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id	Vol-3194/paper56
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title	Query Answer Explanations under Existential Rules
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dblpUrl	https://dblp.org/rec/conf/sebd/CeylanLMV22
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Query Answer Explanations under Existential Rules

Query Answer Explanations under Existential Rules⋆
(Discussion Paper)

İsmail İlkan Ceylan1 , Thomas Lukasiewicz1 , Enrico Malizia2 and
Andrius Vaicenavičius1
1
    Department of Computer Science, University of Oxford, UK
2
    DISI, University of Bologna, Italy


                                         Abstract
                                         Ontology-mediated query answering is an extensively studied paradigm, which aims at improving
                                         query answers with the use of a logical theory. In this paper, we focus on ontology languages based on
                                         existential rules, and we carry out a thorough complexity analysis of the problem of explaining query
                                         answers in terms of minimal subsets of database facts and related tasks.

                                         Keywords
                                         Knowledge representation, Existential rules, Query answering, Explanations, Computational complexity




1. Introduction
Ontology-based data access (OBDA) is a popular paradigm in knowledge representation and
reasoning [2]. In this paradigm, an ontology enriches the user query with commonsense
knowledge, to form a so-called ontology-mediated query (OMQ) [3]. The task of evaluating OMQs
is called ontology-mediated query answering (OMQA). Description logics (DLs) [4] and existential
rules [5] are the most widely used ontology languages. Tracking down the causes of entailment
in DL ontologies has attracted extensive interest. In the DL context, a prominent approach is
identifying explanations as ontology axioms subsets [6, 7, 8], which are called justifications [9, 10].
Axiom pinpointing is extensively studied in DLs, and some implementations exist as well [6, 11].
However, most of the existing approaches to explanations in the DL framework focus on classical
reasoning tasks. Explaining query entailments has only been investigated for the DL-Lite family
of languages [12, 13], and negative answers to OMQs in DLs [14].
   In this paper, we study explanations in the existential rules context. The only prior work
related to explanations in existential rules is given in [15] in the context of probabilistic databases,
and hence of a very different flavor. Here, given an OMQ, we are interested in explaining the
OMQ in terms of the minimal satisfying subsets, called minimal explanations of a given database.
   We analyze the complexity for each of the problems introduced in terms of data, fp-combined,
ba-combined and combined complexity, for a representative set of existential rules.

