Tractable Query Answering for Expressive Ontologies and Existential Rules

Abstract

The disjunctive skolem chase is a sound and complete (albeit non-terminating) algorithm that can be used to solve conjunctive query answering over DL ontologies and programs with disjunctive existential rules. Even though acyclicity notions can be used to ensure chase termination for a large subset of real-world knowledge bases, the complexity of reasoning over acyclic theories still remains high. Hence, we study several restrictions which not only guarantee chase termination but also ensure polynomiality. We include an evaluation that shows that almost all acyclic DL ontologies do indeed satisfy these general restrictions.

M. Krötzsch—The author thanks the competent and friendly staff of trauma surgery ward OUC-S2 at the University Hospital Carl Gustav Carus, Dresden, where some of this research has been executed.

1 Introduction

Answering conjunctive queries (CQs) over knowledge bases is an important reasoning task with many applications in data management and knowledge representation. A flurry of research efforts have significantly improved our understanding of this problem, and led to different solutions for description logics (DL) ontologies [2, 6, 25] and programs with disjunctive existential rules [1, 5]. One such proposed approach is the use of acyclicity notions [9, 10, 19, 21]; i.e., sufficient conditions that guarantee termination of the disjunctive chase algorithm [3]—a sound and complete materialization-based procedure where all relevant consequences of a knowledge base are precomputed, allowing queries to be directly evaluated over materialized sets of facts. As shown in [9, 10], acyclicity notions can be used to determine that the chase will indeed terminate over a large subset of real-world DL ontologies.

Nevertheless, even if a knowledge base is characterized as acyclic, CQ answering still remains a problem of high theoretical complexity: CQ answering over acyclic programs with disjunctive existential rules is coN2ExpTime-complete [7]. For acyclic Horn-$\mathcal {SROIQ}$ ontologies, it is ExpTime-complete [10].

Fig. 1.

Graphical representation of the chase of $\mathcal {P}$.

Example 1

Let $\mathcal {R} _n = \{D_{i-1}(x) \rightarrow \exists y_i . L_i(x, y_i) \wedge D_i(y_i), D_{i-1}(x) \rightarrow \exists z_i . R_i(x, z_i) \wedge D_i(z_i) \mid i = 1, \ldots , n\}$. The chase of the program $\mathcal {P} = \langle {\mathcal {R} _n, \{D_0(c)\}}\rangle$, depicted in Fig. 1, is exponentially large in n. Note that, $\mathcal {P}$ is acyclic with respect to all notions described in [10] and can be expressed in most DL fragments.

In this paper, we study the limits of tractable reasoning using the chase and propose a series of restrictions that, if combined, prevent the exponential blow-up highlighted in the previous example. Moreover, we define a novel acyclicity notion, namely tractable acyclicity, tailored for DL ontologies, which ensures that the size of the chase stays polynomial. In turn, this implies that CQ answering over deterministic “tractably acyclic” ontologies is (theoretically) as hard as solving the same problem over a given set of facts. On the practical side, we assess the generality of tractable acyclicity using two different corpuses of real-world ontologies. As it turns out, our notion does characterize almost all acyclic ontologies, thus showing that CQ answering may be quite efficient in practice.

In summary, our main contributions are as follows:

• We consider five general restrictions on the expressivity of rules and ontologies, and thoroughly study the complexity of CQ answering when combinations of these restrictions are satisfied (Sect. 3).

• Using some of these restrictions, we define tractable acyclicity, a notion specially tailored for DL ontologies which guarantees tractability of reasoning over expressive deterministic ontologies (Sect. 4). To the best of our knowledge, the use of notion is the only approach to guarantee tractable CQ answering over ontologies besides the combined approach [12, 17, 18, 20, 25].

• We empirically study the generality of tractable acyclicity on two large corpuses of real-world ontologies with encouraging results (Sect. 5).

2 Preliminaries

Let P, V and $\mathbf {F}$ be some infinite countable and pairwise disjoint sets of predicates, variables and function symbols, respectively, such that every $S \in \mathbf P \cup \mathbf {F}$ is associated with some arity $\textit{ar}(S) \ge 0$. Constants are function symbols of arity 0. Terms are built from variables and function symbols as usual. We abbreviate a sequence of terms $t_1, \ldots , t_n$ with ${\mathbf {t}}$, and identify such a sequence with the set $\{{\mathbf {t}} \}$. An atom is a formula of the form $P({\mathbf {t}})$ with P a $\vert {\mathbf {t}} \vert$-ary predicate. With $\varphi [{\mathbf {x}} ]$ we stress that $\mathbf {x}$ are the free variables in the formula $\varphi$. We identify a conjunction of formulas with the set of all the formulas in the conjunction and vice-versa.

