Exploring Non-Regular Extensions of Propositional Dynamic Logic with Description-Logics Features

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.


Introduction
Formal ontologies play a crucial role in artificial intelligence, serving as the backbone of various applications such as the Semantic Web, ontology-based information integration, and peer-to-peer data management.In reasoning about graph-structured data, a significant role is played by description logics (DLs) [BHLS17], a robust family of logical formalisms serving as the logical foundation of contemporary standardised ontology languages, including OWL 2 by the W3C [GHM + 08, HKP + 12].Among many features present in extensions of the basic description logic ALC, an especially useful one is • reg , supported by the popular Z-family of description logics [CEO09].With • reg one can specify regular path constraints, allowing the user to navigate graph-structured data.In recent years, many extensions of ALC reg for ontology-engineering were proposed [BCOS14,COS16,Ort10], and the complexity landscape of their reasoning problems is now mostly well-understood [CEO09, BR19,BK22].In fact, the logic ALC reg was already studied in 1979 by the formal-verification community [FL79], under the name of Propositional Dynamic Logic (PDL).Relationship between (extensions of) PDL and ALC reg were investigated by De Giacomo and Lenzerini [DL94].
The spectrum of recognizable word languages is relatively wide.Hence the question of whether regular constraints in path expressions of ALC reg can be lifted to more expressive classes of languages received a lot of attention from researchers.We call such extensions of PDL non-regular.After the first undecidability proof of satisfiability of ALC reg by contextfree languages [HPS81b, Cor.2.2], several decidable cases were identified.For instance, Koren and Pnueli [KP83, Sec.Decidability] proved that ALC reg extended with the simplest non-regular language r # s # := {r n s n | n ∈ N} for fixed roles r , s is decidable; while combining it with s # r # leads to undecidability [HPS81a,Thm. 3.2].This surprises at first glance, but as it was shown later [LLS07, Thm.18], PDL extended with a broad class of input-driven context-free languages, called visibly pushdown languages [AM09, Sec.5], remains decidable.This generalises all previously known decidability results, and partially explains the reason behind known failures (e.g. the languages r # s # and s # r # cannot be both visibly-pushdown under the same partition of the alphabet).Three years ago, the decidability boundary was pushed even further [BL21, Ex. 1], by allowing for mixing modalities in visibly-pushdown expressions (for instance, allowing the user to specify that "for all positive integers n ∈ N, all t-successors of r n -reachable elements can s n -reach an element fulfilling φ").
Our motivation and our contribution.Despite the presence of a plethora of various results concerning non-regular extensions of PDL [KP83, HP84, HS96, HR93, BL21], to the best of our knowledge the extensions of non-regular PDL with popular features supported by W3C ontology languages are yet to be investigated.Such extensions include, among others, nominals (constants), inverse roles (inverse programs), functionality or counting (deterministic programs or graded modalities), and the Self operator (self-loops).The honourable exception is the unpublished undecidability result for ALC reg extended with the language {r n s(r − ) n | n ∈ N}, where r − denotes the converse of r , from Göller's thesis [Göl08] (answering an open problem of Demri [Dem07, Probléme ouvert 29]).The lack of results on entailment of non-regular queries over ontologies is also intriguing, taking into account positive results for conjunctive visibly-pushdown queries in the setting of relational-databases [LL15,Thm. 2].
In this paper we contribute to the further understanding of the aforementioned questions.Our results are negative.Section 3 establishes undecidability of the concept satisfiability of ALC vpl extended with the seemingly innocent Self operator.Section 4 establishes undecidability of the concept satisfiability of ALC vpl extended with nominals.More specifically, the undecidability arises already if the only non-regular language present in concepts is r # s # .Finally, Section 5 establishes undecidability of the query entailment problem over ALC-TBoxes, in which our queries can employ atoms involving the language r # s # .On the positive side, a combination of existing works shows that the entailment problem of REG-PEQs over ALC vpl -TBoxes is 2ExpTime-complete.We conclude with an open problem list.This work is a revised and extended version of our JELIA 2023 paper [Bed23].In addition to the inclusion of full proofs and minor corrections, the paper contains improved and new undecidability results concerning entailment of non-regular queries (Section 5).
Basics on ALC.We recall ingredients of the well-known description logic ALC [BHLS17, Sec.2.1-2.3]. 1 We fix countably infinite pairwise disjoint sets of individual names N I , concept names N C , and role names N R and introduce the description logic ALC.Starting from N C and N R , the set C of ALC-concepts is built using the following concept constructors: A knowledge base (KB) K := (A, T ) is composed of a finite set A (ABox ) of assertions and a finite set T (TBox ) of GCIs.We call the elements of A ∪ T axioms.
The semantics of ALC is defined via interpretations I := (∆ I , • I ) composed of a nonempty set ∆ I , called the domain of I and an interpretation function • I , mapping individual names to elements of ∆ I , concept names to subsets of ∆ I , and role names to subsets of ∆ I × ∆ I .This mapping is then extended to concepts (see Table 1 if it has a model, and inconsistent (or unsatisfiable) otherwise.In the consistency (or satisfiability) problem we ask, whether an input concept has a model.Similarly, we say that I satisfies a KB K := (A, T ) (written: I |= K) if I satisfies all axioms of A ∪ T .The knowledge base satisfiability problem is defined analogously.
1 To guide people with modal logic background: concept names are propositional letters, interpretations are Kripke structures, ABoxes (a.k.a.Assertion Boxes) are partial descriptions of Kripke structures, and TBoxes (a.k.a.Terminological Boxes) are limited forms of the universal modality.Consult Section 2.6.2 of the DL Book [BHLS17] for a more detailed comparison.
We say that ρ starts from (resp.ends in) ALC with extras.In the next sections we consider several popular description-logics features: nominals (O), and the Self operator (• Self ).Their semantics is recalled in Table 2, in which r ∈ N R , ℓ ∈ N, a ∈ N I , and C is a concept.
as an infinite alphabet.Let ALL and REG be classes of all Turing-recognizable (resp.all regular) languages of finite words over any finite subset of Σ all .Given a language L ∈ ALL and a path ρ := ρ 1 ρ 2 . . .ρ n ρ n+1 in an interpretation I, we say that ρ is an L-path, if there exists a word w := w 1 w 2 . . .w n ∈ L such that for all indices i ≤ n we have either (i) Intuitively the word w either traverses roles between the elements of ρ or loops at an element to check the satisfaction of concepts.For convenience, we will also say that e ∈ ∆ I is L-reachable from d ∈ ∆ I (or alternatively that d L-reaches e) if there is an L-path ρ that starts from d and ends in e.The logic ALC all extends ALC with concept constructors of the form ∃L.C, where L ∈ ALL and C is an ALC all -concept.Their semantics is as follows: (∃L.C) I is the set of all d ∈ ∆ I that can L-reach some e ∈ C I , and ∀L.C stands for ¬∃L.¬C.The logic ALC reg is a restriction of ALC all to regular languages.It is a notational variant of Propositional Dynamic Logic [FL79], popular in the community of formal verification.