SEBD 2022: The 30th Italian Symposium on Advanced Database Systems, June 19-22, 2022, Tirrenia (PI), Italy
⋆
 This is a short version of the IJCAI-19 paper [1].
$ ismail.ceylan@cs.ox.ac.uk (İ. İ. Ceylan); thomas.lukasiewicz@cs.ox.ac.uk (T. Lukasiewicz);
enrico.malizia@unibo.it (E. Malizia); andrius.vaicenavicius@cs.ox.ac.uk (A. Vaicenavičius)
� 0000-0003-4118-4689 (İ. İ. Ceylan); 0000-0002-7644-1668 (T. Lukasiewicz); 0000-0002-6780-4711 (E. Malizia)
                                       © 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)
�2. Preliminaries
We here briefly recall some basics on existential rules from the context of Datalog± [5].
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, null, or variable. We assume a set of predicates, each associated with an arity.
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 called fact. Conjunctions of atoms are also identified with
the sets of their atoms. An instance 𝐼 is a (possibly infinite) set of atoms defined over constants
and nulls. A database 𝐷 is a finite instance containing 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. With
a slight abuse of notation, homomorphisms are applied also to (sets/conjunctions of) atoms. A
conjunctive query (CQ) 𝑞 has the form ∃Y𝜑(X, Y), where 𝜑(X, Y) is a conjunction of atoms
without nulls. The answer to 𝑞 over an instance 𝐼, denoted 𝑞(𝐼), is the set of all |X|-tuples
t over C for which there is a homomorphism ℎ such that ℎ(𝜑(X, Y)) ⊆ 𝐼 and ℎ(X) = t. A
Boolean CQ (BCQ) 𝑞 is a CQ ∃Y𝜑(Y), i.e., all variables are existentially quantified; 𝑞 is true
over 𝐼, denoted 𝐼 |= 𝑞, if 𝑞(𝐼) ̸= ∅, i.e., there is a homomorphism ℎ with ℎ(𝜑(Y)) ⊆ 𝐼.
Dependencies. A tuple-generating dependency (TGD) 𝜎 is an FO 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 𝜎, written 𝐼 |= 𝜎,
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. Given a set Σ of TGDs, 𝐼
satisfies Σ, written 𝐼 |= Σ, if 𝐼 satisfies each TGD of Σ. For brevity, we omit the universal
quantifiers in front of TGDs, and use the comma (instead of ∧) for conjoining atoms. Finite sets
of TGDs are called programs (or, ontologies), and TGDs are also called existential rules.
   The Datalog± languages ℒ that we consider to guarantee decidability are among the most
frequently analyzed in the literature, namely, linear (L) [5], guarded (G) [16], sticky (S) [17], and
acyclic TGDs (A), along with the “weak” (proper) generalizations weakly sticky (WS) [17] and
weakly acyclic TGDs (WA) [18], as well as their “full” (i.e., existential-free) proper restrictions
linear full (LF), guarded full (GF), sticky full (SF), and acyclic full TGDs (AF), respectively,
and full TGDs (F) in general. We also recall the following further inclusions: L ⊂ G and
F ⊂ WA ⊂ WS. We refer to [19] for a more detailed overview.
Ontology-Mediated Query Answering. The OMQA paradigm generalizes query answering
over databases by incorporating additional background knowledge via an ontology.
   Formally, an ontology-mediated query (OMQ) is a pair (𝑞, Σ), where 𝑞 is a query, and Σ is
an ontology. Let ℒ be a Datalog± language. If Σ ∈ ℒ, then we say that (𝑞, Σ) is an ℒ-OMQ.
Consider a database 𝐷 and an OMQ (𝑞, Σ). The set of models of (𝐷, Σ), denoted mods(𝐷, Σ),
is the set of instances {𝐼 | 𝐼 ⊇ 𝐷 ∧ 𝐼 |= Σ}. We say that 𝐷 entails (𝑞, Σ), denoted 𝐷 |= (𝑞, Σ),
if 𝐼 |= 𝑄 for every 𝐼 ∈ mods(𝐷, Σ). Ontology-mediated query answering (OMQA) is the task of
deciding whether 𝐷 |= (𝑞, Σ) for a given database 𝐷 and an OMQ (𝑞, Σ). A different way to
define the existential rules semantics is via the concept of the Chase (see, e.g., [20, 21]).
   Following Vardi [22], the combined complexity of BCQ answering considers the database,
the set of dependencies, 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 an
�Table 1                                                  Figure 1
Complexity of OMQA under existential rules [19].         Protein containment in complexes.

    ℒ       Data     fp-comb. ba-comb.      Comb.                     𝑝1           𝑝2           𝑝3
                 0                                          𝑐1
L, LF, AF   in ac      np          np       pspace
  S, SF     in ac0     np          np         exp
    A       in ac0     np         nexp       nexp           𝑐2
    G          p       np          exp       2exp                     𝑝4                        𝑝5
  F, GF        p       np          np         exp           𝑐3
WS, WA         p       np         2exp       2exp


integer constant. The fixed-program-combined (or fp-combined) complexity considers the sets
of TGDs as fixed; the data complexity also assumes the query fixed. Table 1 summarizes the
complexity results for OMQA in the different TGD classes here considered; OMQA (ℒ) denotes
the OMQA problem when restricted over ontologies belonging to ℒ.
   An OMQ (𝑞, Σ) is FO-rewritable, if there exists a query 𝑞Σ such that, for all databases 𝐷,
we have that 𝐷 |= (𝑞, Σ) iff D|= 𝑞Σ . In this case, 𝑞Σ is an FO-rewriting of (𝑞, Σ). A class of
programs ℒ is FO-rewritable, if it admits an FO-rewriting for every query and program in ℒ. All
languages from Table 1 with ac0 data complexity are FO-rewritable.