A (disjunctive existential) rule is a first-order logic (FOL) formula of the form

$$\begin{aligned} \forall {\mathbf {x}}, \mathbf {y}. (B [{\mathbf {x}},\mathbf {y} ]&\rightarrow \vee _{i=1}^n \exists \mathbf {v} _i.H _i[{\mathbf {x}},\mathbf {v} _i]) \end{aligned}$$

(1)

where $B$ (the body) and $H _i$ (the heads) are conjunctions of atoms with $H _i \ne \emptyset$ for all $i = 1, \ldots , n$; and $\mathbf {v} _1, \ldots , \mathbf {v} _n$, $\mathbf {y}$ and $\mathbf {x}$ are pairwise disjoint. For the sake of brevity, we omit universal quantifiers when writing rules. The variables in $\mathbf {x}$ are called frontier variables. A rule is Horn if $n=1$ and non-Horn otherwise. A fact is a ground atom; i.e., an atom without occurrences of variables. An instance $\mathcal {I}$ is a finite set of facts only containing constants as terms. A program is a pair $\langle {\mathcal {R},\mathcal {I}}\rangle$ with $\mathcal {R}$ a rule set and $\mathcal {I}$ an instance. Without loss of generality, we assume that every existentially quantified variable occurs in at most one rule ($\dagger$).

The main reasoning task we are studying in this paper is CQ answering. Nevertheless, without loss of generality, we restrict our attention to the simpler task of entailment of Boolean conjunctive queries (BCQs). A BCQ, or simply a query, is a formula of the form $\exists \mathbf {y}. Q [\mathbf {y} ]$ with $Q$ a conjunction of atoms.

A substitution is a partial function defined over the set of terms. The application of a substitution $\sigma$ to an atom $\alpha$, denoted with $\alpha \sigma$, is the atom that results from replacing all occurrences of every term t in the domain of $\sigma$ with $\sigma (t)$. We denote the substitution $\{(t_1, u_1), \ldots , (t_n, u_n)\}$ with $[t_1 / u_1, \ldots , t_n / u_n]$.

The skolemization $\textit{sk}(\rho )$ of a rule $\rho$ as in (1) is the formula $B \rightarrow \bigvee _{i = 1}^n \textit{sk}(H _i)$ where, for every $i = 1, \ldots , n$, $\textit{sk}(H _i)$ is the conjunction that results from replacing every (existentially quantified variable) $v \in \mathbf {v} _i$ by the term $f_v({\mathbf {x}})$ with $f_v$ a fresh function symbol specific to v (which, by assumption ($\dagger$) and the definition of a rule, is also specific to the i-th disjunct in the head of the rule $\rho$).

Definition 2

Consider a rule $\rho$ of the form (1), a substitution $\sigma$ defined only on ${\mathbf {x}} \cup \mathbf {y}$, and a set of facts $\mathcal {F}$. Then, $\langle {\rho ,\sigma }\rangle$ is applicable to $\mathcal {F}$ if $B \sigma \subseteq \mathcal {F}$. In this case, the result of applying $\langle {\rho ,\sigma }\rangle$ to $\mathcal {F}$ is $\{ \mathcal {F} \cup \textit{sk}(H _i) \sigma \mid i = 1, \ldots , n\}$.

A chase tree of ${\langle \mathcal {R}, \mathcal {I} \rangle }$ is a (possibly infinite) tree where each node is labeled by a set of facts, such that all of the following conditions hold.

• (1) The root is labeled with $\mathcal {I}$.

• (2) If a node labeled with $\mathcal {F}$ has n children labeled with $\mathcal {F} _1, \ldots , \mathcal {F} _n$, then there is some rule $\rho \in \mathcal {R}$ and some substitution $\sigma$ such that $\{\mathcal {F} _1, \ldots , \mathcal {F} _n\}$ is the result of applying $\langle {\rho , \sigma }\rangle$ to $\mathcal {F}$.

• (3) (Fairness) If there is a node $\alpha$ labeled with a set $\mathcal {F}$, a rule $\rho \in \mathcal {R}$, and a substitution $\sigma$ such that $\langle {\rho , \sigma }\rangle \in \mathcal {R}$ is applicable to $\mathcal {F}$; then, in all paths starting from $\alpha$, there is some node $\beta$ with n children, each of them labeled with a different set in the result of applying $\langle {\rho , \sigma }\rangle$ to the label of $\beta$.

The result of the (Skolem) chase is the (possibly infinite) set of all (possibly infinite) sets of facts obtained as the union of all sets of facts along some path.