Visibly-pushdown languages.The class VPL of Visibly-pushdown languages [AM09, Sec.5] (VPLs) is a well-behaved family of context-free languages, in which the usage of the stack in the underlying automaton model is input-driven.For the exposition of VPLs we follow the work of Löding et al. [LLS07, Sec.2.2].A pushdown alphabet Σ is an alphabet equipped with a partition (Σ c , Σ i , Σ r ).The elements of Σ c , Σ i , and Σ r are called, respectively, call letters, internal letters, and return letters.A visibly-pushdown automaton (VPA) A over a pushdown alphabet Σ is a tuple (Q, I, F, Γ, T), where Q is a finite set of states, I is a finite subset of initial states, F is a finite subset of final states, Γ is a finite stack alphabet that contains a bottom-of-stack symbol ¡, and T is a transition relation of type A configuration of a VPA A is a pair (q, σ) ∈ Q × (Γ \ ¡) * ¡ of a state q and a stack content σ.Given a letter a and a configuration (q, σ) we say that (q ′ , σ ′ ) is an a-successor of (q, σ) if one of the following cases hold: • a ∈ Σ c , σ ′ = γσ and there is a transition (q, a, q ′ , γ) ∈ T.
We denote this fact by (q, σ) → a (q ′ , σ ′ ).Given a finite word w := a 1 . . .a n , a run of A on w is a sequence (q 0 , ¡) → a 1 (q 1 , σ 1 ) → a 2 . . .→ an (q n , σ n ) where q 0 ∈ I.We call w accepted by A if there is a run in which the last configuration contains a final state.The language L(A) of A is composed of all words accepted by A. A language L (i.e. a set of words) over Σ is visibly-pushdown if there is a VPA A over Σ for which L(A) = L.
Example 2.1.Suppose that r is a call letter and s is a return letter.Then the languages r # s # := {r n s n | n ∈ N} and r # s ># := {r n s n+m+1 | n, m ∈ N} are visibly-pushdown.Under such a choice of r and s, the language s # r # is not visibly-pushdown.Moreover, every regular language is visibly-pushdown.
The logic ALC vpl .Throughout the paper, Σ all is presented as a pushdown alphabet where the sets (N R ) c , (N R ) i , (N R ) r form a partition of N R .Hence, we define ALC vpl as the restriction of ALC all to visibly-pushdown languages over finite subsets of Σ vpl , in which languages in existential and universal restrictions are represented by means of nondeterministic VPA.The logic ALC vpl generalises many other logics with non-regular path expressions, and has a 2ExpTime-complete [LLS07, Thm.18-19] concept satisfiability problem.As a special case of ALC vpl , we also consider a, rather minimalistic, extension of ALC reg with the language r # s # for one fixed call letter r ∈ (N R ) c and one fixed return letter s ∈ (N R ) r .We denote it here by ALC r #s# reg .The concept satisfiability problem for ALC r #s# reg was shown to be decidable [KP83, Sec.Decidability] already 40 years ago by Koren and Pnueli, but its extensions with popular features like nominals or functionality are still unexplored.
The query entailment problem.Given a class C of languages, the class of C-enriched Positive Existential Queries (abbreviated as C-PEQs) is defined with the following syntax: and x , y are variables from a countably-infinite set N V .Their semantics is defined as expected, e.g.L(x , y) evaluates to true under a variable assignment η : N V → ∆ I if and only if η(x ) can L-reach η(y) in I. ∅-PEQs (or Positive Existential Queries) are a well-known generalization of unions of conjunctive queries (in which disjunction is allowed only at the outermost level).REG-PEQs (or Positive Conjunctive Regular Path Queries) are among the most popular query languages nowadays [FLS98,OS12].Finally, VPL-PEQs recently received some attention in [LL15].A C-CQ is a disjunction-free C-PEQ.We say that an interpretation I satisfies a query q (written I |= q), if there exists an assignment η of variables from q to ∆ I under which q evaluates to true (a match).A concept C entails a query q (written C |= q) if every model of C satisfies q.In total analogy we define the notion of entailment and overload the "|=" symbol for the case of TBoxes and knowledge-bases in place of concepts.For a given description logic DL, the C-PEQ entailment problem over DL-knowledge-bases asks to decide if an input DL-knowledge-base entails an input C-PEQ.The problem's definition for concepts and TBoxes is analogous.
Tree models.Löding et al. established that ALC vpl possesses a tree-model property.An interpretation I is tree-like if its domain is a prefix-closed subset of N * , and for all r ∈ N R and d, e ∈ ∆ I the condition "if (d, e) ∈ r I , then e = dn for some n ∈ N" holds.An interpretation is single-role if any two domain elements are connected by at most one role.The following lemma follows immediately from the proof of Proposition 8 by Löding et al. [LLS07].
Corollary 2.2 (Consequence of the proof of Prop.8 of [LLS07]).Every satisfiable ALC vpl -TBox T has a single-role tree-like model. 2 Moreover, for any query q preserved under homomorphisms we have that T ̸ |= q if and only if there exists a single-role tree-like model I |= T such that I ̸ |= q.
Undecidability results for extensions of ALC vpl that follow from the literature.Other popular (not yet mentioned) features supported by W3C ontology languages are inverse roles and role hierarchies.For bibliographical purposes, we would like to use the extra space given here to briefly discuss how these features result in the undecidability of the respective extensions of ALC vpl .To define the first feature, we associate with each role name r ∈ N R a role name r − that is enforced to be interpreted as the inverse of the interpretation of r .It was shown by Stefan Göller in his PhD thesis [Göl08, Prop.2.32] that ALC reg extended with the single visibly-pushdown language {r n s(r − ) n | n ∈ N} is undecidable.
Corollary 2.3.The concept satisfiability problem for ALC vpl with inverses is undecidable, even if the only allowed non-regular language is {r n s(r The second feature allows for specifying containment of roles by means of expressions of the form for r ⊆ s.The semantics is that I |= r ⊆ s if and only if r I ⊆ s I .Fix r and r ′ to be call letters, and s and s ′ to be return letters.Suppose that an interpretation I satisfies all of the statements s ⊆ r ′ , r ′ ⊆ s, s ′ ⊆ r , and r ⊆ s ′ .Clearly, for all elements d ∈ ∆ I and concepts C we have that d ∈ (∃s # r # .C) I if and only if d ∈ (∃r ′ # s ′ # .C) I .Thus ALC vpl with role-hierarchies can express concepts of ALC reg extended with both r # s # and s # r # .By undecidability of the concept satisfiability [KP83, Sec.Decidability] of the latter we conclude: Corollary 2.4.The concept satisfiability problem for ALC vpl with role-hierarchies is undecidable, even if the only allowed non-regular languages are r # s # and r ′ # s ′ # for fixed call letters r , r ′ and return letters s, s ′ .

Negative results I: The Seemingly innocent Self operator
We start our series of negative results, by showing (in our opinion) a rather surprising undecidability result.Henceforth we employ the Self operator, a modelling feature supported by two profiles of the OWL 2 Web Ontology Language [HKS06,KRH08].Recall that the Self operator allows us to specify the situation when an element is related to itself by a binary relationship, i.e. we interpret the concept ∃r .Self in an interpretation I as the set of all those elements d for which (d, d) belongs to r I .We provide a reduction from 2 The original work considers concepts only.However, all their results transfer immediately to the case of TBoxes, as TBoxes can be internalised in concepts in the presence of regular expressions [BCM + 03, p. 186].The queries are not mentioned either: the so-called tree model property is established with a suitable notion of unravelling, which produces interpretations that can be then homomorphically mapped to the original interpretations (entailing the preservation of the non-satisfaction of query).
the undecidable problem of non-emptiness of the intersection of deterministic one-counter automata (DOCA) [Val73,p. 75].Such an automaton model is similar to pushdown automata, but its stack alphabet is single-letter only.The Self operator will be especially useful to introduce "disjunction" to paths.
