Statistical EL is ExpTime-complete

Statistical EL is ExpTime -complete

Bartosz Bednarczyk

Computational Logic Group, Technische Universität Dresden, Germany Institute of Computer Science, University of Wrocław, Poland

Abstract

We show that the consistency problem for StatisticalELontologies, defined by Peñaloza and Potyka, is ExpTime-hard. Together with existingExpTime upper bounds, we concludeExpTime-completeness of the logic. Our proof goes via a reduction from the consistency problem forELextended with negation of atomic concepts.

1 Introduction

Description logics (DLs) [BHLS17] are a prominent family of logical formalisms tailored to knowledge represen- tation. Nowadays, real-world problems require the ability to handle uncertain knowledge. To deal with this issue, several probabilistic extensions of description logics were proposed in the past [CLC17,Luk08,GBJLS17,PP17]

Among such extensions, the authors of [PP17] proposed StatisticalEL, a statistical variant of the well-known description logicEL[BBL05] famous for tractability of most of its reasoning task.

In this note we establish tight complexity bounds for the consistency problem for statistical EL, closing the complexity gaps from [PP17]. We show that in sharp contrast to its non-probabilistic version, StatisticalELis ExpTime-complete and hence, provably intractable. The main novelty here is theExpTimelower bound, while theExpTimeupper bound follows from recent work by Baader and Ecke [BE17, Corollary 15] or, alternatively, from work on probabilisticALC by Lutz and Schröder [LS10, Theorem 9].

2 Preliminaries

In this section, we recall the basics on description logics (DLs) ELandEL^(¬). For readers unfamiliar with DLs we recommend consulting the textbook [BHLS17], especially Chapters 2.1–2.3, 5.1 and 6.1.

We fix countably-infinite disjoint sets ofconcept namesNCandrole namesNR. Starting fromNCandNR, the set C_EL of EL concept descriptions(or simply ELconcepts) [BBL05] is built usingconjunction (CuD), existential restriction(∃r.C) and the top concept (>), with the grammar below:

C,D ::= > | A | CuD | ∃r.C,

where C,D∈CEL, A∈NCand r ∈NR. AnELgeneral concept inclusion(GCI) has the form CvD forEL concepts C,D∈C_EL. AnELontologyis a finite non-empty set ofELGCIs. The sizeof anELontology is the total number of>, role names, concept names and connectives occurring in it.

Table 1: Concepts and roles inEL.

Name Syntax Semantics

top > ∆^I

atomic concept A A^I⊆∆^I

role r r^I ⊆∆^I×∆^I

concept intersection CuD C^I∩D^I existential restriction ∃r.C

d| ∃e.(d,e)∈r^I∧e∈C^I

The semantics ofELis defined viainterpretationsI= (∆^I,·^I) composed of afinitenon-empty set ∆^I called the domainofI and aninterpretation function ·^I mapping concept names to subsets of ∆^I, and role names to subsets of ∆^I ×∆^I. This mapping is extended to concepts,roles (cf. Table 1) and finally used to define satisfaction of GCIs, namelyI |= CvD iff C^I ⊆D^I. We say that an interpretationI satisfiesan ontology O (orI is a model ofO, written:I |=O) if it satisfies all GCIs fromO. An ontology isconsistentif it has a model and inconsistent otherwise. In theconsistency problemforELwe ask if an inputELontology is consistent. Note that the consistency problem forELis trivial,i.e. everyELontology is consistent.

2.1 EL with atomic negation

The next definitions concern EL^(¬), the extension of ELwith negation of atomic concepts. More precisely, the setC_EL(¬) ofEL^(¬) conceptsis defined by a slight extension of the BNF grammar forEL:

C,D ::= > | A | A¯ | CuD | ∃r.C,

where C,D∈C_EL(¬), A∈NCandr ∈NR. The semantics ofEL^(¬)concepts is defined as in Table1with the exception that the concepts of the form ¯A have the semantics ¯A^I = ∆^I\A^I. The notions of GCIs, ontologies and the consistency problem are lifted toEL^(¬) in an obvious way. We stress that in the presence of negation the consistency problem forEL^(¬)is no longer trivial and actually isExpTime-complete [BBL05, Theorem 6].¹ Proposition 2.1. The consistency problem forEL^(¬)ontologies is ExpTime-hard.