Due to the order of rule applications, a program $\mathcal {P}$ may admit many different chase trees but, nevertheless, the result of the Skolem chase of $\mathcal {P}$ is always unique.

Fact 3

A program $\mathcal {P}$ entails a query $\exists \mathbf {v}. Q$ if and only if $\mathcal {F} \models \exists \mathbf {v}. Q$ holds for every set of facts $\mathcal {F}$ in the result of the chase of $\mathcal {P}$.

If the chase terminates for some program, then the result of the chase is the set of all (finite) leaf labels. In this case, Fact 3 leads to an effective decision procedure for BCQ entailment. Therefore, in the subsequent section, we study several restrictions on a set of rules which ensure efficient chase termination.

3 Tractable Reasoning for Disjunctive Existential Rules

In this section we present and study several restrictions, which can ensure tractability of BCQ entailment over rule sets. These insights will be the basis for our investigation of tractable query answering for ontologies in Sect. 4. An important concept for predicting the behaviour of the chase procedure is the dependency graph of a rule set:

Definition 4

The dependency graph $G(\mathcal {R})$ of a rule set $\mathcal {R}$ has the existential variables in $\mathcal {R}$ as nodes, and an edge $y \rightarrow z$ if the skolem chase of some program $\langle \mathcal {R}, \mathcal {I} \rangle$ contains terms of the form $f_z({\mathbf {t}})$ and $f_y({\mathbf {s}})$ such that $f_y({\mathbf {s}}) \in {\mathbf {t}}$.

The key to our tractability results is the notion of a braid, which, intuitively speaking, consists of a possibly large number of intertwined paths.

Definition 5

Consider a directed graph G. A path is a sequence of nodes $\alpha _1, \ldots , \alpha _n$ with $\alpha _i \rightarrow \alpha _{i+1} \in G$ for all $i = 1, \ldots , n-1$. The graph G is acyclic if, for every path $\alpha _1, \ldots , \alpha _n$ with $n \ge 2$, $\alpha _1 \ne \alpha _n$. A simple path is a path which does not contain two occurrences of the same node. A braid is a sequence of nodes $\alpha _1, \ldots , \alpha _n$ such that, for all $i = 1, \ldots , n-1$, there are at least two different simple paths from $\alpha _i$ to $\alpha _{i+1}$.

A number of natural conditions on a set of rules $\mathcal {R}$ might be considered in order to reduce the complexity of the chase. We will consider the following five:

• (a) The graph $G(\mathcal {R})$ is acyclic.

• (f) The arity of all function symbols in $\textit{sk}(\mathcal {R})$ is at most 1.

• (b) The length of the braids in $G(\mathcal {R})$ is bounded.

• (w) The treewidth of the rules in $\mathcal {R}$ is bounded.

• (p) The arity of the predicates in $\mathcal {R}$ is bounded.

Fig. 2.

Complexity of BCQ entailment with respect to the size of the rule set satisfying some combination of (a), (f), (b), (w) and (p). All of the above results are tight and refer to the combined complexity of BCQ entailment over nondeterministic and deterministic rule sets, respectively.

Most of these conditions are self-explanatory and straightforward to check. The treewidth of rules is the treewidth of the graph that has the terms of a rule as nodes, and an undirected edge whenever two terms appear in the same atom [13]. It is a well-known measure for “tree-likeness”, which is bounded by the number of terms per rule.¹ Checking if a graph G has treewidth at most k for a given constant k is polynomial in G. Both acyclicity and the maximal braid length can be computed efficiently if the dependecy graph is known. We present ways of approximating these conditions efficiently in Sect. 4.

In the remainder of the section, we characterize the (combined) complexity of BCQ entailment over sets of rules satisfying every possible combination of the above restrictions. We summarize our findings in Fig. 2, which only includes cases that satisfy (a), since its omission leads to undecidability (Theorem 11). Moreover, as indicated in Theorem 7, the “coNP/ P ” result refers to the complexity regarding the size of the rule set, with the query considered fixed.

Whilst restrictions (a), (f), (w), and (p) have been considered in previous work [10], (b) is a novel notion instrumental to ensure tractability of reasoning. See how the rule set from Example 1 may not satisfy such a restriction.

Example 6

Let $\mathcal {R} _n$ be the set of rules presented in Example 1 and let $G(\mathcal {R} _n)$ be the graph depicted in Fig. 3. Note how, for every every odd $n \ge 1$, there is a braid of length $(n+1)/2$ in $G(\mathcal {R} _n)$; e.g., $z_1, z_3, \ldots , z_n$ or $y_1, y_3, \ldots , y_n$.

Fig. 3.

Dependency graph of the set of rules $\mathcal {R} _n$ from Example 6.