Let Σ be an alphabet and w := (a 1 , b 1 ) . . .(a n , b n ) be a word over Σ × {c, r, i}.We call the word π 1 (w) := a 1 . . .a n from Σ * the projection of w.An important property of DOCAs is that they can be made visibly pushdown in the following sense.
Lemma 3.1.For any deterministic one-counter automaton A over the alphabet Σ, we can construct a visibly-pushdown automaton Ã over a pushdown alphabet Σ := (Σ × {c}, (Σ × {i}) ∪ {x}, Σ × {r}) with a fresh internal letter x such that all words in L( Ã) are of the form ã1 x ã2 x . . .x ãn for ã1 , . . ., ãn ∈ Σ × {c, i, r}, and Proof sketch.Alur and Madhusudan proved [AM09, Thm.5.2] that for any context-free language L over Σ there exists a VPL L, over the pushdown alphabet (Σ × {c}, (Σ × {i}), Σ × {r}), for which L = {π 1 (w) | w ∈ L} holds. 3Suppose now that a one-counter automaton A is given.By means of the previous construction, we obtain a visibly-pushdown automaton Â := (Q, I, F, Γ, T) for which L(A) = {π 1 (w) | w ∈ L( Â)} holds.What remains to be done is to "insert" the internal letter x after every position of a word accepted by Â.As reading internal letters by visibly-pushdown automata do not affect the content of their stacks, we may proceed as in standard constructions from the theory of regular languages [Sip13,Ex. 1.31].As the first step, we expand the set of states Q with fresh states of the form q δ for all δ ∈ T. As the second step, we "split" every transition δ in T into two "parts".Suppose that δ leads from q to q ′ after reading the letter a.We thus (i) replace δ in T with the transition that transforms q into q δ after reading a (and has the same effect on the stack as δ has), and (ii) append the transition (q δ , x, q ′ ) to T. Call the resulting automaton Ã.It can now be readily verified that L( Ã) = { â1 x â2 x . . .x ân | ŵ := â1 . . .ân , ŵ ∈ L( Â)} holds, and thus, by the relationship between A and Â, the automaton Ã is as desired.
Let us fix a finite alphabet Σ ⊆ N R .We also fix two deterministic one-counter automata A 1 and A 2 over Σ, and let C 1 and C 2 be deterministic one-counter automata recognizing the complement of the languages of A 1 and A 2 (they exist as DOCA are closed under complement [Val73,p. 76]).Finally, we apply Lemma 3.1 to construct their visiblypushdown counterparts Ã1 , Ã2 , C1 , C2 over the same pushdown alphabet Σ.We stress that the letter x, playing the role of a "separator", is identical for all of the aforementioned visibly-pushdown automata.Moreover, note that the non-emptiness of L( Ã1 ) ∩ L( Ã2 ) is not equivalent to the non-emptiness of L(A 1 ) ∩ L(A 1 ), as the projection of a letter a ∈ Σ may be used by A 1 and A 2 in different contexts (e.g. both as a call or as a return).
We are going to encode words accepted by one-counter automata by means of word-like interpretations.A pointed interpretation (I, d) is Σ-friendly if for every element e ∈ ∆ I that is x * -reachable from d in I there exists a unique letter a ∈ Σ so that e carries all ã-self-loops for all ã ∈ Σ with π 1 (ã) = a, and no self-loops for all other letters in Σ (also including x).
Observe that Σ-friendly interpretations can be axiomatised with an ALC Self -concept C Σ fr : Moreover, every x * -path ρ in a Σ-friendly (I, d) represents a word in Σ * in the following sense: the i-th letter of such a word is a if and only if the i-th element of the path carries an (a, c)self-loop.This is well-defined, by the fact that every x * -reachable element in Σ-friendly (I, d) carries a (a, c)-self-loop for a unique letter a ∈ Σ. Consult Figure 1 for a visualization.As a special class of Σ-friendly interpretations we consider Σ-metawords.We say that (I, d) is a Σ-metaword if it is a Σ-friendly interpretation of the domain Z n for some positive n ∈ N, the role name x is interpreted as the set {(i, i+1) | 0 ≤ i ≤ n−2}, and all other role names are either interpreted as ∅ or are subsets of the diagonal {(i, i) | i ∈ Z n } (or, put differently, they appear only as self-loops).The example Σ-friendly (I, d) from Figure 1 is actually a Σ-metaword.Note that for every word w ∈ Σ + there is a Σ-metaword representing w.A crucial observation regarding Σ-metawords is as follows.
Observation 3.2.Let ℓ ∈ {1, 2}, Σ-metaword (I, d), and w be in the language of Ãℓ .Then the element d can {w}-reach an element e via a path ρ if and only if for all odd indices i we have ρ i = ρ i+1 and for all even indices i we have As the next step of the construction, we are going to decorate Σ-friendly interpretations with extra information on whether or not words represented by paths are accepted by A 1 .This is achieved by means of the following concept , for a fresh concept name Acc A 1 .We define the concept C A 2 analogously.We have that: then (I, d) is Σ-friendly and for every element e ∈ ∆ I that is x * -reachable from d via a path ρ we have e ∈ (Acc A ℓ ) I if and only if the Σ-word represented by ρ belongs to L(A ℓ ).Moreover, after reinterpreting the concept name Acc A ℓ , every Σ-metaword becomes a model of C A ℓ .
Proof.The proof relies on Observation 3.2.Suppose that (I, d) is a pointed interpretation and d ∈ (C A ℓ ) I .We know that (I, d) is Σ-friendly by the satisfaction of C Σ fr .For the remainder of the proof, consider any element e ∈ ∆ I that is x * -reachable from d, say, via a path ρ := ρ 1 . . .ρ n .Let w be the word represented by ρ.This implies, that for every index i, the element ρ i in I is equipped with a family of self-loops involving (a decorated) letter w i .We consider two cases: • Assume that e ∈ (Acc A ℓ ) I .We will show that w ∈ L(A ℓ ).Ad absurdum, suppose that w ̸ ∈ L(A ℓ ).Then, by definition of C ℓ , we have that w belongs to L(C ℓ ).By the construction of Cℓ there exists a sequence ⋆ 1 , . . ., ⋆ n ∈ {c, i, r}, for which the word u := (w 1 , ⋆ 1 )x . . .x(w n , ⋆ n ) is accepted by Cℓ .But then the path ρ ′ := ρ 1 ρ 1 ρ 2 ρ 2 . . .ρ n ρ n witnesses {u}-reachability (and thus L( Cℓ )-reachability) of e from d. Due to the satisfaction of ∀L( Cℓ ).¬AccA ℓ by (I, d) we infer e ∈ (¬Acc A ℓ ) I = ∆ I \ (Acc A ℓ ) I .A contradiction. • Assume that w ∈ L(A ℓ ).We proceed analogously to the previous case.By the construction of Ãℓ there exists a sequence ⋆ 1 , . . ., ⋆ n ∈ {c, i, r}, for which the word u := (w 1 , ⋆ 1 )x . . .x(w n , ⋆ n ) is accepted by Ãℓ .Once again, the path ρ ′ := ρ 1 ρ 1 ρ 2 ρ 2 . . .ρ n ρ n witnesses {u}-reachability (and thus L( Ãℓ )-reachability) of e from d. Due to the satisfaction of ∀L( Ãℓ ).Acc A ℓ by (I, d) we infer e ∈ (Acc A ℓ ) I , as desired.For the last statement of the proof, take a Σ-metaword I that represents a word w ∈ Σ * .We alter the interpretation of the concept name Acc A ℓ in I so that (Acc Equipped with Lemma 3.3, we are ready to prove correctness of our reduction. Proof.For one direction, take w ∈ L(A 1 )∩L(A 2 ).Let (I, d) be a Σ-metaword representing w.By Lemma 3.3 we decorate I with concepts Acc A 1 and Acc A 2 so that (I, d) |= C A 1 ⊓C A 2 , and the interpretations of concepts Acc A ℓ contain precisely the elements k for which the k-letter prefix of w belongs to L(A ℓ ).In particular, this means that (|w|−1) ∈ (Acc A 1 ⊓ Acc A 2 ) I .As (|w|−1) is x * -reachable from 0, we conclude the satisfaction of ∃x * .(Acc A 1 ⊓ Acc A 2 ) by (I, 0).Hence, the concept from the statement of Lemma 3.4 is indeed satisfiable.
