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How to Agree to Disagree
Managing Ontological Perspectives using Standpoint Logic
Luc´
ıa G´
omez ´
Alvarez(B
), Sebastian Rudolph ,
and Hannes Strass
Computational Logic Group, Faculty of Computer Science, TU Dresden, Dresden, Germany
{lucia.gomez alvarez,sebastian.rudolph,
hannes.strass}@tu-dresden.de
Abstract. The importance of taking individual, potentially conflicting perspec-
tives into account when dealing with knowledge has been widely recognised.
Many existing ontology management approaches fully merge knowledge per-
spectives, which may require weakening in order to maintain consistency; others
represent the distinct views in an entirely detached way.
As an alternative, we propose Standpoint Logic, a simple, yet versatile multi-
modal logic “add-on” for existing KR languages intended for the integrated rep-
resentation of domain knowledge relative to diverse, possibly conflicting stand-
points, which can be hierarchically organised, combined, and put in relation with
each other.
Starting from the generic framework of First-Order Standpoint Logic (FOSL),
we subsequently focus our attention on the fragment of sentential formulas, for
which we provide a polytime translation into the standpoint-free version. This
result yields decidability and favourable complexities for a variety of highly
expressive decidable fragments of first-order logic. Using some elaborate encod-
ing tricks, we then establish a similar translation for the very expressive description
logic SROIQbsunderlying the OWL 2 DL ontology language. By virtue of this
result, existing highly optimised OWL reasoners can be used to provide practical
reasoning support for ontology languages extended by standpoint modelling.
Keywords: knowledge integration ·ontology alignment ·conflict management
1 Introduction
Artefacts of contemporary knowledge representation (ontologies, knowledge bases, or
knowledge graphs) serve as means to conceptualise specific domains, with varying
degrees of expressivity ranging from simple classifications and database schemas to fully
axiomatised theories. Inevitably, such specifications reflect the individual points of view
of their creators (be it on a personal or an institutional level) along with other contex-
tual aspects, and 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.
This paper proposes a way to address the interoperability challenge while at the
same time preserving the varying perspectives of the original sources. This is partic-
ularly important in scenarios that require the simultaneous consideration of multiple,
potentially contradictory, viewpoints.
c
The Author(s) 2022
U. Sattler et al. (Eds.): ISWC 2022, LNCS 13489, pp. 125–141, 2022.
https://doi.org/10.1007/978-3-031-19433-7_8
126 L. G´
omez ´
Alvarez et al.
Example 1. A broad range of conceptualisations and definitions for the notion of forest
have been specified for different purposes, giving rise to diverging or even contradic-
tory statements regarding forest distributions. Consider a knowledge integration scenario
involving two sources adopting a land cover (LC)and a land use (LU)perspective on
forestry. LC characterises a forest as a “forest ecosystem” with a minimum area (F1)
where a forest ecosystem is specified as an ecosystem with a certain ratio of tree canopy
cover (F2). LU defines a forest with regard to the purpose for which an area of land is
put to use by humans, i.e. a forest is a maximally connected area with “forest use” (F3).1
Both sources LC and LU agree that forests subsume broadleaf, needleleaf and trop-
ical forests (F4), and they both adhere to the Basic Formal Ontology (BFO,[1]), an
upper-level ontology that formalises general terms, stipulating for instance that land and
ecosystem are disjoint categories (F5). Using standard description logic notation and
providing “perspective annotations” by means of correspondingly labelled box operators
borrowed from multi-modal logic, the above setting might be formalised as follows:
(F1) LC[Forest ≡ForestEcosystem ∃hasLand.Area≥0.5ha]
(F2) LC[ForestEcosystem ≡Ecosystem TreeCanopy≥20% ]
(F3) LU[Forest ≡ForestlandUse MCON]∧∗[ForestlandUse Land]
(F4) LC∪LU[(BroadleafForest NeedleleafForest TropicalForest)Forest]
(F5) (LC BFO)∧(LU BFO)∧BFO[Land Ecosystem ⊥]
In the case of Example 1,ecosystem and land are disjoint categories according to
the overarching BFO (F5), yet forests are defined as ecosystems according to LC (F1)
and as lands according to LU (F3). These kinds of disagreements result in well-reported
challenges in the area of Ontology Integration [6,21] and make ontology merging a
non-trivial task, often involving a certain knowledge loss or weakening in order to avoid
incoherence and inconsistency [22,28]. In Example 1, to merge LU and LC, there are
two typical options to resolve the issue: (Opt-Weak) one may give up on the disjointness
axiom (F5), or (Opt-Dup) one could duplicate all the conflicting predicates [20], in this
case not only Forest (into Forest LC and Forest LU), but also the forest subclasses
in (F4): BroadleafForest,NeedleleafForest and TropicalForest. In contrast,
we advocate a multi-perspective approach that can represent and reason with many
– possibly conflicting – standpoints, instead of focusing on combining and merging
different sources into a single conflict-free conceptual model.
Standpoint logic [8] is a formalism inspired by the theory of supervaluationism [7]
and rooted in modal logic that supports the coexistence of multiple standpoints and
the establishment of alignments between them, by extending the base language with
labelled modal operators. Propositions LC φand ♦LC φexpress information relative to
the standpoint LC and read, respectively: “according to LC,itisunequivocal/conceivable
that φ”. In the semantics, standpoints are represented by sets of precisifications,2such
that LC φand ♦LC φhold if φis true in all/some of the precisifications in LC.
The logical statements (F1)–(F5), which formalise Example 1by means of a stand-
point-enhanced description logic, are not inconsistent, so all axioms can be jointly
1“Forest use” areas may qualify for logging and mining concessions as well as be further
classified into, e.g. agricultural or recreational land use.
2Precisifications are analogous to the worlds of frameworks with possible-worlds semantics.
How to Agree to Disagree 127
represented. Let us now illustrate the use of standpoint logic for reasoning with the
individual perspectives. First, assume the following (globally agreed) facts about three
instances, an ecosystem e, a parcel of land l, and a city c:
(F6) ForestEcosystem(e)hasLand(e, l)ForestlandUse(l)
(F7) Area≥0.5ha(l)MCON(l)in(l, c)City(c)
It is clear from (F1) that according to LC,eis a forest, written as LC[Forest(e)],
since it is a forest ecosystem (F6) with an area larger than 0.5ha (F7). On the other hand,
it is clear from (F3) that according to LU,lis a forest, LU[Forest(l)], since it has a for-
est land use (F6) and it is a maximally connected area (F7). More interestingly, we can
also obtain joint inferences: assuming the (generally accepted) background knowledge
expressed by hasLand ◦in in, we can infer
LC∪LU[(City ∃in−.Forest)(c)],
which means that “according to both LC and LU there is some forest in City c.” Thi s
holds for LU since lis a forest and is in c(F7); and it holds for LC because eis a forest
in the land l, which is in turn in c.