Combining all restrictions allows us to obtain the main result of this section.

Theorem 7

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (f), (b), and (w) is in $\text {co}\textsc {NP}$ provided that the size of the query is fixed. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is in P.

The key for proving this result is a property that relates braid length to the size of the chase. As we will show, if a rule set $\mathcal {R}$ satisfies (f), then every term in the chase of $\langle \mathcal {R}, \mathcal {I} \rangle$ corresponds to some path in $G(\mathcal {R})$ and some constant. In turn, this implies that, if there is a polynomial bound on the number of paths in $G(\mathcal {R})$, then the number of terms introduced during the computation of the chase of $\langle \mathcal {R}, \mathcal {I} \rangle$ is also polynomially bounded. Therefore, we first show that there is indeed such a polynomial upper bound on the number of paths in a graph if the length of the braids in such a graph is fixed. Once this is shown, we can easily verify that, if $\mathcal {R}$ satisfies (b) and (f), then there is a polynomial upper bound on the number of terms that may occur in the chase of a program $\langle \mathcal {R}, \mathcal {I} \rangle$.

Lemma 8

Consider some directed acyclic graph G with n nodes. If there is a bound k on the length of the braids, then there are at most $3k \cdot n^{3k}$ paths in G.

Proof

First, we verify the following intermediate result: $(*)$ Consider two nodes $\alpha$ and $\beta$ in G. If, for every node $\gamma$, the sequence $\alpha , \gamma , \beta$ is not a braid in G; then $P_{G}(\alpha , \beta ) \le 3n^2$ with $P_{G}(\alpha , \beta )$ the number of paths from $\alpha$ to $\beta$.

Let $G'$ be the graph that results from removing every node $\gamma$ not occurring in a path from $\alpha$ to $\beta$. Then, for every node $\gamma$ in $G'$ with $\gamma \not = \alpha$ and $\gamma \not = \beta$, $P_{G'}(\alpha , \gamma ) = 1$ or $P_{G'}(\gamma , \beta ) = 1$. Let $G''$ be the graph obtained from $G'$ via simultaneous application of the following rules to every node $\gamma$ in $G'$: If $P_{G'}(\alpha , \gamma ) = 1$ and $\alpha \rightarrow \gamma \notin G'$, then remove the (only) edge of the form $\delta \rightarrow \gamma \in G'$ and add $\alpha \rightarrow \gamma$. If $P_{G'}(\gamma , \beta ) = 1$ and $\gamma \rightarrow \beta \notin G'$, then remove the edge of the form $\gamma \rightarrow \delta \in G'$ and add $\gamma \rightarrow \beta$.

The previously presented transformation preserves the number of paths from $\alpha$ to $\beta$; i.e., $P_{G}(\alpha , \beta ) = P_{G''}(\alpha , \beta )$. Moreover, the nodes in $G''$ can be fully distributed into four pairwise disjoint s ets $L_1$, $L_2$, $L_3$ and $L_4$ such that all of the following hold: $L_1 = \{\alpha \}$; $L_4 = \{\beta \}$; and, for every pair of nodes $\gamma$ and $\delta$, $\gamma \rightarrow \delta \in G''$ only if (i) $\gamma = \alpha$ and $\delta \in L_2 \cup L_4$, (ii) $\gamma \in L_2$ and $\delta \in L_3 \cup L_4$, or (iii) $\gamma \in L_3$ and $\delta = \beta$. As the sets $L_2$ and $L_3$ may contain at most n nodes, then $P_{G''}(\alpha , \beta ) \le n^2 + n + 1 \le 3n^2$ (as n is at least 2).

We now proceed to show the lemma. Let $B_m$ be the set of paths containing a path p in G iff (i) p contains a braid of length m and (ii) p does not contain a braid of length $m+1$. Then, every path $p \in B_m$ is of the form $\alpha _1, s_1, \alpha _2, s_2, \ldots , s_{m-1}, \alpha _m$ where $\alpha _1, \ldots , \alpha _m$ is a braid; and, for every $i = 2,\ldots , m-1$, $s_i$ is a sequence of nodes not containing a node $\gamma$ such that $\alpha _{i}, \gamma , \alpha _{i+1}$ is a braid in G (as this would imply that p contains a braid of length $m+1$). By $(*)$, there are at most $3n^2$ possible paths in G for every $s_i$. Moreover, there are at most $n^m$ braids of length m. Therefore, $B_m$ contains at most $n^m \cdot 3n^{2(m-1)} \le 3n^{3m}$ paths. The number of paths in G is at most $\sum ^k_{i=0} \vert B_i \vert \le k \vert B_k \vert$ (as every $B_j$ with $j > k$ is empty). Hence, the number of paths in G is necessarily less than $3k \cdot n^{3k}$.    $\square$

Proof (of Theorem 7)

Since $\mathcal {R}$ satisfies (w), we can apply a normalization procedure to compute a conservative extension $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ of $\langle \mathcal {R}, \mathcal {I} \rangle$ with an upper bound on the number of variables per rule [13]. Moreover, this transformation does not modify the dependency graph of $\mathcal {R}$ (i.e., $G(\mathcal {R}) = G(\mathcal {R} ')$).