For the other direction, assume that (I, d) is a model of C A 1 ⊓C A 2 ⊓∃x * .(Acc A 1 ⊓ Acc A 2 ).Then there exists an x * -path ρ from d to some e ∈ (Acc A 1 ⊓Acc A 2 ) I .Hence, by Lemma 3.3, for all ℓ ∈ {1, 2} the word represented by ρ belongs to L(A ℓ ), and thus also L(A 1 )∩L(A 2 ).
By the undecidability of the non-emptiness problem for intersection of one-counter languages [Val73, p. 75], we conclude Theorem 3.5.
Theorem 3.5.The concept satisfiability problem for ALC Self vpl is undecidable, even if only visibly-pushdown languages that are encodings of DOCA languages are allowed in concepts.
There is nothing special about deterministic one-counter automata used in the proof.In fact, any automaton model would satisfy our needs as long as it would (i) have an undecidable non-emptiness problem for the intersection of languages, (ii) enjoy the analogue of Lemma 3.1, and (iii) be closed under complement.We leave it is an open problem to see if there exists a single visibly-pushdown language L that makes the concept satisfiability of ALC Self reg extended with L undecidable.For instance, the decidability status of ALC r #s# reg with Self is open.
4. Negative results II: Nominals meet r # s # We next provide an undecidability proof for the concept satisfiability problem for ALCO r #s# reg .To achieve this, we employ a slight variant of the classical domino tiling problem [Wan61].
A domino tiling system is a triple D := (Col, T, ), where Col is a finite set of colours, T ⊆ Col 4 is a set of 4-sided tiles, and ∈ Col is a distinguished colour called white.For brevity, we call a tile (c l , c d , c r , c u ) ∈ T (i) left-border if c l = , (ii) down-border if c d = , (iii) right-border if c r = , and (iv) up-border if c u = .We also say that tiles t := (c l , c d , c r , c u ) and t We say that D covers Z n ×Z m (where n and m are positive integers) if there exists a mapping ξ : Z n × Z m → T such that for all pairs (x, y) ∈ Z n × Z m with ξ(x, y) := (c l , c d , c r , c u ) the following conditions are satisfied: (TBorders):  ru I lu I W.l.o.g.we assume that T does not contain tiles having more than 2 white sides.A system D is solvable if there exist positive integers n, m ∈ N for which D covers Z n × Z m .The problem of deciding whether an input domino tiling system is solvable is undecidable, which can be shown by a minor modification of classical undecidability proofs [PH23, Lemma 3.9] [Boa97].For a domino tiling system D := (Col, T, ) we employ fresh concept names from C T := {C t | t ∈ T} to encode mappings ξ from some Z n × Z m to T in interpretations I as certain r + -paths ρ from ld I to ru I passing through rd (SLen): There exists a unique positive integer N such that all r + -paths between ld I and rd I are of length N−1.Moreover, rd I is the only element r N−1 -reachable from ld I .(SVerti): For all elements d that are r * -reachable from ld I and labelled by some t ∈ T that is not up-border, we have that (a) there exists a tile t ′ ∈ T such that all elements e r Nreachable (for N guaranteed by (SLen)) from d carry t ′ , (b) t and t ′ are V-compatible, (c) t is left-border (resp.right-border) if and only if t ′ is.Note that tiles are not "deterministic" in the following sense: it could happen that two elements carry the same tile but tiles of their (horizontal or vertical) successors do not coincide.
If I satisfies all but the last two conditions, we call it a D-pseudosnake.The key properties of our encoding are extracted and established in Lemmas 4.3-4.4.Thus I is an n • m element r + -path, labelled accordingly to ξ.As ξ respects (TBorders), (THori) and (TVerti), we can readily verify that I is indeed a D-snake.The only case that requires treatment, is to verify the satisfaction of (SHori) for elements of the form d := (n−1) + y • n for some y ∈ N.Then, by (TBorders), d carries a right-border tile and its r -successor d ′ := 0+(y+1)•n carries a left-border tile.Hence, their tiles are H-compatible.
Lemma 4.4.If there exists a D-snake for a domino tiling system D, then D is solvable.
Proof.Suppose that I is a D-snake.Let ρ := ρ 1 . . .ρ |ρ| be the path guaranteed by (SPath) and let N be the integer guaranteed by (SLen).We show by induction that |ρ| is divisible by N. The inductive assumption states that for all integers k ∈ N with k • N ≤ ∥ρ∥ we have: (i) ρ k•N+1 carries a left-border tile, (ii) There is no 2 ≤ i < N such that ρ k•N+i carries a left-border tile or a right-border tile, (iii) ρ k•N+N carries a right-border tile.Then by Property (iii) and the fact that ρ |ρ| (equal to ru I by (SPath)) carries a right-border tile (by Property (d) of (SSpecTil)), we can conclude that |ρ| is indeed divisible by N.
We heavily rely on the fact that every element of ρ is labelled by precisely one tile, which is due to (SUniqTil).We start with the case of k = 0. Then ρ 0•N+1 = ρ 1 is equal to ld I , by (SPath).Moreover, ρ 1 is labelled with a left-border tile, by Property (a) of (SSpecTil).What is more, ρ 0•N+N = ρ N is equal to rd I (by (SLen)), which carries a right-border tile by Property (b) of (SSpecTil).This resolves Properties (i) and (iii).To establish Property (ii), assume towards a contradiction that there is i between 2 and N for which ρ i carries a left-border tile (the proof for a right-border tile is analogous).Take the smallest such i.By (SHori) we infer that ρ (i−1) carries a right-border tile.In particular, this means that i > 2 because the tile carried by ρ 1 is not right-border.By exhaustive application of (SHori) and the fact that the tile of ρ 1 is down-border, we deduce that the tile of ρ (i−1) is also down-border.Hence, by Property (b) of (SSpecTil) we have that ρ (i−1) is equal to rd I .But then the path ρ (i−1) ρ i . . .ρ N witnesses r + -reachability of rd I from itself, which is forbidden by (SNoLoop).A contradiction.For the inductive step, assume that Properties (i)-(iii) hold true for some k, and consider the case of k+1.Note that Property (i) follows from Property (iii) of the inductive assumption by (SHori).We next show that Property (ii) holds.Assume ad absurdum that there is i for which ρ (k+1)•N+i carries a left-border (resp.right-border) tile.But then, invoking Item (c) of (SVerti), we infer that ρ k•N+i is also leftborder (resp.right-border).This contradicts Property (ii) of the inductive assumption.Hence Property (ii) holds true.By inductive assumption, we know that ρ k•N+N carries a rightborder tile.Then, we apply Property (c) of (SVerti) to infer that ρ k•N+N+N = ρ (k+1)•N+N is right-border, as desired.This establishes Property (iii), and concludes the induction.
