=Paper=
{{Paper
|id=Vol-3915/paper-1
|storemode=property
|title=On the Complexity of Querying Inconsistent Weighted Knowledge Bases (Short paper)
|pdfUrl=https://ceur-ws.org/Vol-3915/Paper-1.pdf
|volume=Vol-3915
|authors=Thomas Lukasiewicz,Enrico Malizia,Cristian Molinaro
|dblpUrl=https://dblp.org/rec/conf/aiia/LukasiewiczMM24
}}
==On the Complexity of Querying Inconsistent Weighted Knowledge Bases (Short paper)==
https://ceur-ws.org/Vol-3915/Paper-1.pdf
On the Complexity of Querying Inconsistent Weighted
Knowledge Bases
Thomas Lukasiewicz1,2 , Enrico Malizia3 and Cristian Molinaro4,*
1
Institute of Logic and Computation, Vienna University of Technology, Austria
2
Department of Computer Science, University of Oxford, UK
3
DISI, University of Bologna, Italy
4
DIMES, University of Calabria, Italy
Abstract
Inconsistency-tolerant semantics are approaches to provide meaningful answers to queries even in the presence
of inconsistent knowledge. Several such semantics rely on the notion of a repair, which is a “maximal” consistent
subset of the database, where different maximality criteria might be adopted depending on the application at
hand. Common maximality criteria assume that all facts in a database are equally important. However, in several
real-world applications, it is often the case that different facts have different importance. In this paper, we consider
Datalog± knowledge bases where database facts are weighted, with weights expressing facts’ importance (or
reliability or some other aspect of interest). We present recent results on the complexity of querying inconsistent
knowledge bases in this setting.
Keywords
Inconsistency-tolerant semantics, computational complexity
1. Introduction
In real-world applications, data possibly coming from different sources may exhibit inconsistencies.
Obtaining meaningful query answers in these scenarios can be achieved by resorting to inconsistency-
tolerant semantics. Popular ones are the ABox repair (AR), first defined for relational databases [1]
and then generalized for description logics (DLs) [2], the intersection of repairs (IAR) [2], and the
intersection of closed repairs (ICR) [3]. All such semantics, as well as others (see, e.g., [2]), are based on
the notion of a repair, which is a “maximal” consistent subset of the knowledge base’s facts. Subset
maximality was adopted upon introduction of the above semantics. However, other maximality criteria
are relevant in practice and have been introduced over the years. For instance, maximum cardinality
is a stronger criterion ruling out subset-maximal repairs not containing the highest number of facts,
which is suitable for settings where all database facts are considered equally reliable. For Datalog±
languages, subset-maximal repairs have been considered in [4, 5] while cardinality-maximal ones in [6];
in the context of querying inconsistent DL knowledge bases, the aforementioned maximality criteria,
as well as others, have been investigated in [7]. Inconsistency-tolerant semantics have been defined
also w.r.t. “preferred” repairs that are selected among the subset-maximal ones on the basis of user
preferences [8, 9, 10, 11, 12].
In this paper, we consider the case where database facts are associated with weights (e.g., quantitatively
measuring their reliability), a scenario arising in many applications. For example, consider a neuro-
symbolic system in which the neuronal part of the system produces some predictions as database facts
associated with a confidence score (see, e.g., [13] and references therein). Then, in case of inconsistencies,
these values can be used in the computation of most reliable repairs. In such a setting, a natural criterion
to define repairs is to select the weight-maximal consistent subsets of the database. In this paper, we
AIxIA 2024 Discussion Papers - 23rd International Conference of the Italian Association for Artificial Intelligence, Bolzano, Italy,
November 25–28, 2024
*
Corresponding author.
$ thomas.lukasiewicz@tuwien.ac.at (T. Lukasiewicz); enrico.malizia@unibo.it (E. Malizia); c.molinaro@dimes.unical.it
(C. Molinaro)
0000-0002-6780-4711 (E. Malizia); 0000-0003-4103-1084 (C. Molinaro)
© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
discuss recent results presented in [14] on the complexity of the AR, IAR, and ICR semantics when
such a notion of repair is adopted in the presence of weighted knowledge bases expressed via Datalog±
languages.