In contrast to the options of the ontology merging approach, using standpoint
logic prevents the multiplication (and corresponding “semantic detachment”) of pred-
icates from (Opt-Dup). It also avoids unintended consequences arising when knowledge
sources are weakened just enough to maintain satisfiability: In the corresponding (Opt-
Weak) scenario, after merging the knowledge sources of LU and LC and removing (F5),
we can consistently infer Forest(e)from the standpoint-free versions of (F1), (F6) and
(F7) and Forest(l)from (F3), (F6) and (F7), similar to the standpoint framework. But
on top of that, reapplying (F1) and (F3) also yields “eis a forest, and its land lis also a
forest and an ecosystem, and has some other associated land, bigger than 0.5ha” through
the following derivable assertions:
Forest(e)hasLand(e, l)Forest(l)ForestEcosystem(l)∃hasLand.Area≥0.5ha(l)
This illustrates how, beyond the problem of inconsistency, naively merging different
models of a domain may lead to erroneous reasoning. In fact, other non-clashing dif-
ferences between the forest definitions (F1) and (F3) respond to relevant nuances that
relate to each standpoint and should also not be naively merged. For instance, from the
land cover perspective (F1), there is no spatial connectedness requirement, since there
are “mosaic forest ecosystems” where the landscape displays forest patches that are
sufficiently close to constitute a single ecosystem. On the other hand, for LU, there is
no minimum tree canopy (F3), since a temporarily cleared area still has a “forest use”.
Standpoint logic preserves the independence of the perspectives and escapes global
inconsistency – without weakening the sources or duplicating entities – because its
model theory (cf. Section 2.1) requires consistency only within standpoints and precisi-
fications. Notwithstanding, it allows for the specification of structures of standpoints
and alignments between them. Natural reasoning tasks over such multi-standpoint spec-
ifications include gathering unequivocal or undisputed knowledge, determining knowl-
edge that is relative to a standpoint or a set of them, and contrasting the knowledge that
can be inferred from different standpoints.
128 L. G´
omez ´
Alvarez et al.
Let us get an idea of the expressivity of the proposed logic. In spite of its simple
syntax, the language is remarkably versatile; it allows for specifying knowledge relative
(a) to a standpoint, e.g. (F1), (b) to the global standpoint, denoted by ∗,e.g.(F3), and (c)
to set-theoretic combinations of standpoints, e.g. (F4). Additional language features can
be defined in terms of the former: ILCφ, which means that, “according to LC, it is inher-
ently indeterminate whether φ” can be defined by ILCφ:= ♦LC φ∧♦LC ¬φ.Thesharper
operator is used to establish hierarchies of standpoints and constraints on the struc-
ture of precisifications, e.g. (F5), and can be defined via s1s2:= s1\s2[⊥].
Intuitively, s1s2expresses that standpoint s1inherits the propositions of s2, by virtue
of “s1⊆s2” holding for the corresponding sets of precisifications. This type of state-
ment comes handy to “import” background knowledge from some ontology, such as the
foundational ontology BFO in our example. In combination, these modelling features
allow for expressing further constraints useful for knowledge integration scenarios, e.g.,
(F8) ∗(LC ∪LU)∧♦LC[]∧♦LU[],
where the first conjunct allows us to specify that no interpretations beyond the stand-
points of interest are under consideration, by stating that the universal standpoint is a
subset of the union of LC and LU. The other two conjuncts enforce the non-emptiness of
the standpoints of interest, LC and LU, ensuring that each standpoint by itself is coherent.
To illustrate a use case, consider the statement ♦∗[Forest(f)∧¬MCON(f)], expressing
that it is conceivable that fis a non-spatially-connected forest. From this, we can infer
together with (F8) and the unfulfilled requirement of connectedness of LU (F3), that f
must be conceivable for LC instead, and thus fmust be a forest ecosystem (F1):
♦LC[ForestEcosystem(f)∧(∃hasLand.Area≥0.5ha )(f)]
G´
omez ´
Alvarez and Rudolph [8] have introduced the standpoint framework over a
propositional base logic. While they showed favourable complexity results (standard rea-
soning tasks are N P-complete just like for plain propositional logic), the framework is not
expressive enough for knowledge integration scenarios employing contemporary ontol-
ogy languages. In this paper, we widen the scope by (1) introducing the very general
framework of first-order standpoint logic (FOSL) and (2) allowing for more modelling
flexibility on the side of standpoint descriptions by introducing support for set-theoretical
combinations of standpoints (Section 2). We provide the syntax and semantics of this
generic framework, before focusing on the identification of FOSL fragments with ben-
eficial computational properties. To this end, we define the sentential fragment, which
imposes restrictions on the use of standpoint operators and guarantees a small model
property (Section 2.2). Tailored to this case, we introduce a polynomial satisfiability-
preserving translation (Section 2.3) that does not affect membership in diverse decidable
fragments of FO. This allows us to immediately obtain decidability and tight complexity
bounds for the standpoint versions of diverse FO fragments (e.g. the 2-variable counting
fragment, the guarded negation fragment and the triguarded fragment) (Section 3). In
addition, it provides a way to leverage off-the-shelf reasoners for practical reasoning in
standpoint versions of popular ontology languages. We demonstrate this by extending
our results to a standpoint logic based on the description logic SROIQbs, a semantic
fragment of FO closely related to the OWL 2 DL ontology language (Section 4). Finally,
we revisit our example to discuss and illustrate properties of our proposal (Section 5).
An extended version of this paper including proofs is available as a technical report [30].
How to Agree to Disagree 129
2 First-Order Standpoint Logic
In this section we introduce the general framework of first order standpoint logic
(FOSL) as well as its sentential fragment and establish model theoretic and computa-
tional properties. In addition to establishing various worthwhile decidability and com-
plexity results, this approach also provides us with a clearer view on the underlying
principles of our arguments, while avoiding distractions brought about by some pecu-
liarities of expressive ontology languages, which we will address separately later on.
2.1 FOSL Syntax and Semantics
Definition 1. The syntax of first-order standpoint logic (SFO) is based on a signature
P,C,S, consisting of predicate symbols P(each associated with an arity n∈N),
constant symbols Cand standpoint symbols S, usually denoted with s,s,aswellasa
set Vof variables, typically denoted with x,y,...(possibly annotated). These four sets
are assumed to be countably infinite and pairwise disjoint. The set Tof terms contains
all constants and variables, that is, T=C∪V.