3. Explanations for Ontology-Mediated Query Answers
In our approach, a minimal explanation for an ontology-mediated query answer is an inclusion-
minimal subset of a database that supports such query answer.
Definition 1. For a database 𝐷 and an OMQ (𝑞, Σ), where 𝑞 is a query and Σ is a program,
such that 𝐷 |= (𝑞, Σ), an explanation for 𝐷 |= (𝑞, Σ) is a subset 𝐸 ⊆ 𝐷 of facts entailing the
OMQ (𝑞, Σ), i.e., 𝐸 |= (𝑞, Σ); 𝐸 is a minimal explanation, or MinEx, for 𝐷 |= (𝑞, Σ) if 𝐸 is
subset-minimal, i.e., there is no proper subset 𝐸 ′ ⊊ 𝐸 that is an explanation for 𝐷 |= (𝑞, Σ).
Example 2. Each protein 𝑝1 , . . . , 𝑝5 belongs to complexes 𝑐1 , 𝑐2 and 𝑐3 , as shown in Fig 1. We
want to find minimal subsets of proteins covering all complexes (a minimal hitting set [23, 24]).
This is tantamount to looking for a MinEx, where the database is 𝐷 = {prt(𝑝𝑖 ) | 1 ≤ 𝑖 ≤
5} ∪ {in(𝑝𝑖 , 𝑐𝑗 ) | 𝑝𝑖 is in 𝑐𝑗 }, the ontology is Σ = {prt(𝑃 ) ∧ in(𝑃, 𝐶) → cov (𝐶)}, and the
query is 𝑞 = (cov (𝑐1 ) ∧ cov (𝑐2 ) ∧ cov (𝑐3 )) ∧ 𝜑in , where 𝜑in is a conjuction of all in facts in 𝐷.
Clearly, the MinExes for (𝑞, Σ) in 𝐷 are in bijection with the protein covers of complexes. ◁
    We analyze these problems. Is-MinEX: given a set of facts, decide if it is a MinEx; All-
MinEX: given sets of facts, decide if they are all the MinExes; MinEX-Rel: given a fact, decide
if it is contained in some MinEx; MinEX-Irrel: given forbidden sets of facts, decide if there is
a MinEx not containing any forbidden set; and Small-MinEX (resp., Large-MinEX): given a
number 𝑛, decide if there is a MinEx with less (resp. more) elements than 𝑛.
Example 3. Consider the following subsets of the database:

        𝐸1 = {prt(𝑝1 ), prt(𝑝3 )} ∪ 𝐷in , and 𝐸2 = {prt(𝑝2 ), prt(𝑝4 ), prt(𝑝5 )} ∪ 𝐷in ,
�Table 2
Complexity results for Is-MinEX (ℒ) and for All-MinEX (ℒ).
                              Is-MinEX                                   All-MinEX
    ℒ        Data    fp-comb.     ba-comb.    Comb.     Data    fp-comb.     ba-comb.    Comb.
L, LF, AF   in ac0       dp          dp       pspace     in p       dp            dp     pspace
  S, SF     in ac0       dp          dp         exp      in p       dp            dp       exp
    A       in ac0       dp         dexp       dexp      in p       dp          dexp      dexp
    G          p         dp         exp        2exp     co-np       dp           exp      2exp
  F, GF        p         dp          dp         exp     co-np       dp            dp       exp
WS, WA         p         dp         2exp       2exp     co-np       dp          2exp      2exp


where 𝐷in is the set of all in facts from 𝐷. Then we have:
• Is-MinEX: Both 𝐸1 and 𝐸2 are MinExes for (𝑞, Σ) in 𝐷 . On the other hand, 𝐸1′ = {prt(𝑝1 ),
  prt(𝑝4 )} ∪ 𝐷in and 𝐸2′ = {prt(𝑝1 ), prt(𝑝2 ), prt(𝑝3 )} ∪ 𝐷in are not MinExes as they do not
  entail the query and are not minimal, respectively.
• All-MinEX: Both 𝐸3 = {prt(𝑝1 ), prt(𝑝5 )} ∪ 𝐷in and 𝐸4 = {prt(𝑝3 ), prt(𝑝4 )} ∪ 𝐷in are
  MinExes, and the set ℰ = {𝐸1 , 𝐸2 , 𝐸3 , 𝐸4 } is the set of all MinExes for (𝑞, Σ) in 𝐷 .
• MinEX-Rel: For each 𝑖 ∈ {1, . . . , 5}, the fact prt(𝑝𝑖 ) is contained in some MinEx for (𝑞, Σ)
  in 𝐷 , and thus the fact is relevant.
• MinEX-Irrel: Let {{prt(𝑝1 ), prt(𝑝3 )}, {prt(𝑝5 )}} be a set of forbidden sets of facts. Note
  that there is an explanation that does not contain any of these sets, which is 𝐸4 . Notice,
  however, 𝐸1 , 𝐸2 and 𝐸3 contain a forbidden set.
• Large-MinEX and Small-MinEX: All MinExes are either of size 2 + |𝐷in | or 3 + |𝐷in |. ◁