We first determine an upper bound on the maximal number of terms T and atoms A that may occur in the chase of $\langle {\mathcal {R} ', \mathcal {I}}\rangle$. By (f), every term in the chase of $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ is of the form $f_{y_n}(\ldots (f_{y_1}(c))\ldots )$ with c a constant. Furthermore, such a term occurs in the chase only if $y_1, \ldots , y_n$ is a path in $G(\mathcal {R} ')$. Hence, every term in the chase of $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ corresponds to some path in $G(\mathcal {R} ')$ and some constant. By Lemma 8 and the fact that $\mathcal {R} '$ satisfies (b), we conclude that the number of paths in $G(\mathcal {R} ')$ is polynomial in the number of nodes in $G(\mathcal {R} ')$ (which coincides with the number of existentially quantified variables in $\mathcal {R} '$). Therefore, T is polynomially large with respect to $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ and, since the number of variables per rule in $\mathcal {R} '$ is fixed, so is A. If $\mathcal {R} '$ is a set of deterministic rules, then we can compute the only branch on some (arbitrarily chosen) chase tree of $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ to solve BCQ entailment. This branch is a sequence of at most A sets of facts; and, as there is an upper bound on the number of variables per rule in $\mathcal {R} '$, each of these sets can be computed in polynomial time. Moreover, checking if the facts in the branch entail a query is in P if the size of the query is fixed.

In the nondeterministic case, we can guess some sequence of facts and then check whether (i) such a sequence is a complete branch in some chase tree of $\langle {\mathcal {R} ', \mathcal {I}}\rangle$. Then, a query is not entailed by $\langle {\mathcal {R} ', \mathcal {I}}\rangle$ iff (ii) it is not entailed by the facts in this branch. Note that, (i-ii) can be checked in P.    $\square$

We proceed by showing the complexity of BCQ answering for any other combination of the restrictions (a), (f), (b), (w), and (p). This shows, in particular, that our chosen set of restrictions is minimal (among these selected conditions) and any other combination leads to intractability.

Theorem 9

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (f), (b), (w), and (p) is coNP-hard. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is P-hard.

Proof

The results stated in the theorem follow from hardness of SAT and propositional Horn logic entailment, respectively.    $\square$

Theorem 10

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (f), (b), and (p) is in $\text {co}\textsc {NP} ^\textsc {NP}$-complete. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is in NP-complete.

Proof

To show membership, we can make an analogous argument to the one in the proof of Theorem 7 to show that there is a polynomial upper bound on the number of terms T that may occur during the computation of the chase $\langle \mathcal {R}, \mathcal {I} \rangle$. Moreover, since the arity of the predicates is bounded by some $\ell$, the number of atoms in the chase is at most $A = P^\ell \cdot T$.

If $\mathcal {R}$ is a set of deterministic rules, then we can guess some sequence of sets $\mathcal {F} _1, \ldots , \mathcal {F} _n$ of facts with $\mathcal {F} _1 = \mathcal {I}$; some sequence $\langle {\rho _1, \sigma _1}\rangle , \ldots , \langle {\rho _{n-1}, \sigma _{n-1}}\rangle$ of pairs of rules and substitutions with $\rho _i \in \mathcal {R}$ for every $i = 1, \ldots , n-1$; and some additional substitution $\sigma$. To determine if $\langle \mathcal {R}, \mathcal {I} \rangle$ entails some query Q, we check that, for every $i = 1, \ldots , n-1$, (i) $\mathcal {F} _{i+1}$ is the result of the application of $\langle {\rho _i, \sigma _i}\rangle$ on $\mathcal {F} _i$; and (ii) $\mathcal {F} _n \models Q \sigma$. Note that, (i-ii) can be verified in polynomial time, and $\mathcal {F} _1, \ldots , \mathcal {F} _n$ may not necessarily be a complete branch in a chase tree of $\langle \mathcal {R}, \mathcal {I} \rangle$.