The set ESof standpoint expressions is defined as follows:
e1,e2::= ∗|s|e1∪e2|e1∩e2|e1\e2
The set SFO of FOSL formulas is then given by
φ, ψ ::= P(t1,...,t
k)|¬φ|φ∧ψ|∀xφ |eφ,
where P∈Pis an k-ary predicate symbol, t1,...,t
k∈T are terms, x∈V, and e∈E
S.
For a formula φ, we denote the set of all of its subformulas by Sub(φ).Thesize of a
formula is |φ|:= |Sub(φ)|. The connectives and operators t,f,φ∨ψ,φ→ψ,∃xφ,
and ♦eφare introduced as syntactic macros as usual. As further useful syntactic sugar,
we introduce sharpening statements e1e2to denote e1\e2f,theindeterminacy
operator via Ieφ:= ♦eφ∧♦e¬φ, and the determinacy operator via Deφ:= ¬Ieφ.
Definition 2. Given a signature P,C,S,afirst-order standpoint structure Mis a
tuple Δ, Π, σ, γwhere:
–Δis a non-empty set, the domain of M;
–Πis the non-empty set of precisifications;
–σis a function mapping each standpoint symbol from Sto a set of precisifications
(i.e., a subset of Π);
–γis a function mapping each precisification from Πto an ordinary first-order struc-
ture Iover the domain Δ, whose interpretation function ·Imaps:
•each predicate symbol P∈P of arity kto an k-ary relation PI⊆Δk,
•each constant symbol a∈C to a domain element aI∈Δ.
Moreover, for any two π1,π
2∈Πand every a∈Cwe require aγ(π1)=aγ(π2).
Note that all first-order structures in all precisifications implicitly share the same inter-
pretation domain Δgiven by the overarching first-order standpoint structure M, that is,
we adopt the constant domain assumption.3Moreover, the last condition of Definition 2
3This is not a substantial restriction, as other variants – expanding domains, varying domains
– can be emulated using constant domains [29, Theorem 6].
130 L. G´
omez ´
Alvarez et al.
also enforces rigid constants, that is, constants denote the same objects in different
standpoints (while clearly their properties could differ).
Definition 3. Let M=Δ, Π, σ, γbe a first-order standpoint structure for the signa-
ture P,C,S and Vbe a set of variables. A variable assignment is a function v:V→Δ
mapping variables to domain elements. Given a variable assignment v, we denote by
v{x→δ}the function mapping xto δ∈Δand any other variable yto v(y).
An interpretation function ·Iand a variable assignment specify how to interpret
terms by domain elements: We let tI,v =v(x)if t=x∈V, and tI,v =aIif t=a∈C.
To interpret standpoint expressions, we lift σfrom Sto all of ESby letting σ(∗)=Π
and σ(e1 e2)=σ(e1) σ (e2)for ∈{∪,∩,\}
The satisfaction relation for formulas is defined in the usual way via structural
induction. In what follows, let π∈Πand let v:V→Δbe a variable assignment;
we now establish the definition of the satisfaction relation |=for first-order standpoint
logic using pointed first-order standpoint structures:
M,π,v |=P(t1,...,t
k)iff (tγ(π),v
1,...,t
γ(π),v
k)∈Pγ(π)
M,π,v |=¬φiff M,π,v |=φ
M,π,v |=φ∧ψiff M,π,v |=φand M,π,v |=ψ
M,π,v |=∀xφ iff M,π,v{x→δ}|=φfor all δ∈Δ
M,π,v |=eφiff M,π
,v |=φfor all π∈σ(e)
M,π |=φiff M,π,v |=φfor all v:V→Δ
M|=φiff M,π |=φfor all π∈Π
As usual, Mis a model for a formula φiff M|=φ. As an aside, note that the modal-logic
nature of FOSL may become more evident upon realizing that an alternative definition of
its semantics via Kripke structures can be given (with einterpreted in the standard way)
by assigning every e∈E
Sthe accessibility relation {(π, π)|π, π∈Π, π∈σ(e)}.
Later in this paper, we will consider cases where the number of precisifications is
fixed. Thus, we conclude this section by a corresponding definition.
Definition 4. For a natural number n∈N, a FOSL formula φis n-satisfiable iff it has
a model Δ, Π, σ, γwith |Π|=n.
2.2 Small Model Property of Sentential Formulas
One interesting aspect of standpoint logic is that its simplified Kripke semantics brings
about convenient model-theoretic properties that do not hold for arbitrary (multi-)modal
logics. For propositional standpoint logic, it is known that standard reasoning tasks (such
as checking satisfiability) are NP-complete [8], in contrast to PSPA CE-completeness in
related systems such as K45n, KD45nand S5n. This result is in fact linked to a small
model property, according to which every satisfiable formula has a model with a “small”
number of precisifications. This beneficial property only holds in the single-modal K45,
KD45 and S5 [23] but applies to the multi-modal propositional standpoint logic because
of its stronger modal interaction. Fortunately, it can also be shown to carry over to some
fragments of FOSL and to the use of standpoint expressions. In particular, in this section,
How to Agree to Disagree 131
we will show that if we restrict the language to those formulas with no free variables in
subformulas of the form eφ, then we can indeed guarantee that every satisfiable FOSL
formula has a model whose number of precisifications is linear in the size of the formula.
Definition 5. Let φbe a formula of FOSL. We say that φis sentential iff for all subfor-
mulas of φof the form eψ, all variables occurring in ψare bound by a quantifier.
Theorem 1. A sentential FOSL formula φis satisfiable iff it has a model with at most |φ|
precisifications. That is, for sentential FOSL, satisfiability and |φ|-satisfiability coincide.
In the following, it will be convenient to assume that formulas are in standpoint
standard normal form (SSNF), where no modal operator eoccurs inside the scope of
another. Any FOSL formula φcan be transformed into SSNF in polynomial time.
2.3 Translation to Plain First-Order Logic
In this section, we present a translation Transnmapping any FOSL formula φto a
plain FO formula Transn(φ)such that n-satisfiability of φcoincides with satisfiabil-
ity of Transn(φ). The translation will make explicit use of a fixed, finite set Πnof
precisifications with |Πn|=n.