2.2 Statistical EL

Statistical EL, abbreviated asSEL, is a probabilistic DL introduced recently by Peñaloza and Potyka [PP17, Section 4] to reason about statistical properties over finite domains. Statistical ELontologies are composed of probabilistic conditionalsof the form (C|D) [k, l], where C,D are ELconcepts from C_EL and k, l ∈Q are rational numbers satisfying 0 ≤k ≤ l ≤ 1. The size of SEL ontologies is defined as in EL except that the numbers in probabilistic conditionals also contribute to the size and are measured in binary.

We say that an interpretationI satisfies a probabilistic conditional (C|D) [k, l] if:

either D^I=∅ork≤ |(CuD)^I|

|D^I| ≤l.

Note that usual EL GCIs C vD are equivalent to (D|C) [1,1] (cf. [PP17, Proposition 4]). Hence, eachEL ontology can be seen as anSELontology and we can freely use GCIs in place of probabilistic conditionals.

3 Main result

After introducing all the required definitions, we are ready to prove the main result of this note, namely:

Theorem 3.1. The consistency problem forSELisExpTime-complete.

The ExpTime upper bound follows from [BE17, Corollary 15] or [LS10, Theorem 9], hence we focus on the lower bound only. LetObe an arbitraryEL^(¬) ontology. WithC_O we denote the set of all concept names that appear (possibly under negation) inO. We next design anSELontologyOredsuch that Oredis consistent iff O is and thatOred is only polynomially larger thanO. It will be composed of twoSELontologies,Otr andOcorr, responsible respectively for “translating”Ointo SELand for guaranteeing the correctness of the translation.

The main idea of the encoding is as follows. We first produce for each concept name A fromCO two fresh, different, concepts A+,A− 6∈CO intuitively intended to contain, respectively, all members of A and from its complement. Due to the lack of negation, we clearly are not able to fully formalise the above intuition, but the best we can do is to enforce, with the ontologyOcorr, that these concepts are interpreted as disjoint sets and each of them contains exactly half of the domain. This is sufficient for our purposes, since with fresh, pair-wise different, concepts Real,Real₊,Real₋ 6∈ C_O we can separate the “real” model of O from the auxiliary parts required for the encoding. Finally, in the “translation” ontology O_tr we state that the restriction of a model ofO_red to Real₊ satisfiesO. The translation simply changes all occurrences of A (resp. ¯A) into A₊ (resp. A₋) and employs Real₊ to relativise concepts.

We start with the definition ofO_corr.

Ocorr:={(A+| >) [0.5,0.5],(A₋ | >) [0.5,0.5],(A+|A₋) [0,0] | A∈ {Real} ∪C_O} By unfolding the definition of probabilistic conditionals we immediately conclude the following facts.

Fact 3.2. For any concept nameA we have thatI |= (A| >) [0.5,0.5]iff|∆^I|is even and |A^I|=¹₂|∆^I|.

Fact 3.3. For any different concept names A,Bwe have thatI |= (A|B) [0,0]iff A^I∩B^I=∅.

1In the setting of [BBL05] the domains of interpretations might have unrestricted (i.e. not necessarily finite) sizes, but the result is also applicable to our scenario sinceEL^(¬)has finite model property, which follows from [BHLS17, Corr. 3.17].

Now we focus on the “translating” ontology Otr. Lettrbe a translation function defined bytr(>) = Real+, tr(A) = A+uReal+ andtr( ¯A) = A₋uReal+ for all concept names A∈NCas well astr(CuD) =tr(C)utr(D) and tr(∃r.C) = Real+u ∃r.(tr(C)uReal+) for complex concepts. The ontologyOtr is obtained by replacing each GCI CvD fromOwithtr(C)vtr(D). Finally, we put Ored :=Ocorr∪ Otr.