2. Preliminaries
General. We assume a set C of constants, a set N of labeled nulls, and a set V of variables. A term 𝑡 is a
constant, a null, or a variable. We also assume a set of predicates, each associated with an arity, i.e., a
non-negative integer. An atom has the form 𝑝(𝑡1 , . . . , 𝑡𝑛 ), where 𝑝 is an 𝑛-ary predicate, and 𝑡1 , . . . , 𝑡𝑛
are terms. An atom containing only constants is also called a fact. Conjunctions of atoms are often
identified with the sets of their atoms. An instance 𝐼 is a (possibly infinite) set of atoms containing only
constants and nulls. A database 𝐷 is a finite instance that contains only constants. A homomorphism is
a substitution ℎ : C ∪ N ∪ V → C ∪ N ∪ V that is the identity on C and maps N to C ∪ N. A Boolean
conjunctive query (BCQ) 𝑞 has the form ∃Y𝜑(Y), where 𝜑(Y) is a conjunction of atoms without nulls.
A BCQ 𝑞 is true over an instance 𝐼, denoted 𝐼 |= 𝑞, if there is a homomorphism ℎ with ℎ(𝜑(Y)) ⊆ 𝐼.
Dependencies. A tuple-generating dependency (TGD) 𝜎 is a first-order formula
∀X∀Y (𝜙(X, Y) → ∃Z 𝑝(X, Z)), where X, Y, and Z are pairwise disjoint sets of variables,
𝜙(X, Y) is a conjunction of atoms, and 𝑝(X, Z) is an atom, all without nulls. An instance 𝐼 satisfies
a TGD 𝜎, written 𝐼 |= 𝜎, if the following holds: whenever there exists a homomorphism ℎ such
that ℎ(𝜙(X, Y)) ⊆ 𝐼, then there exists ℎ′ ⊇ ℎ|X , where ℎ|X is the restriction of ℎ on X, such that
ℎ′ (𝑝(X, Z)) ∈ 𝐼. A negative constraint (NC) 𝜈 is a first-order formula ∀X (𝜙(X) → ⊥), where X ⊆ V,
𝜙(X) is a conjunction of atoms without nulls, and ⊥ denotes the truth constant false. An instance 𝐼
satisfies an NC 𝜈, written 𝐼 |= 𝜈, if there is no homomorphism ℎ such that ℎ(𝜙(X)) ⊆ 𝐼. We will use
𝑞𝜈 to denote the BCQ ∃X 𝜙(X). Given a set Σ of TGDs and NCs, 𝐼 satisfies Σ, written 𝐼 |= Σ, if 𝐼
satisfies each TGD and NC of Σ. For a class C of TGDs, C⊥ denotes the combination of C with arbitrary
NCs. Finite sets of TGDs and NCs are called programs. The Datalog± languages we consider are among
the most frequently analyzed in the literature, namely, linear (L) [15], guarded (G) [16], sticky (S) [17],
and acyclic TGDs (A), the “weak” generalizations weakly sticky (WS) [17] and weakly acyclic TGDs
(WA) [18], their “full” restrictions linear full (LF), guarded full (GF), sticky full (SF), and acyclic full
TGDs (AF), respectively, and full TGDs (F) in general. We refer to [12, 5] for a detailed overview.
Knowledge Bases. A knowledge base is a pair (𝐷, Σ), where 𝐷 is a database and Σ is a program. For
a program Σ, Σ𝑇 and ΣNC denote the subsets of Σ containing the TGDs and NCs of Σ, respectively.
The set of models of KB = (𝐷, Σ), denoted mods(KB ), is the set of instances {𝐼 | 𝐼 ⊇ 𝐷 ∧ 𝐼 |= Σ}.
We say that KB is consistent if mods(KB ) ̸= ∅, otherwise KB is inconsistent. The answer to a BCQ
𝑞 relative to KB is true, denoted KB |= 𝑞, if 𝐼 |= 𝑞 for every 𝐼 ∈ mods(KB ). Another way to define
ontological query answering is via the concept of the Chase (see, e.g., [16, 19]).
The BCQ answering problem is: given a knowledge base KB and a BCQ 𝑞, decide whether KB |= 𝑞.