Our translation will map any φinto a formula of (standpoint-free) first-order logic. The
basic idea is to “emulate” standpoint structures Δ, Πn,σ,γin plain first-order struc-
tures over Δby means of a “superposition” of all γ(π), which requires to introduce n
“copies” of the original set of predicates. To this end, we define our first-order vocabulary
by V
FO(P,C,S,Π
n)=P,C where Pcontains
– for each predicate P∈Pand precisification π∈Πn, a predicate of the form Pπof
the same arity as P, intuitively expressing that Pπshould capture Pγ(π);
– for each standpoint constant s∈Sand every precisification π∈Πn, a nullary
predicate of the form sπ, intuitively expressing that π∈σ(s).
The top-level translation is then defined to set:
Transn(φ)=π∈Πntransn(π, φ)∧π∈Πn∗π,
where transnis inductively defined by
transn(π, P(t1,...,t
k)) = Pπ(t1,...,t
k)
transn(π, ¬ψ)=¬transn(π, ψ)
transn(π, ψ1∧ψ2)=trans
n(π, ψ1)∧transn(π, ψ2)
transn(π, ∀xψ)=∀x(transn(π, ψ))
transn(π
,eψ)=π∈Πn(transE(π, e)→transn(π, ψ))
Therein, transEimplements the semantics of standpoint expressions, providing for each
expression e∈E
Sa propositional formula transE(π, e)over {sπ|π∈Πn}as follows:
transE(π, s)=sπ
transE(π, e1∪e2)=trans
E(π, e1)∨transE(π, e2)
transE(π, e1∩e2)=trans
E(π, e1)∧transE(π, e2)
transE(π, e1\e2)=trans
E(π, e1)∧¬transE(π, e2)
132 L. G´
omez ´
Alvarez et al.
A routine inspection of the translation ensures that it can be done in polynomial
time and its output is of polynomial size, provided it is applied to formulas in SSNF.
Theorem 2. Aformulaφis n-satisfiable in FOSL if and only if the formula Tra n s n(φ)
is satisfiable in first-order logic.
In fact, Theorem 2provides us with a recipe for a satisfiability-preserving trans-
lation for any formula that comes with a “small model guarantee”, whenever a bound
on the number of precisifications can be computed upfront. In particular, leveraging
Theorem 1, we obtain the following corollary.
Corollary 1. Aformulaφis satisfiable in sentential first-order standpoint logic if and
only if the formula Trans|φ|(φ)is satisfiable in first-order logic.
3 Expressive Decidable FOSL Fragments
We note that even for the sentential version, first-order standpoint logic is still a gener-
alization of plain first-order logic, whence reasoning in it is undecidable. Therefore, we
will next look into some popular decidable FO fragments and establish decidability and
complexity results for reasoning in their sentential standpoint versions.
Definition 6. Let Fdenote some FO fragment. Then the logic sentential Standpoint-F,
denoted S[F], contains the sentential FOSL formulas φwhere:
– all variables inside φare bound by some quantifier,
– for every subformula ψ∈Sub(SSNF(φ)) preceded by a quantifier, ψ∈F holds.
Fragment Fis standpoint-friendly iff every φ∈S[F]satisfies Tra n s |φ|(SSNF(φ)) ∈F.
Lemma 1. Let Fbe a standpoint-friendly fragment of FOL. Then the following hold:
1. Satisfiability for S[F]is decidable if and only if it is for F.
2. If the satisfiability problem in Fis at least NP-hard, then the satisfiability problem
in S[F]is of the same complexity as in F.
It turns out that many popular formalisms are standpoint-friendly. For propositional
logic (PL), this is straightforward: quantifiers and variables are absent altogether, which
is also the reason why sentential Standpoint-PL and proper Standpoint-PL coincide.
Thus, Lemma 1yields an alternative argument for the NP-completeness of the latter,
which was established previously [8].
On the expressive end of the logical spectrum, it is worthwhile to inspect fragments
of FO that are still decidable (as a minimal requirement for the feasibility of automated
reasoning). In fact, standpoint-friendliness can be established by structural induction
for many of those. Notable examples are:
–thecounting 2-variable fragment C2[10,24], which subsumes many description
logics and serves as a mathematical backbone for related complexity results,
–theguarded negation fragment GNFO [2,3], which encompasses both the popular
guarded fragment as well as the ubiquitous class of (unions of) conjunctive queries
also known as existential positive FO, and
How to Agree to Disagree 133
– the triguarded fragment TGF [14,27] (a more recent formalism subsuming both the
two-variable and the guarded fragment without equality).
Intuitively, standpoint-friendliness for all these (and presumably many more) fragments
follows from the fact that they are closed under Boolean combinations of sentences
and that the transformation does not affect the structure of quantified formulas. We
therefore immediately obtain that these four popular decidable fragments of FOL allow
for accommodating standpoints without any increase in complexity.
Corollary 2. The sentential FOSL fragments S[PL] (= SPL),S[C2],S[GNFO], and S[TGF]
are all decidable and the complexity of their satisfiability problem is complete for NP,
NEXPTIME,2EXPTIME, and N2EXPTIME, respectively.
As an aside, we note that all these results remain valid when considering finite satis-
fiability (i.e., restricting to models with finite Δ), because for all considered fragments,
companion results for the finite-model case exist and the equisatisfiability argument for
our translation preserves (finiteness of) Δ.
4 Sentential Standpoint-SROIQbs
We next present the highly expressive yet decidable logic (Sentential) Standpoint-
SROIQbs, which adds the feature of standpoint-aware modelling to SROIQbs,a
description logic (DL) obtained from the well-known DL SROIQ[13] by a gentle exten-
sion of its expressivity, allowing safe Boolean role expressions over simple roles [26].4
The SROIQ family serves as the logical foundation of popular ontology languages
like OWL 2 DL. In view of the fact that SROIQbsis a semantic fragment of FO, we
can leverage the previously established results and present a satisfiability-preserving
polynomial translation from Standpoint-SROIQbsinto plain SROIQbsknowledge
bases. On the theoretical side, this will directly provide us with favourable and tight
complexity results for reasoning in Standpoint-SROIQbs. On the practical side, this
paves the way towards practical reasoning in “Standpoint-OWL”, since it allows us to
use highly optimised OWL 2 DL reasoners off the shelf.
4.1 SROIQbs: Syntax and Semantics
Let C,P1, and P2be finite, mutually disjoint sets called individual names,concept
names and role names, respectively. P2is subdivided into simple role names Ps
2and
non-simple role names Pns
2, the latter containing the universal role uand being strictly
ordered by some strict order ≺. In the original definition of SROIQbs, simplicity of
roles and ≺are not given a priori, but meant to be implicitly determined by the set of
axioms. Our choice to fix them explicitly upfront simplifies the presentation without
restricting expressivity. Then, the set Rsof simple role expressions is defined by
r1,r
2::= s|s−|r1∪r2|r1∩r2|r1\r2,
with s∈Ps
2, while the set of (arbitrary) role expressions is R=Rs∪Pns
2. The order ≺
4Focusing on the mildly stronger SROIQbsinstead of the more mainstream SROIQallows for
a more coherent and economic presentation, without giving up the good computational properties
and the availability of optimised algorithms and tools.