4. Recognizing (Sets of All) Minimal Explanations
First, we provide general results for Is-MinEX and All-MinEX. These general algorithms use
the procedure for the OMQA problem. For Is-MinEX, checking that a set 𝐸 is a MinEx requires
checking that 𝐸 entails the OMQ, and that 𝐸 is minimal, if removing any element from 𝐸 yields
a set that does not entail the OMQ. For All-MinEX, we need to check that each set of facts in
the input entails the OMQ, and that in the input there is no missing explanation.

Theorem 4. For a class ℒ of TGDs, if OMQA (ℒ) is in the complexity class 𝒞, then:
• Is-MinEX(ℒ) can be decided by a check in 𝒞 and a linear number of checks in co-𝒞;
• All-MinEX(ℒ) can be decided by a check in p𝒞 and a check in co-(np𝒞 ).

Theorem 5. If ℒ is a class of TGDs for which OMQA is FO-rewritable, then Is-MinEX(ℒ) is in
ac0 in the data complexity.

Theorem 6. For a class ℒ of TGDs, if OMQA(ℒ) is in np, then All-MinEX(ℒ) is in dp .

Theorem 7. Let ℒ be class of TGDs for which OMQA is FO-rewritable. Then, computing all
MinExes for 𝐷 |= (𝑞, Σ), where Σ is from ℒ, is feasible in polynomial time in the data complexity.
�Table 3
Complexity results for MinEX-Irrel/Small-MinEX (ℒ) and for MinEX-Rel/Large-MinEX (ℒ).

                    MinEX-Irrel / Small-MinEX                   MinEX-Rel / Large-MinEX
    ℒ        Data    fp-comb.    ba-comb.     Comb.     Data    fp-comb.     ba-comb.    Comb.
L, LF, AF    in p       np           np       pspace     in p       Σp2         Σp2       pspace
  S, SF      in p       np           np         exp      in p       Σp2         Σp2         exp
    A        in p       np          nexp       nexp      in p       Σp2        pnexp       pnexp
    G         np        np          exp        2exp       np        Σp2         exp        2exp
  F, GF       np        np           np         exp       np        Σp2         Σp2         exp
WS, WA        np        np          2exp       2exp       np        Σp2        2exp        2exp


   The upper-bounds stated above can be shown tight. For Is-MinEX, the p-hardness results are
obtained via a reduction from OMQA for GF, which is p-complete in those cases; the dp -hardness
results via a reduction from the classical Sat-UnSat problem; and the dexp -hardness results
via a reduction from the dexp -complete problem ExpTiling-NonExpTiling: given two tiling
problems TP 1 and TP 2 for the exponential square 2𝑛 × 2𝑛 , and two initial tiling conditions
𝑤1 and 𝑤2 , decide whether TP 1 have no solution with 𝑤1 and TP 2 have a solution with 𝑤2 .
   All the remaining hardness results in Table 2 for Is-MinEX follow from the observation that
Is-MinEX is no easier than OMQA in the ba- and combined complexity.
   For All-MinEX, the co-np-hardness results are via a reduction from UnSat, while all other
hardness results are obtained with minimal variations on the Is-MinEX’s hardness proofs.


5. Existence of an Explanation Excluding/Including Facts
For MinEX-Irrel, to check the existence of a MinEx not intersecting any of the forbidden sets
provided in input, it suffices to guess a set of fact entailing the OMQ (we do not need to check
the minimality of the guessed set in this case). For MinEX-Rel, checking the existence of a
MinEx including a fact we need to guess a set of facts that subsequently needs to be checked to
be a MinEx (we do need to check the minimality of the guessed set in this case).