Let M := |ρ|/N.By the previous claim, we know that M ∈ N. Consider a function ξ : Z N × Z M → T that maps all (x, y) to the unique tile carried by ρ x+N•y+1 .(Note that we number paths from 1!) This function is well-defined by (SUniqTil) and it satisfies (THori) and (TVerti) due to the satisfaction of (SHori) and (SVerti).The satisfaction of the first two statements of (TBorders) by ξ is guaranteed by Properties (i)-(iii) from the induction above.Finally, the last two statements of (TBorders) are due to straightforward induction that employs (SSpecTil) and the last statement of (SHori).As we proved that ξ covers Z N × Z M , we conclude that D is indeed solvable.
While D-snakes do not seem to be directly axiomatizable even in ALC vpl , we at least see how to express D-pseudosnakes in ALCO r #s# reg .The next lemma is routine.We can now define C D as the conjunction of all the concept definitions presented above.It follows immediately from the semantics of ALCO r #s# reg that the presented concept definition is consistent if and only there exists an element starting a pseudosnake.
Note that the property that pseudosnakes are missing in order to be proper snakes, is the ability to measure.We tackle this issue by introducing a gadget called a "yardstick".The forthcoming lemma explains the name "yardstick".Intuitively it says that in any T-yardstick I, all s * -paths from md I to all end I t have equal length.Lemma 4.7.Let I be a T-yardstick.Then there exists a unique positive integer N such that: (i) for all t ∈ T we have that end I t is s N -reachable from md I , and (ii) for all t ∈ T we have that end I t is s N−1 -reachable from md I t .We will call N the length of I.
Proof.Fix t ⋆ ∈ T. By (YEqDst) we know that st I r # s # -reaches end I t⋆ , and let ρ := ρ 1 . . .ρ 2N+1 be a path witnessing it.We claim that this is the desired length of I. First, note that N > 0 by (YNom).Second, by the semantics of r # s # , for all i ≤ N we have (ρ i , ρ i+1 ) ∈ r I and (ρ N+i , ρ N+i+1 ) ∈ s I .Thus ρ N+1 is r * -reachable from st I and has an s-successor.These two facts imply (by (YMid)) that ρ N+1 is equal to md I .It remains to show that all the paths leading from md I to some end t are of length N. Towards a contradiction, assume that there is t ′ ∈ T and an integer M ̸ = N such that md I s M -reaches end I t ′ via a path ρ ′ := ρ ′ 1 . . .ρ ′ M .We stress that ρ ′ 1 = md I and ρ ′ M = end I t ′ (by design of ρ ′ ), and (by a conjunction of (YSuccOfMid) and (YReachMidT)).To conclude the proof, it suffices to resolve the following two cases.
• Suppose that M > N. Consider a path ρ 1 . . .ρ N ρ ′ 1 . . .ρ ′ N .By construction, such a path witnesses the fact that st I (r N s N )-reaches (and thus also r t ′ (otherwise we would have a contradiction with (YReachMidT)).But then end I t ′ s + -reaches itself via a path ρ ′ N . . .ρ M , which is of positive length by the fact that M > N.This yields a contradiction with (YNoLoop).
The next lemma proves the existence of arbitrary large yardsticks.
Lemma 4.8.For every finite non-empty set T and a positive integer N, there exists a T-yardstick of length N.
Proof.Consider the following interpretation I with ∆ I := Z N ∪ {N} ∪ (Z N−1 × T): • st I := 0, md I := N, (md t ) I := (0, t), (end t ) I := (N−1, t) for all t ∈ T, and a I := 0 for all names a ∈ N I \ N T .and (r ′ ) I = ∅ for all role names r ′ ∈ N R \ {r , s}.An example such I for N = 3 and T = {♡, ♠} is depicted above (in restriction to grey nodes only).It is routine to verify that I satisfies all the properties in Definition 4.6, as well as conditions (i) and (ii) from the statement of Lemma 4.7.This concludes the proof.
To make use of yardsticks in our proofs, we need to axiomatise them inside ALCO r #s# reg .Lemma 4.9.There exists an ALCO r #s# reg -concept C T , that employs only role names r , s and individual names from N T , such that for all interpretations I we have: Proof.The following concepts are written from the point of view of the interpretation of st.
We define C T as the conjunction of {st} and all the concept definitions presented above.By semantics of ALCO r #s# reg we have that C T is consistent if and only if (C T ) I = {st I }.
We next put pseudosnakes and yardsticks together, obtaining metricobras.The intuition behind their construction is fairly simple: (i) we take a disjoint union of a pseudosnake and a yardstick, (ii) we then connect (via the role s) every element carrying a tile t with the interpretation of the corresponding nominal md t , and finally (iii) we synchronise the length of the underlying yardstick, say N, with the length of the path between the interpretations of ld and rd.After such "merging", retrieving (SVerti) is easy: rather than testing if every N-reachable element from some d carries a suitable tile t (for an a priori unknown N) we can check instead whether d can r # s # -reach the interpretation of end t .
A formal definition and a picture come next.Lemma 5.2.The white-bordered Octant Tiling Problem is undecidable.
Proof.We say that a domino tiling system D almost covers the octant if there is a mapping ξ : O → T such that all pairs (n, m) ∈ O fulfil (OVerti) and (OHori).It was stated by Bresolin et al. [BDG + 10, Sec.3.1] (and follows from the classical work of van Emde Boas [Boa97]) that deciding whether D, that does not contain white-bordered tiles, almost covers the octant is undecidable.To prove undecidability of the White-bordered Octant Tiling Problem we provide a reduction from the tiling problem of Bresolin et al.Thus, let D := (Col, T, ) be a domino tiling system that does not have white-bordered tiles, and consider D ′ := (Col, T ′ , ) to be the tiling system obtained by putting T We claim that D ′ covers the octant if and only if D almost covers the octant.Indeed: • The "if" direction is relatively straightforward.Let ξ be a map witnessing that D almost covers the octant.We alter the tiles assigned by ξ to the diagonal to make them leftand up-border, then we shift ξ right, and fill the remaining places with .Formally, let ξ ′ be defined as: (i) ξ ′ (i, i) := for all i ∈ N, (ii) ξ ′ (i, i−1) := ( , c d , c r , ), where ξ(i, i) = (c l , c d , c r , c u ), for all positive i ∈ N, and (iii) ξ ′ (i, j) = ξ ′ (i, j−1) for all remaining i, j ∈ N. It can be readily verified that ξ ′ satisfies the required conditions.• For the "only if" direction, let ξ ′ be a map witnessing that D ′ covers the octant.It suffices to take ξ : (i, j) → ξ ′ (i+2, j).By Observation 5.1) none of the tiles assigned by ξ is white-bordered.As ξ ′ covers the octant, we infer that ξ (almost) covers the octant.