If $\mathcal {R}$ is a set of nondeterministic rules, then we simply guess some sequence of sets $\mathcal {F} _1, \ldots , \mathcal {F} _n$ of facts with $\mathcal {F} _1 = \mathcal {I}$. To determine that $\langle \mathcal {R}, \mathcal {I} \rangle$ does not entail some query Q, we check that, for every $i = 1, \ldots , n-1$, $(i) \mathcal {F} _{i+1}$ is the result of the application of some rule in $\mathcal {R}$ and some substitution on $\mathcal {F} _i$; (ii) no rule in $\mathcal {R}$ and substitution is applicable to $\mathcal {F} _n$; and (iii)$\mathcal {F} _n \not \models Q$. (i-iii) can be polynomially checked using an NP oracle.

For $\text {co}\textsc {NP} ^\textsc {NP}$-hardness, we reduce from the valuation problem of quantified Boolean formulas (QBF) of the form $\forall \mathbf {X}.\exists \mathbf {Y}.\varphi$, where $\mathbf {X}, \mathbf {Y}$ are lists of propositional variables and $\varphi$ is in 3CNF, i.e., $\varphi =(L^1_1\vee L^1_2\vee L^1_3)\wedge \ldots \wedge (L^n_1\vee L^n_2\vee L^n_3)$, such that the literals $L^i_j$ are variables or negated variables from $\mathbf {X}\cup \mathbf {Y}$.

We construct a set of nondeterministic without existential variables rules using constants t (true) and f (false). We add two facts $\mathsf {tf}(t)$ and $\mathsf {tf}(f)$. For every $i\in \{1,\ldots ,n\}$, we add all (polynomially many) facts of the form $c_i(v_1,v_2,v_3)$ with $v_1,v_2,v_3\in \{t,f\}$ such that $(L^i_1\vee L^i_2\vee L^i_3)$ is true when assigning the values $v_1,v_2,v_3$ to the (at most) three variables in the clause. In addition, for each universally quantified $X\in \mathbf {X}$, we add a disjunctive fact $\mathsf {val}_X(t)\vee \mathsf {val}_X(f)$. Finally, QBF valuation is encoded in the rule:

$$\begin{aligned} \bigwedge _{1\le i\le n} c_i(x^i_1,x^i_2,x^i_3)\wedge \bigwedge _{X\in \mathbf {X}}\mathsf {val}_X(v_X) \wedge \bigwedge _{Y\in \mathbf {Y}}\mathsf {tf}(v_Y)\rightarrow \mathsf {trueQBF} \end{aligned}$$

(2)

where each variable has the form $v_Z$ for $Z\in \mathbf {X}\cup \mathbf {Y}$, and $x^i_j$ denotes $v_Z$ for the propositional variable Z that occurs in $L^i_j$. Then $\mathsf {trueQBF}$ is entailed iff, for all models (i.e., all assignments of universal variables $X\in \mathbf {X}$), there is an assignment for the variables $Y\in \mathbf {Y}$, such that each clause in $\varphi$ is true.

The hardness result for deterministic rules follows when considering QBF without universally quantified variables; i.e., propositional satisfiability.    $\square$

Theorem 11

BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a set of deterministic rules satisfying (f), (b), (w), and (p) is undecidable.

Proof

We use a reduction from a known undecidable problem described as follows (see Sect. 2.5.1 of [16] for a very similar and more detailed argument). A context-free grammar is a tuple $\langle {S, P}\rangle$ with S a non-terminal, and P a set of production rules of the form $A \rightarrow BC$ or $A \rightarrow a$ where A, B and C are non-terminals and a is a terminal. The language generated by a grammar $\langle {S, P}\rangle$ is the set of all strings of terminals which can be produced by rewriting S applying the production rules in P. The following problem is undecidable [14]: Given two context-free grammars $G_1 = \langle {P_1, S_1}\rangle$ and $G_2 = \langle {P_2, S_2}\rangle$, with disjoint sets of non-terminals and common terminal symbols 0 and 1, determine whether there is some word in the intersection of the languages generated by $G_1$ and $G_2$.

Consider two binary predicates $T_0$ and $T_1$, a specific binary predicate $NT_A$ for every non-terminal A occurring in $G_1$ or $G_2$, a unary predicate X, and a constant c. For all $i \in \{1, 2\}$, let $\mathcal {R} _i = \{T_a(x, y) \rightarrow NT_A(x, y) \mid A \rightarrow a \in P_i\} \cup \{NT_B(x, y) \wedge NT_C(y, z) \rightarrow NT_A(x, z) \mid A \rightarrow BC \in P_i\}$. Moreover, let $\mathcal {R} = \mathcal {R} _1 \cup \mathcal {R} _2 \cup \{X(x) \rightarrow \exists y . T_0(x, y) \wedge X(y), X(x) \rightarrow \exists z . T_1(x, z) \wedge X(z)\}$. Then, the intersection of the languages generated by $G_1$ and $G_2$ is empty iff $\langle {\mathcal {R}, \{X(c)\}}\rangle$ does not entail the query $\exists x . NT_{S_1}(c, x) \wedge NT_{S_2}(c, x)$.