134 L. G´
omez ´
Alvarez et al.
Table 1. SROIQbsrole, concept expressions and axioms. C≡Dabbreviates CD,DC.
Name Syntax Semantics
inverse role s−{(δ, δ)∈Δ×Δ|(δ,δ)∈sI}
role union r1∪r2rI
1∪rI
2
role intersection r1∩r2rI
1∪rI
2
role difference r1\r2rI
1\rI
2
universal role uΔI×ΔI
nominal {a}{aI}
top ΔI
bottom ⊥∅
negation ¬CΔ
I\CI
conjunction CDC
I∩DI
disjunction CDC
I∪DI
univ. restriction ∀r.C {δ|∀y.(δ, δ)∈rI→δ∈CI}
exist. restriction ∃r.C {δ|∃y.(δ, δ)∈rI∧δ∈CI}
Self concept ∃r.Self {δ|(δ, δ)∈rI}
qualified number nr.C {δ|#{δ∈CI|(δ,δ )∈rI}≤n}
restrictions nr.C {δ|#{δ∈CI|(δ, δ)∈rI}≥n}
Name Syntax Semantics
concept assertion C(a)aI∈CI
role assertion r(a, b)(aI,b
I)∈rI
equality a.
=ba
I=bI
inequality a.
=ba
I=bI
general concept CDC
I⊆DI
inclusion (GCI)
role inclusion r1◦...◦rnrrI
1◦...◦rI
n⊆rI
axioms r1◦...◦rn◦rrrI
1◦...◦rI
n◦rI⊆rI
(RIAs) r◦r1◦...◦rnrr
I◦rI
1◦...◦rI
n⊆rI
r◦rrr
I◦rI⊆rI
In RIAs, r∈P
ns
2, while ri∈Rand ri≺rfor all
i∈{1,...,n}.
is then extended to Rby making all elements of Rs≺-minimal. The syntax of concept
expressions is given by
C, D ::= A|{a}||⊥|¬C|CD|CD|∀r.C |∃r.C |∃r
.Self |nr
.C |nr
.C,
with A∈P
1,a∈C,r∈R,r∈R
s, and n∈N. We note that any concept expression
can be put in negation normal form, where negation only occurs in front of concept
names, nominals, or Self concepts. The different types of SROIQbssentences (called
axioms) are given in Table 1.5
Similar to FOL, the semantics of SROIQbsis defined via interpretations I=(Δ, ·I)
composed of a non-empty set Δcalled the domain of Iand a function ·Imapping indi-
vidual names to elements of Δ, concept names to subsets of Δ, and role names to subsets
of Δ×Δ. This mapping is extended to role and concept expressions and finally used to
define satisfaction of axioms (see Table 1).
4.2 Standpoint-SROIQbs
The set S[SROIQbs
]of sentential Standpoint-SROIQbssentences is now defined induc-
tively as follows:
–ifAx is a SROIQbsaxiom then Ax ∈S[SROIQ bs
],
–ifφ, ψ ∈S[SROIQbs
]then ¬φ,aswellasφ∧ψand φ∨ψare in S[SROIQ bs
],
–ifφ∈S[SROIQbs
]and e∈E
Sthen eφ∈S[SROIQbs
]and ♦eφ∈S[SROIQbs
].
The semantics of sentential Standpoint-SROIQbsis defined in the obvious way,
by “plugging” the semantics of SROIQbsaxioms into the semantics of S[FO].Wesay
aS[SROIQbs
]sentence φis in negation normal form (NNF), if negation occurs only inside
or directly in front of SROIQbsaxioms; obviously every Standpoint-SROIQbssen-
tence can be efficiently transformed into an equivalent one in NNF.
5The original definition of SROIQ contained more axioms (role transitivity, (a)symmetry,
(ir)reflexivity and disjointness), but these are syntactic sugar in our setting.
How to Agree to Disagree 135
4.3 Coping with Peculiarities of SROIQbs
In the following, we will provide a polynomial translation, mapping any S[SROIQ bs
]
sentence φto an equisatisfiable set of SROIQbsaxioms. This translation is very much
in the spirit of the one presented for sentential FOSL, however, SROIQbscomes with
diverse syntactic impediments that we need to circumvent. Thus, before presenting the
translation, we will briefly discuss these issues and how to solve them.
First, SROIQbsdoes not provide nullary predicates (i.e., propositional symbols).
As a surrogate, we use concept expressions of the form ∀u.Awhich have the pleasant
property of holding either for all domain individuals or for none. Second, SROIQbs
does not directly allow for arbitrary Boolean combinations of axioms. For all non-RIA
axioms, a more or less straightforward equivalent encoding is possible using nominals
and the universal role; for instance the expression ¬[r(a, b)] ∨[AB]can be converted
into ¬∃u.({a}∃r.{b})∀u.(¬AB).
Dealing with RIAs requires auxiliary vocabulary; for negated RIAs, we introduce
a fresh nominal, say {x}, to mark the end of a “violating” role chain, so ¬[s◦sr]
essentially becomes ∃u.(∃s.∃s.{x})(¬∃r.{x}).
Unnegated RIAs are even trickier. There is no way of converting them into GCIs, so
we have to keep them, but we attach an additional “guard”, which allows us to disable
them whenever necessary. This guard can then be triggered from within a GCI. For an
example, consider the expression [t◦tr]∨[t◦tr]. Then, introducing fresh
“guard roles” s1and s2, we assert the three axioms (∀u.∃s1.Self )(∀u.∃s2.Self )
as well as s1◦t◦trand s2◦t◦tr. With this arrangement, the first axiom
will ensure that all domain elements carry an s1-loop or all domain elements carry
an s2-loop. Depending on that choice, the corresponding RIA in the second line will
behave like its original, unguarded version, while the other one may be entirely disabled.
The introduced strategy for handling positive RIAs has a downside: due to the
restricted shapes of RIAs (governed by ≺), axioms of the shape r◦rr(expressing
transitivity) cannot be endowed with guards. In order to overcome this nuisance, every
nonsimple role rhas to be accompanied by a subrole r, which acts as a “lower approx-
imation” of rand – whenever ris defined transitive – “feeds into” rvia tail recursion.
This way of reformulating r◦rrallows to attach the wanted guard, but requires
adjustments in some axioms that mention r.