Our undecidability proof relies on concepts from C T and the non-regular language r # s # .We fix and enumerate a set of tiles T as t 1 , . . ., t N for N := |T|.As the first building block, we introduce octant interpretations.An interpretation I is called T-octant if there exists a function ξ : O → T fulling (OInit) and (OVerti) (but not necessarily (OHori)) that satisfies the following conditions: (i) ∆ I = O, (ii) r I = {((n, 0), (n+1, 0)) | n ∈ N}, (iii) s I = {((n, m), (n, m+1)) | n, m ∈ N, m < n}, and (iv) C I t = {(n, m) | ξ(n, m) = t} for all tiles t ∈ T. In this case I represents ξ.Due to the fact that ξ is a function, every domain element carries precisely one tile.Moreover, for every such ξ we can easily find a T-octant representing ξ (just employ the above definition).For more intuitions consult Figure 3.
To avoid disjunction in the forthcoming query, we need to extend octant interpretations with yet another way of representing tiles, which will be based on distances.Suppose that an element (n, m) from an octant interpretation I carries a tile t i , and that the tile assigned to its horizontal predecessor (n−1, m), if exists, is equal to t j .We equip (n, m) with an outgoing s + -path ρ of length N, composed of fresh elements (we employ fresh concept names In and In to make a distinction between elements in octant and the ones present in "extra paths").The i-th element of ρ will be the unique element of ρ that belongs to the interpretation of a concept Cur.Similarly, the j-th element of ρ will be the unique element of ρ that belongs to the interpretation of a concept Prev.Thus, the distance from (n, m) to an element labelled by the concept Cur (resp.Prev) uniquely determines the tile of a current node (resp.its horizontal predecessor).As a mere technicality, needed for query design, we enrich the element (0, 0) with an incoming r + -path of length N. A formalization comes next.
Definition 5.3.Let I be an interpretation with a domain O×Z N+1 ∪{(−i−1, 0, 0) | i ∈ Z N }, and let I O be the restriction of I to the set O × {0}.We call I a T-hyperoctant if: and • for every (n, m) ∈ O there is precisely one positive k for which (n, m, k) ∈ Prev I holds, and for such a number k we have that t k and the tile carried by (n, m, 0) are H-compatible.Note that, in addition to what is present in the definition of T-hyperoctant, we employed fresh concept names, namely In, Prev, Cur, In, and Prev.The purpose of "overlined" concepts is to help with a design of a query, as the use of negation is not allowed there.We call I proper if for all (n, m, k) ∈ Cur I we have (n+1, m, k) ∈ Prev I .Note that properness is not definable in ALC, but we will enforce it with a query.The map ξ : O → T represented by a hyperoctant I is the map represented by its T-octant substructure I O .Proof.Take a proper T-hyperoctant I, and the map ξ represented by I.By definition of a T-octant, we have that ξ satisfies (OInit) and (OVerti).To establish (OHori) take any (n, m) ∈ O, and suppose that ξ(n, m) = t i and ξ(n+1, m) = t j hold for some integers i, j ∈ N. By definition of a hyperoctant and properness of I, we have that (n+1, m, 0) carries the tile t j , and (n+1, m, i) ∈ Prev I .Thus, by the 6th item of Definition 5.3 we infer that tiles t i and t j are H-compatible.The proof of the other statement is routine: take I O to be a T-octant representing ξ.W.l.o.g.assume that I O interprets all role names and concept names that are not mentioned in its definitions as empty sets.We next rename the domain of I O to O × {0} and append fresh elements to make the domain equal to Then, we enlarge the interpretations of r , s, In, and Cur in a minimal way according to the first five items of Definition 5.3.Finally, we interpret Prev as the set composed of all triples (n+1, m, k) for all (n, m) ∈ O carrying a tile t k , and all triples (n, n, ℓ) for x ∈ N and ℓ denoting the index of in T. Call the resulting interpretation I. Clearly, I is T-hyperoctant due to the fact that ξ is a map and respects conditions (OInit), (OVerti), and (OHori).
We employ a VPL-CQ q D ▲ (u 1 , u 2 , v 1 , v 2 , w 1 , w 2 , x 1 , x 2 , y 1 , y 2 , z 1 , z 2 ) as a tool for detecting whether a given T-hyperoctant I is proper.Observe that I is not proper if and only if there is a position (n, m) ∈ O and a number 1 ≤ k ≤ N, for which we have (n, m, k) ∈ Cur I and (n+1, m, k) ̸ ∈ Prev I .This is precisely the condition that is going to be expressed with q D ▲ , informally presented at Figure 5.The intuition behind q D ▲ is as follows.
We first ensure that z 1 , z 2 are mapped to some elements representing the coordinates (n, m) and (n ′ , m ′ ) of the octant for some integers satisfying n ′ = n+1 and m = m ′ .The fact that they belong to the octant is handled by means of the In concept.The equality n ′ = n+1 is achieved by introducing variables y 1 , y 2 , stating their r -connectedness, and the s * -reachability of z 1 and z 2 from, respectively, y 1 and y 2 .Thus y 1 , y 2 are placed "at the bottom" of the variables z 1 and z 2 .Next, the equi-hight of z 1 , z 2 is ensured with extra r -connected variables x 1 , x 2 located to the left of y 1 , and non-regular atoms r # s # (x 1 , z 1 ) and r # s # (x 2 , z 2 ), enforcing equality of the distance between x i and y i , and the distance between y i and z i , for all i ∈ {1, 2}.Once we know that the variables z 1 , z 2 are mapped by a query as desired, we need to express that they violate properness of I. Recall that we want to establish (n, m, k) ∈ Cur I and (n+1, m, k) ̸ ∈ Prev I for some k.Such elements will be represented, respectively, by variables v 1 and v 2 .To express the mentioned constraint, we introduce fresh r -connected variables u 1 , u 2 that are located to the left of x 1 , x 2 , and whose distance to u 1 , u 2 will be precisely the k that we are looking for.We stress that we do not "hardcode" the value of k in the query.As the variable u 1 is free to map whenever it wants, this mimics a disjunction over possible values of k.We ensure the variables v 1 , v 2 are mapped to elements outside the octant by expressing that they are r * -reachable from the s-successors w 1 , w 2 of z 1 , z 2 , that are labelled with In.Note that just expressing that v 1 , v 2 belong to In does not suffice, as the path leading from some of z i to v i could contain elements in In (which we implicitly forbid).With non-regular atoms r # s # (u 1 , v 1 ) and r # s # (u 2 , v 2 ) we make sure that the distance between z 1 and v 1 (respectively z 2 and v 2 ) is indeed k.
As the final step of our construction, we define an ALC-TBox T D ▲ whose intended treelike models will contain T-hyperoctants.Most of the axioms written below are formalizations of straightforward properties satisfied by T-hyperoctants and grids.As we are aiming to design an ALC-TBox, all the properties expressed in T D ▲ are interpreted "globally".Hence, to express existence of a starting point of a T-hyperoctant we employ a fresh role name aux and say that every element has an aux -successor (with the intended meaning that such a successors "starts" a T-hyperoctant).For brevity, we say that an element is inner (resp.outer) if it belongs (resp.does not belong) to the interpretation of In.For a language L we also say that an element e in I is L-outer-reachable from d if there exists a L-path from d to e composed solely of outer elements (with a possible exception of d).