Note that the size ofOred is polynomial in|O|. For more intuitions, consult the picture below.

reduction

d2: A,B¯ d3: A,B

d1: ¯A,B d⁰₁: A+,B−

d⁰₂: A₋,B₊

d⁰₃: A₋,B₋ d3: A+,B+

d2: A+,B₋ d₁: A₋,B₊

Real+

Real−

J |=O_red I |=O

Lemma3.6

Lemma3.5

3.1 Correctness of the reduction

Let us start with an auxiliary notion of interpretations that aregood-for-encoding. We say thatJ is good-for- encoding if for all concept names A∈C_O it satisfies A^J = A^J₊ and A^J₋ = ∆^J\A^J. The following lemma relates the translation functiontr, good-for-encoding interpretations and their submodels.

Lemma 3.4 (Agreement lemma). LetJ be good-for-encoding and let I be its induced subinterpretation with domain Real^J₊. Then all EL^(¬) concepts C employing only concept names from C_O satisfy C^I = tr(C)^J. Moreover for such concepts C,D we have:J |=tr(C)vtr(D)iffI |= CvD.

Proof. We proceed by inductively on the shape of concepts C. The cases for C =>,A or ¯A for A∈NCfollow immediately from the definition oftrand the assumptions A^J = A^J₊ and A^J₋ = ∆^J\A^J. The case of C = DuE follows from the fact thattris homomorphic foru. Hence, the only interesting case is when C =∃r.D. Assuming D^I =tr(D)^J we will show two inclusions.

• For the first inclusion, take d ∈(∃r.D)^I. Thus d∈∆^I(= Real^J₊) and there exists an e∈∆^I satisfying both (d,e)∈r^I and e∈D^I. Note that e∈∆^I implies e∈Real^J₊. Moreover, by the equality D^I =tr(D)^J we have e∈(Real₊utr(D))^J. Sincer^I⊆r^J we infer d∈(∃r.(Real₊utr(D)))^J, but because d∈Real^J₊ we can conclude that d∈(Real+u ∃r.(Real+utr(D)))^J =tr(∃r.D)^J.

• For the opposite direction take d ∈ tr(∃r.D)^J = (Real+ u ∃r.(Real+ utr(D)))^J. It implies that d ∈ Real^J₊(= ∆^I) as well as that there is an e∈Real^J₊(= ∆^I) witnessing (d,e)∈r^J and e∈tr(D)^J(= D^I).

Since both d,e belong to ∆^I we infer that (d,e)∈r^I and hence d∈(∃r.D)^I.

For the last statement of the lemma: to show thatJ |=tr(C)vtr(D) iffI |= CvD hold, it suffices to invoke C^I=tr(C)^J and D^I =tr(D)^J to see that the inclusions C^I⊆D^I andtr(C)^J ⊆tr(D)^J are equivalent.

The agreement lemma can now be used to show that the consistency ofOredimplies the consistency ofO.

Lemma 3.5. IfOred is consistent then so isO.

Proof. Let J be a model of O_red with A^J := A^J₊ (since A does not appear in O_red this can be assumed w.l.o.g.) for all concept names A fromC_O. By the satisfaction ofOcorrwe know that A^J₊ and A^J₋ are disjoint and thus J is good-for-encoding. Hence, takeI to be its induced subinterpretation with domain Real^J₊. By applying Lemma3.4we know that for each GCI CvD from Othe satisfaction ofJ |=tr(C)vtr(D) implies I |= CvD. Thus we getI |=O, which implies thatOis consistent.

We next show that the consistency of Oimplies the consistency of Ored. In the proof we basically take a model ofO, duplicate each domain element and define the memberships of fresh concepts introduced byO_red. Such concepts are defined in such a way that if an element from a modelI ofO is a member of A^I then the corresponding element in a constructed modelJ forO_red is a member of A^J₊ while its copy belongs to A^J₋. In this way, the total number of elements in every concept is always equal to half of the domain.