4.4 Translation into Plain SROIQbs
We now assume a given S[SROIQbs
]sentence φ, w.l.o.g. in NNF, and provide the formal
definition of the translation. As before, we fix Π|φ|and let our translation’s vocabulary
V[SROIQbs
](φ)consist of all individual names inside φ, plus, for each π∈Π|φ|,the
following symbols: (a) a concept name Aπfor each A∈P
1; (b) a simple role name
sπfor each s∈Ps
2; (c) non-simple role names rπand rπfor each r∈Pns
2\{u};(d)a
simple role name sπ
ρfor each unnegated RIA ρinside φ; (e) a fresh constant name aπ
ρ
for each negated RIA ρinside φ; (f) a concept name Ms
πfor each s∈S. Thereby, the
non-simple role names inherit their ordering ≺from Pns
2and we also let rπ≺rπfor
each r∈Pns
2\{u}.
136 L. G´
omez ´
Alvarez et al.
The translation Trans(φ)of φis then a set of SROIQ axioms defined as follows:
First, Trans(φ)contains the RIA rπrπfor every r∈Pns
2\{u}and each π∈Π|φ|.
Second, for every unnegated RIA ρinside φand each π∈Π|φ|,Trans(φ)contains the
RIA BGπ(ρ), with BGπdefined by
r1◦...◦rnr→ sπ
ρ◦rπ
1◦...◦rπ
nrπr1◦...◦rn◦rr→ sπ
ρ◦rπ
1◦...◦rπ
n◦rπrπ
r◦r1◦...◦rnr→ rπ◦rπ
1◦...◦rπ
n◦sπ
ρrπr◦rr→ sπ
ρ◦rπ◦rπrπ,
whereby the role expression rπis obtained from rby substituting every role name s
with sπ(except u, which remains unaltered). Third and last, Tra n s ( φ)contains the
GCI
π∈Π|φ|trans(π, φ)π∈Π|φ|∀u.M∗
π
where, by inductive definition,
trans(π, Ax) = trans+(π, Ax)
trans(π, ¬Ax) = trans−(π, Ax)
trans(π, ψ 1∧ψ2) = trans(π, ψ 1)trans(π, ψ2)
trans(π, ψ 1∨ψ2) = trans(π, ψ 1)trans(π, ψ2)
trans(π
,eψ)=π∈Π|φ|(¬transE(π, e)trans(π, ψ ))
trans(π
,♦eψ)=π∈Π|φ|(transE(π, e)trans(π, ψ ))
We next present the translation of unnegated and negated SROIQ axioms (ρstands
for an RIA r1◦...◦rmr):
trans+(π, ρ)=∀u.∃sπ
ρ.Self trans−(π, ρ)=∃u.(∀rπ
.¬{aπ
ρ})(∃rπ
1...∃rπ
m.{aπ
ρ})
trans+(π, C D)=∀u.(¬CD)πtrans−(π, C D)=∃u.(C¬D)π
trans+(π, C (a)) = ∃u.{a}Cπtrans−(π, C (a)) = ∃u.{a}(¬C)π
trans+(π, r (a, b)) = ∃u.{a}∃rπ.{b}trans−(π, r (a, b)) = ∃u.{a}∀rπ.¬{b}
trans+(π, a .
=b)=∃u.{a}{b}trans−(π, a .
=b)=∃u.{a}¬{b}
Therein, for any role expression r,weletrdenote rif r=ris a non-simple role name,
and otherwise r=r. Moreover, Cπdenotes the concept expression that is obtained
from Cby transforming it into negation normal form, replacing concept names Awith
Aπand role expressions rby rπ, and replacing every ∃rfor non-simple rwith ∃r.
As before, transEimplements the semantics of standpoint expressions, but now
adjusted to the new framework: each expression e∈E
Sis transformed into a concept
expression transE(π, e)over the vocabulary Ms
π|s∈S,π∈Π|φ|as follows:
transE(π, s)=∀u.Ms
π
transE(π, e1∪e2) = transE(π, e1)transE(π, e2)
transE(π, e1∩e2) = transE(π, e1)transE(π, e2)
transE(π, e1\e2) = transE(π, e1)¬transE(π, e2)
With all definitions in place, we obtain the desired result.
Theorem 3. Given φ∈S[SROIQ bs
],thesetTrans(φ)(i) is a valid SROIQbsknowledge
base, (ii) is equisatisfiable with φ,(iii) is of polynomial size wrt. φ, and (iv) can be
computed in polynomial time.
How to Agree to Disagree 137
5 Example in the Forestry Domain
We consider an extension of Example 1in Sentential Standpoint-SROIQbsto illustrate
the main reasoning tasks in more detail. The following additional axiom specifies that
forest land use and urban land use are disjoint subclasses of land (F9).
(F9) ∗[ForestlandUseUrbanLandUse Land ∧ForestlandUseUrbanLandUse ⊥]
Now, let us see how, through inferences in S[SROIQ bs
], we can gather unequivocal
knowledge (Uneq), obtain knowledge that is relative to a standpoint (Rel), and contrast
the knowledge that can be inferred from different standpoints (Cont). For unequivocal
knowledge (Uneq), we can infer unambiguously that forests are no urban-use lands:
∗[Forest ¬UrbanLandUse]
This holds because each precisification must comply with LC or LU (F8), and we have
LC[Forest ¬UrbanLandUse]from (F1)and(F5), and LU[Forest ¬UrbanLandUse]
from (F3)and(F9). Regarding relative (Rel) and contrasting (Cont) knowledge, if we
now wanted to query our knowledge base for instances of forest, we would obtain
LC[Forest(e)] ∧LC [¬Forest(l)] I∗[Forest(e)] ∧I
∗[Forest(l)]
The first deduced formula contains knowledge relative to LC, showing its stance on
whether the instances constitute a forest, which happens to be conclusive in both cases.
The second formula states the global indeterminacy of both l’s and e’s membership to
the concept Forest. This stems from the disagreement between the interpretations LC
and LU, whose overall incompatibility (LC∩LU[⊥]) can also be inferred.
Finally, it is worth looking at the limitations of the sentential fragment of Standpoint
SROIQbs. In a non-sentential setting, where modalities can be used at the concept
level, “complex alignments” or bridges can be established between concepts according
to possibly many standpoints. For instance, one can write
LU[Forest]LC [∃hasLand−.Forest]∗[Cleared]
to express that the areas classified as forest according to LU belong to a forest according
to LC or have been cleared (in which case LC does not recognise them as forest). It is an
objective of future work of ours to study decidable fragments for which the restrictions
on the use of modalities are relaxed to express such kinds of axioms.