The rules in $\mathcal {R}$ satisfy (f), (b), (w), and (p): The arity of all of the symbols in $\textit{sk}(\mathcal {R})$ (i.e., $f_y$ and $f_z$) is one, $G(R)$ contains two nodes, and the arity of every predicate is at most two. Moreover, the number of variables per rule is bounded and hence, so is the treewidth.    $\square$

Theorem 12

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a) is in coN2ExpTime. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is in 2ExpTime.

Proof

We first determine the maximal number of ground (skolem) terms and corresponding facts that may occur in the chase. Let n be the number of skolem functions in $\textit{sk}(\mathcal {R})$, and let m be the maximal arity of such functions. The maximal nesting depth of ground terms in the chase is n, since every term of greater depth is cyclic and, by (a), such terms may not occur in the chase of $\langle \mathcal {R}, \mathcal {I} \rangle$. Ground terms then correspond to trees of depth at most n, fan-out at most m, and with leaves from the set C of constants in $\langle \mathcal {R}, \mathcal {I} \rangle$. Such trees have most $n\cdot m^n$ nodes in total. As each node is assigned a constant or function symbol, there are at most $T=(\vert C \vert +n)^{n\cdot m^n}$ trees, and hence ground terms, overall. Now, if $\langle \mathcal {R}, \mathcal {I} \rangle$ contains k different predicate symbols of arity at most $\ell$, then the maximal number of ground facts based on T terms is $A = k T^\ell =k (C_\mathcal {I} +n)^{\ell \cdot n\cdot m^n}$. The number of facts A is therefore double exponential in the size of $\langle \mathcal {R}, \mathcal {I} \rangle$ and hence, so is the length of every branch in a chase tree of a program $\langle \mathcal {R}, \mathcal {I} \rangle$.

If $\mathcal {R}$ is a set of deterministic rules, then there is only one branch in every possible chase tree of a program $\langle \mathcal {R}, \mathcal {I} \rangle$ which can be computed in double-exponentially many steps. Then, a query is entailed by $\langle \mathcal {R}, \mathcal {I} \rangle$ iff such query is entailed by the set of facts in the branch. If $\mathcal {R}$ only contains nondeterministic rules, membership in coN2ExpTime follows from the fact that BCQ non-entailment can be shown by guessing some branch of the tree, and then checking that the set of facts in such branch does not entail the query.    $\square$

Theorem 13

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a) and (f) is in coNExpTime. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is in ExpTime.

Proof

We determine that the maximal number of facts that may occur in the chase of $\langle \mathcal {R}, \mathcal {I} \rangle$ is exponential in the size of the program. The remainder of the proof is analogous to that of Theorem 12.

Let n be the number of skolem functions in $\textit{sk}(\mathcal {R})$ which, by (f), have an arity of at most 1. The maximal nesting depth of ground terms in the chase is n, since every term of greater depth is cyclic and, by (a), such terms may not occur in the chase of $\langle \mathcal {R}, \mathcal {I} \rangle$. Ground terms then correspond to sequences of depth at most n and, since each element in the sequence is assigned a constant or function symbol, there are at most $T = (C+n)^n$ ground terms, overall. In turn, the maximal number of facts in the chase is $A = k T^\ell =k (C + n)^{\ell \cdot n}$ with k the number of predicates and $\ell$ the maximal arity of a predicate in $\langle \mathcal {R}, \mathcal {I} \rangle$.    $\square$

Theorem 14

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (b), (w), and (p) is coN2ExpTime-hard. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is 2ExpTime-hard.

Proof

For the first result, we present a reduction of the word problem of double-exponentially time-bounded non deterministic Turing machines (TMs) to BCQ non-entailment. Given such reduction, it is clear how to produce a similar reduction to prove the second result stated in the theorem.