Following [20], the combined complexity of BCQ answering considers the database, the program, and the
query as part of the input. The bounded-arity-combined (or ba-combined) complexity assumes that the
arity of the underlying schema is bounded by constant. The fixed-program-combined (or fp-combined)
complexity considers the program fixed; in the data complexity the query is fixed as well. We refer to
[5] for an overview of the complexity of BCQ answering for the languages in this paper. For more on
computational complexity theory we refer the reader to any textbook on the topic, such as [21].
3. Inconsistency-Tolerant Semantics for Weighted KBs
From now on, we implicitly assume that the database 𝐷 of any knowledge base comes along with a
weight function 𝑤 : 𝐷 → N assigning weights to its facts. For every 𝐷′ ⊆ 𝐷, 𝑤 assigns a weight to
𝐷′ defined as 𝑤(𝐷′ ) = 𝑓 ∈𝐷′ 𝑤(𝑓 ) (with a slight abuse of notation, 𝑤 applies to both facts and sets
∑︀
of facts). For every 𝐷1 , 𝐷2 ⊆ 𝐷, we write 𝐷1 ≤𝑤 𝐷2 (resp., 𝐷1 <𝑤 𝐷2 ) iff 𝑤(𝐷1 ) ≤ 𝑤(𝐷2 ) (resp.,
𝑤(𝐷1 ) < 𝑤(𝐷2 )).
Given a knowledge base KB = (𝐷, Σ), a selection of KB is a database 𝐷′ such that 𝐷′ ⊆ 𝐷. A
selection 𝐷′ of KB is consistent iff (𝐷′ , Σ) is consistent. Symmetrically, the concept of consistent
selection is linked to that of culprit, which is a subset 𝐶 of 𝐷 s.t. (𝐶, Σ𝑇 ) |= 𝑞𝜈 for some 𝜈 ∈ ΣNC . By
deleting from 𝐷 a hitting set ([22, 23, 24]) of facts 𝑆 intersecting all culprits, we obtain a consistent
selection 𝐷′ = 𝐷 ∖ 𝑆.
Definition 3.1. A ≤𝑤 -repair of a knowledge base KB is a consistent selection 𝐷′ of KB such that there
is no consistent selection 𝐷′′ of KB with 𝐷′ <𝑤 𝐷′′ .
For a knowledge base KB = (𝐷, Σ), Rep ≤𝑤 (KB ) denotes the set of all ≤𝑤 -repairs of KB , and the
closure of KB , denoted Cl (KB ), is the set of all facts built from constants in 𝐷 and Σ, entailed by 𝐷
and the TGDs of Σ.
Definition 3.2. Let KB be a knowledge base and let 𝑞 be a BCQ.
• KB entails 𝑞 under the ≤𝑤 -AR semantics, denoted KB |=≤𝑤 -AR 𝑞, if (𝐷′ , Σ) |= 𝑞 for all 𝐷′ ∈
Rep ≤𝑤 (KB ).
• KB entails
⋂︀ ′𝑞 under the ≤𝑤 -IAR semantics, denoted KB |=≤𝑤 -IAR 𝑞, if (𝐷𝐼 , Σ) |= 𝑞, where
′
𝐷𝐼 = {𝐷 | 𝐷 ∈ Rep ≤𝑤 (KB )}.
⋂︀ 𝑞 under
• KB entails the ≤𝑤 -ICR semantics, denoted KB |=≤𝑤 -ICR 𝑞, if (𝐷𝐶 , Σ) |= 𝑞, where
𝐷𝐶 = {Cl ((𝐷 , Σ)) | 𝐷′ ∈ Rep ≤𝑤 (KB )}.
′
4. Discussion of Complexity Results
The problems whose complexity we are interested in are denoted as ≤𝑤 -𝑆(ℒ), with 𝑆 ∈
{AR, IAR, ICR}, and are defined as follows: Given a knowledge base (𝐷, Σ) with Σ ∈ ℒ, and a
BCQ 𝑞, does (𝐷, Σ) |=≤𝑤 -S 𝑞 hold?
The complexity results are summarized in Tables 1 and 2. All entries are completeness results. The
complexity ranges from Δp2 - to 2exp-completeness. For more details on how the results have been
derived, we refer the reader to [14]. Here we focus on the main takeaways the complexity analysis
provides.