6 Related Work
A variety of formal representation systems have been proposed to model perspectives in
rather diverse areas of research and with heterogeneous nomenclatures. Standpoint logic
bears some similarities to context logic in the style proposed by McCarthy and Buvac
[19], which has also been applied in a description logic setting [15]. This tradition treats
contexts as “first-class citizens” of the logic, i.e., full-fledged formal objects over which
one can express first-order properties. In contrast, standpoint logic is suitable when a
formalisation of the contexts involved is unfeasible, or when the interest resides in the
content of the standpoints rather than the context in which they occur.
Another related notion is that of ontology views, where some works consider poten-
tially conflicting viewpoints [11,12,25]. Ribi´
ere and Dieng [25] and Heman et al. [11,12]
138 L. G´
omez ´
Alvarez et al.
implement the intuition of “viewpoints” via ad-hoc extensions of the syntax and se-
mantics of description logics, in a style similar to the work on contextuality by Bensli-
mane et al. [4]. Gorshkov et al. [9] implement them using named graphs. Instead, the
standpoint approach extends the base language with modalities and provides a Kripke-
style semantics for it. This leads to a simpler, more recognisable and more expressive
framework that supports, for instance, hierarchies and combinations of standpoints, infer-
ences of partial truths, the preservation of consistency with the established alignments
and inferences about the standpoints themselves. On a technical level, first-order stand-
point logic can be seen as a many-dimensional (multi-)modal logic [16], whence results
from that area apply to our setting. In particular, the search for non-trivial fragments
of first-order modal logics that are still decidable and even practically relevant is an
important endeavour, for which we believe that standpoint logic can play useful role.
Finally, in the area of ontology modularity, different formalisms such as DDL bridge
rules [5] and ε-connections [17,18] have been proposed to specify the interaction between
independent knowledge sources. These can be related to the present framework in that
they provide mechanisms to establish links between conceptual models that do not need
to be entirely coherent with each other. Yet the motivation is inherently different: while
the standpoint framework focuses on integrating possibly overlapping knowledge into a
global source (while preserving “standpoint-provenance” and thus enabling a peaceful
coexistence of conflicting information), DDL bridge rules and ε-connections have been
devised to establish a certain synchronisation between modules that are and will remain
separate. DDL bridge rules could, however, be simulated within a standpoint framework.
7 Conclusions and Future Work
The diversity of human world views along with the semantic heterogeneity of natural
language are at the heart of well-recognised knowledge interoperability challenges. As
an alternative to the common strategy of merging, we proposed the use of a logical
formalism based on the notion of standpoint that is suitable for knowledge representa-
tion and reasoning with sets of possibly conflicting characterisations of a domain.
Using first-order logic as an expressive underlying language, we proposed a multi-
modal framework by means of which different agents can establish their individual
standpoints (which typically involves specifying constraints and relations), but which
also allows for combining standpoints and establishing alignments between them. Rea-
soning tasks over such multi-standpoint specifications include gathering unequivocal
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.
Remarkably, the simplified Kripke semantics allows us to establish a small model
property for the sentential fragment of FOSL. This result gave rise to a polynomial,
satisfiability-preserving translation into the base logic, which also maintains member-
ship in diverse decidable fragments, immediately implying that for a range of logics,
reasoning in their standpoint-enhanced versions does not increase their computational
complexity. This indicates that the framework can be applied to ontology alignment,
concept negotiation, and knowledge aggregation with inference systems built on top of
existing, highly optimised off-the-shelf reasoners.
How to Agree to Disagree 139
Future work includes the study of the complexity of FOSL fragments allowing the
presence of free variables within the scope of modalities. Note however that in the
general case of FOSL, this leads to the loss of the small model property.
Example 2. Consider the following (non-sentential) FOSL sentence, axiomatising
”better” (Btt) to be interpreted as a non-well-founded strict linear order and requiring
for every domain element x(of infinitely many) the existence of some precisification
where xis the (one and only) “best”:
∀xyz(Btt(x, y )∧Btt(y,z )) →Btt(x, z)∧∀xy¬(Btt(x, y)∧Btt(y, x)) ∧
∀xyx=y→(Btt(x, y)∨Btt(y, x))∧∀x∃yBtt(x, y)∧∀x♦∗¬∃yBtt(y, x)
Obviously, this sentence is satisfiable, but only in a model with infinitely many precisi-
fications; that is, the small model property is violated in the worst possible way.
On the other hand, it is desirable from a modelling perspective to allow for some
interplay between FO quantifiers and standpoint modalities. E.g., the non-sentential
FOSL sentence ∀x1···xkP(x1,...,x
k)→∗P(x1,...,x
k)expresses the rigidity
of a predicate P, thereby “synchronising” it over all precisifications.
Consequently,we will study how by imposing syntactic restrictions, we can guarantee
the existence of small (or at least reasonably-sized) models for non-sentential standpoint
formulas. Results in the field of many-dimensional modal logics [16] show that reasoning
is decidable for diverse fragments of first-order modal logic such as the monodic fragment
(where modalities occur only in front of formulas with at most one free variable). However,
Example 2already shows (by virtue of being within the monodic fragment) that we cannot
hope for a small model property even for this slight extension of the sentential fragment.
A detailed analysis of these issues as they apply to the simplified semantics of standpoint
logic is the object of current work.
Additionally, we intend to implement the proposed translations and perform exper-
iments to test the performance of the standpoint framework in scenarios of Knowledge
Integration. While sentential standpoints can be added at no extra cost in complexity
for the discussed fragments in this paper, we intend to run experiments to assess the
runtime impact on large knowledge bases with off-the-shelf reasoners.
As another important topic toward the deployment of our framework, we will look
into conceptual modelling aspects. Reviewing documented recurrent scenarios and pat-
terns in the area of knowledge integration, we intend to establish guiding principles for
conveniently encoding those by using novel strategies possible with structures of stand-
points. Examples for such scenarios include the disambiguation of knowledge sources
by using combinations of standpoints, and the establishment of bridge-like rules for
alignment. For the latter we will investigate their relationship to similar constructs from
other frameworks such as ε-connections, distributed description logics, and others.
Supplemental Material Statement: Proofs can be found in the extended version [30].
Acknowledgments. Luc´
ıa G´
omez ´
Alvarez was supported by the Bundesministerium f¨
ur Bildung
und Forschung (BMBF) 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).
140 L. G´
omez ´
Alvarez et al.