(GStart): Every element has an aux -successor.Every such successor has an outgoing r N -path composed of outer elements that leads to an inner element carrying a tile that has an inner r -successor carrying a tile different from .(GCompl): Concept name In (resp.Prev) is interpreted as the complement of the interpretation of concept name In (resp.Prev).(GTil): Every inner element is labelled with precisely one concept name from C T , and there are no outer elements labelled with such concept names.(GSuc): If an element has an inner r -successor then such a successor also has an inner r -successor.Every inner element has an inner s-successor. 4(GPath): Every inner element has an outgoing s N -path composed solely of outer elements.(GCur): For every inner element carrying the tile t k we have that: (i) all s k -outer-reachable elements are labelled with Cur, (ii) there is no s ℓ -outer-reachable element, for ℓ ≤ N and ℓ ̸ = k, that is labelled with Cur.(GPrev): For every inner element there exists a number k ≤ N such that: (i) all s k -outerreachable elements are labelled with Prev, (ii) there is no s ℓ -outer-reachable element, for ℓ ≤ N and ℓ ̸ = k, that is labelled with Prev.(GVerti): Every pair of tiles carried by s-successors is V-compatible.(GHori): For all tiles t ∈ T and all elements carrying t that can s ℓ -outer-reach an element labelled with Prev for some ℓ, we have that t ℓ and t are H-compatible.
The definition of T D ▲ is provided in the proof of the following lemma.
It should be clear that the above GCIs formalise the required properties.
We first see that T-hyperoctant can be extended to (counter)models of T D ▲ and q D ▲ .Lemma 5.7.Every proper T-hyperoctant can be extended to a model of T D ▲ that violates q D ▲ .Proof.Let I be a proper T-hyperoctant.Consider an interpretation J with the domain ∆ J := ∆ I ∪ ((N × N) \ O) × Z N+1 defined as follows: • aux J := ∆ J × {(−N, 0, 0)}, and • For each concept name A, each tuple (n, m, k) with (n, m) ̸ ∈ O belongs A J iff (0, 0, k) ∈ A I .Intuitively, we enlarged I to a grid and filled fresh places with copies of the diagonal of I (this is needed to fulfil the second conjunct of the axiomatisation of (GSuc)).By construction of J (and the fact that I is a T-hyperoctant) it follows J is a model of T D ▲ .Call J proper if for all (n, m, k) ∈ Cur J we have (n+1, m, k) ∈ Prev J .Note that as I is proper, the above condition holds for all triples (n, m, k) with (n, m) ∈ O.For other tuples, we simply use the fact that the interpretation of concepts for (n, m, k) with (n, m) ̸ ∈ O is inherited from the elements of the form (0, 0, k).Thus J is indeed proper.Now, without any further changes, the proof of the "only if" direction of Lemma 5.5 establishes J ̸ |= q D ▲ .As a handy lemma used in the forthcoming proof, we need to establish that: Lemma 5.8.Every single-role tree-like model of T D ▲ contains a substructure that is isomorphic to a T-hyperoctant.
Proof.The proof idea is simple but tedious.We take single-role tree-like model I of T D ▲ , and select elements that will later constitute a hyperoctant branch-by-branch.Let d (−N,0,0) be any element of I which is an aux -successor of some element of I (it exists as ∆ I ̸ = ∅ and I satisfies (GStart)).Thus there exists an r * -path of elements d (−N,0,0) , d (−N+1,0,0) , . . ., d (0,0,0) , d (1,0,0) witnessing the satisfaction of (GStart), where only d (0,0,0) , d (1,0,0) belong to In J (by (GStart) and (GCompl)), d (0,0,0) carries a tile and d (1,0,0) carries a tile that differs from .After employing a routine induction, we see that the first conjunct of (GSuc) yields the existence of an infinite r * -path d (1,0,0) , d (2,0,0) , . . . of inner elements.By the second conjunct of (GSuc) we know that every d (i,0,0) has an outgoing s i -path d (i,1,0) , . . ., d (i,i,0) composed of inner elements.Finally, by (GPath), every element d (i,j,0) has an outgoing s N -path d (i,j,1) ,. .., d (i,j,N) composed of outer elements.Note that by tree-likeness of I we have that all of selected elements d (i,j,k) are pairwise-different, and any pair of elements is connected by at most one role.Let J be the restriction of I to the set of selected element, with the domain renamed with a map d (i,j,k) → (i, j, k).It follows from (GTil) and (GCompl) that J restricted to In J is a T-octant.Other properties of T-hyperoctants can be readily verified with (GCur), (GPrev), (GVerti), (GHori).Thus J is a T-hyperoctant, as desired.
The next lemma summarises our reduction.
Lemma 5.9.D covers the octant if and only if T D ▲ ̸ |= q D ▲ .Proof.Suppose that D covers the octant.Then, by Lemma 5.4, there exists a proper Thyperoctant I. Hence, from Lemma 5.7 we conclude existence of a model J of T D ▲ such that J ̸ |= q D ▲ .Thus T D ▲ ̸ |= q D ▲ .For the other direction, suppose that there exists an interpretation I such that I |= T D ▲ but I ̸ |= q D ▲ .By Corollary 2.2 we can assume that I is single-role tree-like.From Lemma 5.8 we know that I contains a substructure J that is a T-hyperoctant.As I ̸ |= q D ▲ , we know that J ̸ |= q D ▲ .Thus J is a proper T-hyperoctant.Hence, by Lemma 5.4 we conclude that D covers the octant.
As the octant tiling problem is undecidable, we conclude the main theorem of this section.
Interestingly enough5 , our proof technique can be adjusted (with little effort) to derive related results for query languages involving inverses.Before moving to the undecidability result, let us define the class of Visibly-Pushdown Path Queries [LL15, p. 330] (VPQs) as the class of single-atom VPL-CQs (i.e.VPL-CQs without conjunction).We stress that the entailment problem of such queries is decidable.The main idea is that the non-satisfaction of a VPQ of the form L(x , y) can be expressed with an ALC vpl -GCI ⊤ ⊑ ¬∃L.⊤.Indeed: Corollary 5.11.The entailment problem of VPQs over ALC vpl -TBoxes is 2ExpTimecomplete (assuming that the VPLs from the query and the TBox are over the same alphabet).
Proof.The lower bound is inherited from the concept satisfiability problem for ALC vpl [LLS07,Thm. 19].Let T be an ALC vpl -TBox and q be a VPQ of the form L(x , y).The crucial observation is that by the semantics of queries we have that T ̸ |= q if and only if T ′ := T ∪ {⊤ ⊑ ¬∃L.⊤} has a model.Hence, by a well-known internalisation of TBoxes as concepts in the presence of regular expressions [BCM + 03, p. 186], we can compute (in time polynomial w.r.t.|T |) an ALC vpl -concept C that is satisfiable if and only if T ′ is.Now it suffices to check the satisfiability of C.This in turn can be done in doubly-exponential time by the result of Löding et al. [LLS07,Thm. 18], finishing the proof.