Lemma 3.6. IfO is consistent then so isOred.

Proof. LetI |=O and let ∆^I={d1,d2, . . . ,dn}. We define an interpretationJ as follows:

1. ∆^J :={d1,d⁰₁,d2,d⁰₂, . . . ,dn,d⁰_n}.

2. For all concept names A∈C_O we put

• A^J₊ := A^J :={di | d_i ∈A^I} ∪ {d⁰_i | d_i6∈A^I}and A^J₋ :={di | d_i6∈A^I} ∪ {d⁰_i | d_i∈A^I},

• Real^J₊ := ∆^I and Real^J₋ := ∆^J \∆^I,

and for all other concept names B we put B^I := ∆^I. 3. For each role namer we put r^J :=r^I.

We first showJ |=O_corr. To this end, take any name A∈ {Real} ∪C_O. We proveJ |= (A₊ |A₋) [0,0], which by Fact3.3is equivalent to showing disjointness of A^J₊ and A^J₋. Assume the contrary,i.e. that there is a domain element d∈A^J₊∩A^J₋. If d = d_i for some index ithen, by Item2, it means that d_i ∈A^I and d_i 6∈A^I at the same time, which is clearly not possible. The case when d = d⁰_i for some indexiis treated similarly. Next, by invoking Item2 of the definition ofJ we can perform some basic calculations:

|A^J₊|=|{di | d_i∈A^I}|+|{d⁰_i | d_i6∈A^I}|=|{di | d_i∈A^I}|+|{di | d_i6∈A^I}|=|∆^I|= 0.5· |∆^J|.

|A^J₋|=|{di | di6∈A^I}|+|{d⁰_i | di∈A^I}|=|{d⁰_i | di6∈A^I}|+|{d⁰_i | di∈A^I}|=|{d⁰₁, . . . ,d⁰_n}|= 0.5·|∆^J|.

Hence by Fact3.2we getJ |= (A₊| >) [0.5,0.5] andJ |= (A₋| >) [0.5,0.5], finishing the proof of J |=O_corr. To proveJ |=O_tr we take any GCI tr(C)vtr(D) fromO_tr. FromI |=O, we know thatI |= CvD. Note thatJ is good-for-encoding, hence by Lemma3.4it follows that J |=tr(C)vtr(D). ThusJ |=O_tr.

Lemma 3.6and Lemma3.5show that the presented reduction is correct. Since our reduction is polynomial for everyEL^(¬)ontologyO, from Proposition2.1we can conclude the main theorem of this note.

Theorem 3.7. The consistency problem for StatisticalELisExpTime-hard and remainsExpTime-hard even if the only numbers used in probabilistic conditionals are0,0.5 and1.

With the already mentioned ExpTimeupper bound we conclude Theorem 3.1.

4 Conclusions

We have proved that the consistency problem for Statistical ELontologies isExpTime-complete. While the upper bound was derived from the works of Baader and Ecke [BE17] or Lutz and Schröder on (the extensions of)ALCwith cardinality constraints, the main contribution of the paper is the lower bound. Our proof went via a reduction from the consistency problem forEL^(¬) ontologies and heavily relied on the fact that probabilistic conditionals can express that exactly half of the domain elements belong to a certain concept.

An interesting direction for future work is to consider extensions of other well-known decidable fragments of first-order logic with probabilistic conditionals. Promising candidates are the guarded fragment [ANvB98], the guarded negation fragment [BtCS15], the two-variable logic [GKV97] and tamed fragments of existential rules [MT14]. Another idea is to study query answering [OS12] in the presence of probabilistic conditionals.

Some initial results were obtained in [BBR20].

Acknowledgements

This work was supported by the Polish Ministry of Science and Higher Education program “Diamentowy Grant”

no. DI2017 006447. The author would like to thank Ania Karykowska and Elisa Böhl for reading the draft of this note as well as Piotr Witkowski for serving as a rubber duck while “rubber duck debugging” the proof ideas.

He also grateful for many grammar corrections and improvement ideas given by anonymous IPL reviewers.

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