Theorem 8. For a class ℒ of TGDs, if OMQA (ℒ) (resp., Is-MinEX (ℒ)), is in the complexity class
𝒞 (resp., 𝒟), then MinEX-Irrel(ℒ) (resp., MinEX-Rel (ℒ)) is in np𝒞 (resp., np𝒟 ).

Theorem 9. For a class ℒ of TGDs, if OMQA (ℒ) is in np, then MinEX-Irrel(ℒ) is in np.

   Also these complexity results are tight. In particular, for MinEX-Irrel, the np-hardness results
in the data complexity are obtained via a reduction from the problem of deciding whether in a
graph there is a path between two vertices avoiding a set of pairs of edges [25, Problem GT54].
   For MinEX-Rel, the np-hardness results are via a reduction from the problem of deciding
whether in a directed graph there is a path from a node 𝑠 to a node 𝑡 passing through a node
𝑚 [26]; the Σp2 -hardness results via a reduction from the validity problem ∃𝑋∀𝑌 𝜑(𝑋, 𝑌 );
and the pnexp -hardness results via a reduction from the following problem [19]: for a triple
(𝑚, TP 1 , TP 2 ), where 𝑚 is an integer in unary notation, and TP 1 and TP 2 are two tiling
�problems for the exponential square 2𝑛 × 2𝑛 , decide whether there is an initial condition 𝑤 of
length 𝑚, such that TP 1 has no solution with 𝑤, and TP 2 has a solution with 𝑤.
   The other hardness results for MinEX-Irrel (resp., MinEX-Rel) follow via a reduction from
OMQA in the fp-, ba-, and combined complexity (resp., in the ba- and combined complexity).


6. Cardinality-Based Explanation Problems
The intuitions for the membership results of Small-MinEX and Large-MinEX problems are as
follows. For Small-MinEX, deciding if there is a MinEx not bigger than a given size, it suffices
to guess a small enough set of facts, and then test that this set entails the OMQ (in this case,
we do not need to ensure the minimality of the guessed set). For Large-MinEX, to decide the
existence of a big enough MinEx, we can guess a proper set of facts and check that it is actually
a MinEx (in this case, we do need to check the minimality of the guessed set).
Theorem 10. For a class ℒ of TGDs, if OMQA (ℒ) (resp., Is-MinEX (ℒ)), is in the complexity class
𝒞 (resp., 𝒟), then Small-MinEX (ℒ) (resp., Large-MinEX (ℒ)) is in np𝒞 (resp., np𝒟 ).
  Some entries in Table 3 need the result below; p membership results follow from Theorem 7.
Theorem 11. For a class ℒ of TGDs, if OMQA (ℒ) is in np, then Small-MinEX (ℒ) is in np.
  We have completeness results also for these problems. For Small-MinEX, the np-hardness
results in the data complexity are via a reduction from Vertex Cover.
  For Large-MinEX, the np-hardness results are obtained via a reduction from the problem
Hamiltonian Path; the Σp2 -hardness results and the pnexp -hardness are obtained via small
variations of the corresponding MinEX-Rel’s hardness proofs.
  The other hardness results for Small-MinEX (resp., Large-MinEX) are via a reduction from
OMQA in the fp-, ba-, and combined complexity (resp., in the ba- and combined complexity).


7. Summary and Outlook
In this paper, we have studied the problem of explaining query answers in terms of minimal
subsets of database facts, and provided a thorough complexity analysis for several decision
problems associated with minimal explanations under existential rules.
   Future work can include studying other ontology languages. Our work on explaining OMQA
has been extended to the inconsistent setting [27, 28], to negative OMQA [29], to preferred
explanations [30], and also to DL [31]. The explanation notion, in line with the inconsistent
setting above, can also be extended to the cases of cardinality-based repairs [32], generalized
repairs [33], probabilistic/preferred repairs [34, 35, 36], and repairs based on value updates [37].
Inspired by the idea of exploring preferences over explanations, we can also consider how more
elaborate preference models can be included in this framework [38, 39, 40, 41].


Acknowledgments
This work was supported by the Alan Turing Institute under the UK EPSRC grant EP/N510129/1,
and by the EPSRC grants EP/R013667/1, EP/L012138/1, and EP/M025268/1.
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