We next extend the class of VPQs with an inverse operator, obtaining the class of Two-Way Visibly Pushdown Path Queries (2VPQs).More precisely, such queries are VPQs that allow for letters of the form r − for all role names r ∈ N R in the underlying visibly-pushdown alphabet.When evaluating 2VPQs over interpretations I, the role (r − ) I is interpreted as the set-theoretic inverse of the role r I .In what follows we sketch the proof of the fact that the entailment of 2VPQs over ALC-TBoxes is undecidable.The key ingredients of our reduction are the previously-defined ALC-TBox T D ▲ and a fresh 2VPQ q D 2VPQ := L ↓→↑ (x , y), where L ↓→↑ := Cur? In? s − In? k s (In? s) m (In? r In?) (s In?) m s In? s k In? Prev? | k, m ∈ N .
It is an easy exercise to construct a pushdown automaton for the language L ↓→↑ .Such an automaton becomes visibly-pushdown under the requirement that s and s − are, respectively, return and call symbols, and r is an internal symbol.The forthcoming Lemma 5.12 relates the queries q D ▲ and q D 2VPQ .As it can be established analogously to the other lemmas of this section, we only sketch the proof and leave some minor details to the reader.Lemma 5.12.Let D := (Col, T, ) be a domino tilling system and let I be a T-hyperoctant.We have that I |= q D ▲ if and only if I |= q D 2VPQ .Moreover, by applying the construction from Lemma 5.7, every proper T-hyperoctant can be extended to a model of T D ▲ that violates q D 2VPQ .Proof sketch.We first sketch the proof of the equivalence I |= q D ▲ if and only if I |= q D 2VPQ .First, take any match η witnessing I |= q D ▲ .Then it can be readily verified that the mapping x → η(v 1 ), y → η(v 2 ) is a match for I and q D 2VPQ .For the opposite direction, let d, e ∈ ∆ I be domain elements for which the mapping x → d, y → e is a match for I and q D 2VPQ .Hence, e is [In? s − In?
For the remaining part of the proof, take any proper T-hyperoctant I. We apply the construction from the proof of Lemma 5.7, and obtain a model J of T D ▲ that violates q D ▲ .It suffices to show that J ̸ |= q D 2VPQ .Towards a contradiction, suppose the opposite, and take any match η for J and q D 2VPQ .Then, by the construction of J and the shape of q D 2VPQ , we conclude that the image of η belongs to the T-hyperoctant part I of J .By equivalence between the queries q D 2VPQ and q D ▲ sketched above, we conclude I |= q D ▲ .A contradiction.The second auxiliary lemma has a proof analogous to the proof Lemma 5.9.Lemma 5.13.Let D := (Col, T, ) be a domino tilling system and let I be a T-hyperoctant.We have that T D ▲ ̸ |= q D 2VPQ if and only if D covers the octant.Proof.If D covers the octant, by Lemma 5.4 there exists a proper T-hyperoctant I. Applying the second part of Lemma 5.12, we extend I to a model of T D ▲ violating J ̸ |= q D 2VPQ .Thus T D ▲ ̸ |= q D 2VPQ .For the reverse direction, take any model I of T D ▲ such that I ̸ |= q D 2VPQ .By Corollary 2.2 we can assume that I is single-role tree-like.From Lemma 5.8 we know that I contains a substructure J that is a T-hyperoctant.As I ̸ |= q D 2VPQ , we know that J ̸ |= q D 2VPQ , and thus, by Lemma 5.12, we know that J ̸ |= q D ▲ .Hence, by Lemma 5.5, J is a proper T-hyperoctant.Invoking Lemma 5.4 we conclude that D covers the octant.
), and finally used to define satisfaction of assertions and GCIs.More precisely I |= C ⊑ D if C I ⊆ D I , I |= C(a) if a I ∈ C I , and I |= r (a, b) if (a I , b I ) ∈ r I .We say that an interpretation I satisfies a concept C (or I is a model of C, written:
y) and ξ(x+1, y) are H-compatible.(TVerti): If (x, y+1) ∈ Z n × Z m then ξ(x, y) and ξ(x, y+1) are V-compatible.Intuitively, ξ : Z n × Z m can be seen as a white-bordered rectangle of size n × m coloured by unit 4-sided tiles (with coordinates corresponding to the left, down, right, and upper colour) from T, where sides of tiles of consecutive squares have matching colours.Example 4.1.Suppose that Col = { , , , } and T = Col 4 .Then the map ξ and can be visualised as follows.
I and lu I (where the individual names from N T := {ld, rd, lu, ru} are fresh).Consult the figure in Example 4.1.Definition 4.2.Consider a domino tiling system D := (Col, T, ).An interpretation I is a D-snake whenever all seven criteria listed below are fulfilled: (SPath): There is an r + -path ρ that starts in ld I , then passes through rd I , then passes through lu I and finishes in ru I .More formally, there are indices 1 < i < j < |ρ| such that ρ 1 = ld I , ρ i = rd I , ρ j = lu I and ρ |ρ| = ru I .(SNoLoop): No N T -named element can r + -reach itself.(SUniqTil): For every element d that is r * -reachable from ld I there exists precisely one tile t ∈ T such that d ∈ C I t (we say that d is labelled by a tile t or that d carries t).(SSpecTil): The N T -named elements are unique elements r * -reachable from ld I that are labelled by tiles with two white sides.Moreover, we have that (a) ld I carries a tile that is left-border and down-border, (b) rd I carries a tile that is right-border and down-border, (c) lu I carries a tile that is left-border and up-border, (d) ru I carries a tile that is right-border and up-border.(SHori): For all elements d different from ru I that are r * -reachable from ld I and labelled by some tile t := (c l , c d , c r , c u ), there exists a tile t ′ := (c ′ l , c ′ d , c ′ r , c ′ u ) for which all r -successors e of d carry the tile t ′ and: (i) t, t ′ are H-compatible, (ii) if c d = then (c r ̸ = iff c ′ d = ), and (iii) if c u = then c ′ u = .
Definition 4.6.Let T be a finite non-empty set, let N T := {st, md, md t , end t | t ∈ T} be composed of (pairwise different) individual names.A T-yardstick is any interpretation I satisfying all the conditions below.(YNom): N T -named elements in I are pairwise different and (r + s) * -reachable from st I .(YNoLoop): No N T -named element can (r + s) + -reach itself.(YMid): md I is the unique element with an s-successor that is r * -reachable from st I .(YSuccOfMid): The s-successors of md I are precisely {md t | t ∈ T}-named elements.(YReachMidT): For every t ∈ T we have that md I t can s * -reach end I t but it cannot s * -reach end I t ′ for all t ′ ̸ = t.(YEqDst): The elem.r # s # -reachable from st I are precisely the {end t | t ∈ T}-named ones.(YNoEqDst): No {end t | t ∈ T}-named element is r # s # -reachable from an element (s +r ) +reachable from st I .An example {♡, ♠}-yardstick is depicted below.A "minimal" yardstick contains the grey nodes only.

Figure 2 :
Figure 2: A fragment of an example D-metricobra representing ξ from Example 4.1.The upper part corresponds to a D-snake, and the lower part corresponds to a Tyardstick.The distances between named elements are important.

Figure 4 :
Figure4: A fragment of a proper {t 1 , t 2 , t 3 }-hyperoctant for t 1 = , t 2 = , and t 3 = .Elements in In I are depicted as circles.Elements in Prev I are marked grey, elements in Cur I are marked black, and the lime elements belong to Prev I ∩ Cur I .

Table 1 :
Concepts and roles in ALC.

Table 2 :
Popular description-logics features.} self-operator ∃r .Self {d | (d, d) ∈ r I } ALC with path expressions.We treat the set Σ all