The IAR and ICR semantics have the same complexity, which is a behavior shown by cardinality-
maximal repairs as well [6], while this does not hold for subset-maximal ones [5]. As usual (under other
maximality criteria to define repairs), the IAR and ICR semantics are at most as expensive as the AR
semantics. Indeed, we can see that the complexity increases when moving from the IAR/ICR to the
AR semantics only in the fixed-program combined complexity, while the complexity does not change
across the three inconsistency-tolerant semantics under the remaining complexity measures (namely,
data, bounded-arity combined, and combined complexity).
It is also interesting to compare weight-maximal repairs with subset-maximal and cardinality-maximal
ones, whose complexity results can be found in [5] and [6], respectively. Clearly, weight-maximal repairs
generalize cardinality-maximal ones (the latter can be simply modeled by assigning the same weight to
all facts), and when we move from the latter to the former, the complexity of all inconsistency-tolerant
semantics increases in several cases. Compared with subset-maximal repairs, the complexity of all
inconsistency-tolerant semantics under weight-maximal repairs is always at least as high as the one
under subset-maximal repairs. Overall, we can conclude that while weights give us the flexibility of
assigning different importance to different facts, they incur an increase of complexity compared with
more “standard” notions of repairs.
ℒ Data fp-c. ba-c. Comb. ℒ Data fp-c. ba-c. Comb.
L⊥ , LF⊥ , AF⊥ Δp2 Πp2 Δp3 pspace L⊥ , LF⊥ , AF⊥ Δp2 Δp2 Δp3 pspace
S⊥ , SF⊥ Δp2 Πp2 Δp3 exp S⊥ , SF⊥ Δp2 Δp2 Δp3 exp
nexp
A⊥ Δp2 Πp2 p pnexp A⊥ Δp2 Δp2 pnexp
pnexp
G⊥ Δp2 Πp2 exp 2exp G⊥ Δp2 Δp2 exp 2exp
F⊥ , GF⊥ Δp2 Πp2 Δp3 exp F⊥ , GF⊥ Δp2 Δp2 Δp3 exp
WS⊥ , WA⊥ Δp2 Πp2 2exp 2exp WS⊥ , WA⊥ Δp2 Δp2 2exp 2exp
Table 1 Table 2
Complexity of ≤𝑤 -AR(ℒ). Complexity of ≤𝑤 -IAR(ℒ) and ≤𝑤 -ICR(ℒ).
5. Summary and Outlook
We have considered the problem of querying inconsistent knowledge bases whose database facts are
weighted. We have discussed recent results, presented in [14], on the complexity of inconsistent-tolerant
semantics in such a setting.
Future research includes defining other semantics for inconsistency-tolerant OMQA, by considering
more elaborate user preferences over repairs [25, 26, 27, 28, 29, 30, 31] and also considering compact
representations [32, 33, 34]. Another interesting approach that has been investigated recently in the
context of handling inconsistent knowledge is that of measuring inconsistencies via the Shapley value
[35], it would be interesting to bring to existential rules the ideas implemented for DLs [36, 37, 38].
As a natural extension of the setting considered in this paper, TGDs and NCs might be weighted too,
similarly to what has been recently done in [39], which considers weighted knowledge bases where
both axioms and assertions have weights. Another direction for future work is to devise approximation
algorithms that are practical, as done in the setting of incomplete databases [40, 41], e.g. by resorting to
a logic programming approach [42]. Recently, there has been an increasing interest on explainable AI,
including explaining query answering under existential rules [43, 44, 45] and DLs [46, 47]. In particular,
[46, 48, 49] addressed the problem of explaining why a query is entailed or not under inconsistency-
tolerant semantics, where repairs are subset-maximal. An interesting direction for future work is to
address the same problem for weight-maximal repairs.
Acknowledgments
Enrico Malizia’s work was supported by the European Union – NextGenerationEU programme, through
the Italian Ministry of University and Research (MUR) PRIN 2022-PNRR grant P2022KHTX7 “DISTORT”—
CUP: J53D23015000001, under the Italian “National Recovery and Resilience Plan” (PNRR), Mission 4
Component 1.
Cristian Molinaro acknowledges the support of the PNRR project FAIR - Future AI Research
(PE00000013), Spoke 9 - Green-aware AI, under the NRRP MUR program funded by the NextGen-
erationEU.
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