00485047.
[9] A. Pietruszczak, Simplified Kripke style semantics for modal logics K45, KB4 and KD45, Bul-
letin of the Section of Logic 38 (2009) 163–171. URL: https://philpapers.org/rec/PIESKS-2.
[10] I. Horrocks, O. Kutz, U. Sattler, The even more irresistible SROIQ, in: P. Doherty, J. Mylo-
poulos, C. A. Welty (Eds.), Procs. 10th Int. Conf. on Principles of Knowledge Representation
and Reasoning, AAAI Press, 2006, pp. 57–67.
[11] S. Rudolph, M. Krötzsch, P. Hitzler, Cheap Boolean role constructors for description
logics, in: S. Hölldobler, C. Lutz, H. Wansing (Eds.), Procs. of the 11th European Conf. on
Logics in Artificial Intelligence, volume 5293, Springer, 2008, pp. 362–374. doi:10.1007/
978-3-540-87803-2_30.
[12] F. Wolter, M. Zakharyaschev, On the decidability of description logics with modal operators,
in: A. G. Cohn, L. K. Schubert, S. C. Shapiro (Eds.), Procs. of the Sixth Int. Conf. on Principles
of Knowledge Representation and Reasoning, Morgan Kaufmann, 1998, pp. 512–523.