References
1. Arp, R., Smith, B., Spear, A.D.: Building ontologies with basic formal ontology. MIT Press
(2015)
2. B´
ar´
any, V., Benedikt, M., ten Cate, B.: Some model theory of guarded negation. Journal of
Symbolic Logic 83(4), 1307–1344 (2018)
3. B´
ar´
any, V., ten Cate, B., Segoufin, L.: Guarded negation. Journal of the ACM 62(3) (2015)
4. Benslimane, D., Arara, A., Falquet, G., Maamar, Z., Thiran, P., Gargouri, F.: Contex-
tual ontologies. Motivations, challenges, and solutions. In: Procs. of the 4th Int. Conf. on
Advances in Information Systems. pp. 168–176. Springer (2006)
5. Borgida, A., Serafini, L.: Distributed Description Logics: Assimilating information from peer
sources. Journal on Data Semantics 2800, 153–184 (2003)
6. Euzenat, J., Mocan, A., Scharffe, F.: Ontology alignments. In: Hepp, M., Leenheer, P.D.,
Moor, A.D., Sure, Y. (eds.) Ontology Management. Computing for Human Experience, vol.
7, pp. 177–206. Springer (2008)
7. Fine, K.: Vagueness, truth and logic. Synthese 30(3–4), 265–300 (1975)
8. G´
omez ´
Alvarez, L., Rudolph, S.: Standpoint logic: Multi-perspective knowledge representa-
tion. In: Neuhaus, F., Brodaric, B. (eds.) Procs. of the 12th Int. Conf. on Formal Ontology in
Information Systems. FAIA, vol. 344, pp. 3–17. IOS Press (2021)
9. Gorshkov, S., Kralin, S., Miroshnichenko, M.: Multi-viewpoint ontologies for decision-
making support. In: Ngonga Ngomo, A.C., Kˇ
remen, P. (eds.) Knowledge Engineering and
Semantic Web. pp. 3–17. Springer (2016)
10. Gr¨
adel, E., Otto, M., Rosen, E.: Two-variable logic with counting is decidable. In: Procs.
of the 12th Annual IEEE Symposium on Logic in Computer Science. pp. 306–317. IEEE
(1997)
11. Hemam, M.: An extension of the ontology web language with multi-viewpoints and proba-
bilistic reasoning. Int. Journal of Advanced Intelligence Paradigms 10(3), 247–265 (2018)
12. Hemam, M., Boufa¨
ıda, Z.: MVP-OWL: A multi-viewpoints ontology language for the
Semantic Web. Int. Journal of Reasoning-based Intelligent Systems 3(3–4), 147–155 (2011)
13. Horrocks, I., Kutz, O., Sattler, U.: The even more irresistible SROIQ. In: Doherty, P.,
Mylopoulos, J., Welty, C.A. (eds.) Procs. 10th Int. Conf. on Principles of Knowledge Repre-
sentation and Reasoning. pp. 57–67. AAAI Press (2006)
14. Kieronski, E., Rudolph, S.: Finite model theory of the triguarded fragment and related logics.
In: Procs. of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science. pp.
1–13. IEEE (2021)
15. Klarman, S., Guti´
errez-Basulto, V.: Description logics of context. Journal of Logic and Com-
putation 26(3), 817–854 (2013)
16. Kurucz, A., Wolter, F., Zakharyaschev, M., Gabbay, D.M.: Many-Dimensional Modal Log-
ics: Theory and Applications, 1st edn. Studies in Logics and the Foundations of Mathematics,
Elsevier (2003)
17. Kutz, O., Lutz, C., Wolter, F., Zakharyaschev, M.: E-connections of description logics.
Description Logics Workshop 81 (2003)
18. Kutz, O., Lutz, C., Wolter, F., Zakharyaschev, M.: E-connections of abstract description sys-
tems. Artificial intelligence 156(1), 1–73 (2004)
19. McCarthy, J., Buvac, S.: Formalizing context (expanded notes). CSLI Lecture Notes 81, 13–
50 (1998)
20. Osman, I., Ben Yahia, S., Diallo, G.: Ontology Integration: Approaches and Challenging
Issues. Information Fusion 71, 38–63 (2021)
21. Otero-Cerdeira, L., Rodr´
ıguez-Mart´
ınez, F.J., G´
omez-Rodr´
ıguez, A.: Ontology matching: A
literature review. Expert Systems with Applications 42(2) (2015)
How to Agree to Disagree 141
22. Pesquita, C., Faria, D., Santos, E., Couto, F.M.: To repair or not to repair: Reconciling cor-
rectness and coherence in ontology reference alignments. In: Procs. of the 8th Int. Conf. on
Ontology Matching. vol. 1111, pp. 13–24. CEUR (2013)
23. Pietruszczak, A.: Simplified Kripke style semantics for modal logics K45, KB4 and KD45.
Bulletin of the Section of Logic 38(3/4), 163–171 (2009)
24. Pratt-Hartmann, I.: Complexity of the two-variable fragment with counting quantifiers. Jour-
nal of Logic, Language, and Information 14(3), 369–395 (2005)
25. Ribi`
ere, M., Dieng-Kuntz, R.: A viewpoint model for cooperative building of an Ontology.
In: Procs. of the 10th Int. Conf. on Conceptual Structures. pp. 220–234. Springer (2002)
26. Rudolph, S., Kr¨
otzsch, M., Hitzler, P.: Cheap Boolean role constructors for description log-
ics. In: H¨
olldobler, S., Lutz, C., Wansing, H. (eds.) Procs. of the 11th European Conf. on
Logics in Artificial Intelligence. vol. 5293, pp. 362–374. Springer (2008)
27. Rudolph, S., Simkus, M.: The triguarded fragment of first-order logic. In: Barthe, G., Sut-
cliffe, G., Veanes, M. (eds.) Procs. of the 22nd Int. Conf. on Logic for Programming, Artifi-
cial Intelligence and Reasoning. EPiC Series in Computing, vol. 57, pp. 604–619 (2018)
28. Solimando, A., Jim´
enez-Ruiz, E., Guerrini, G.: Minimizing conservativity violations in
ontology alignments. Knowledge and Information Systems 51(3) (2017)
29. Wolter, F., Zakharyaschev, M.: On the decidability of description logics with modal opera-
tors. In: Cohn, A.G., Schubert, L.K., Shapiro, S.C. (eds.) Procs. of the Sixth Int. Conf. on
Principles of Knowledge Representation and Reasoning. pp. 512–523. Morgan Kaufmann
(1998)
30. ´
Alvarez, L.G., Rudolph, S., Strass, H.: How to Agree to Disagree: Managing Ontological
Perspectives using Standpoint Logic. Tech. rep., https://arXiv.org/abs/2206.06793 (2022)
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