Consider a N2ExpTime Turing Machine (TM) M. We simulate the computation of M on an input string I by constructing a program $\langle \mathcal {R}, \mathcal {I} \rangle$ such that $\langle \mathcal {R}, \mathcal {I} \rangle$ does not entail some nullary predicate $\textit{Reject}$ iff M accepts I. To address computation steps and tape cells, we recall a construction by [4] to (deterministically) construct a chain of double exponentially many elements. Let $\mathcal {I} = \{r_0(0), r_0(a), \textit{Scc}_0(0, 1), \textit{Min}_0(0), \textit{Max}_0(a)\}$. For each $i \in \{0, \ldots , n-1\}$, with n the length of the input I, we add the rules in $\{R_i(x) \wedge R_i(y) \rightarrow \exists v_i . S_i(x, y, v_{i+1}) \wedge R_{i+1}(v_{i+1}), S_i(x, y, z) \wedge S_i(x, y', z')$ $\wedge$ $\textit{Scc}_i(y, y') \rightarrow \textit{Scc}_{i+1}(z, z'), S_i(x,y, z) \wedge S_i(x', y', z') \wedge \textit{Max}_i(y) \wedge \textit{Min}_i(y') \wedge \textit{Scc}_i(x, x') \rightarrow \textit{Scc}_{i+1}(z, z'), \textit{Min}_i(x) \wedge S_i(x, x, y)$ $\rightarrow \textit{Min}_{i+1}(y), \textit{Max}_i(x) \wedge S_i(x, x, y) \rightarrow \textit{Max}_{i+1}(y)\}$ It can be shown, by induction on i, that in any path of any chase tree of $\langle \mathcal {R}, \mathcal {I} \rangle$, the relation $r_n$ contains $2^{2^n}$ elements, which are linearly ordered by $\textit{Scc}_n$.

The remaining TM simulation follows standard constructions (cf. [11]), using elements of the $r_n$ chain to refer to specific time points and tape cells when encoding a run of the TM. Nondeterministic transitions are captured using rules with disjunction. Assuming that the state of M at step s is captured with facts $\textit{State}_q(s)$ for all states Q, we can complete the simulation by adding rules $\textit{State}_q(x)\wedge \textit{Max}_n(x) \rightarrow \textit{Reject}$ for all non-accepting states q of M. We can assume without loss of generality that M runs for the maximum double-exponential number of steps on all rejecting runs, so that the query $\textit{Reject}$ is entailed iff there are no accepting runs.

The rules in $\mathcal {R}$ satisfy (a), (b), (w), and (p): $G(\mathcal {R})$ is the smallest graph containing $v_i \rightarrow v_{i+1}$ for every $i = 1, \ldots , n$ and hence, this graph is acyclic and does not contain any braids. Also, both the arity of the predicates, and treewidth of the rules in $\mathcal {R}$ is fixed. Finally, it can be checked that the rules added to finalize the reduction (cf. [11]) do not violate (a), (b), (w), nor (p).    $\square$

Theorem 15

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (f), (w), and (p) is coNExpTime-hard. If $\mathcal {R}$ is a set of deterministic rules, then it is ExpTime-hard.

Proof

We show that, using a set of rules satisfying (a), (f), (w), and (p), we can define a program that, given some n, can generate an exponentially long chain of terms sorted by some binary predicate. The remainder of the proof is analogous to that of Theorem 14.

Let $\mathcal {R}$ be the set containing the rules in $\{S_i(x) \rightarrow \exists y_{i+1}, z_{i+1} . L_i(x, y) \wedge R_i(x, z_{i+1}) \wedge \textit{Scc}_{i+1}(y_{i+1}, z_{i+1}), R_i(x, z) \wedge \textit{Scc}_i(x, y) \wedge L_i(y, w) \rightarrow \textit{Scc}_{i+1}(z, w)\}$ for each $i \in \{0, \ldots , n-1\}$. We can show, by induction on i, that in any path of any chase tree of $\langle {\mathcal {R}, \{S_0(c)\}}\rangle$, the relation $S_n$ contains $2^n$ elements, which are linearly ordered by $\textit{Scc}_n$.

The rules in $\mathcal {R}$ satisfy (a), (f), (w), and (p): $G(\mathcal {R})$ is the smallest graph containing $y_i \rightarrow y_{i+1}$, $y_i \rightarrow z_{i+1}$, $z_i \rightarrow y_{i+1}$, and $z_i \rightarrow z_{i+1}$ for every $i = 1, \ldots , n$, and hence, this graph is acyclic. Also, the arity of every function symbol in $\textit{sk}(\mathcal {R})$ is 1, and both the arity of the predicates and treewidth of the rules is fixed.    $\square$

Theorem 16

Deciding BCQ entailment for programs $\langle \mathcal {R}, \mathcal {I} \rangle$ with $\mathcal {R}$ a rule set satisfying (a), (f), and (b) is coNExpTime-hard. Moreover, if $\mathcal {R}$ is a set of deterministic rules, then it is ExpTime-hard.

Proof

The first and second parts of the theorem follow from the hardness of fact entailment over disjunctive and non-disjunctive Datalog [11], respectively. Note that, every (possibly disjunctive) Datalog program—a program containing only deterministic rules without existential variables—satisfies (a), (f) and (b).    $\square$

4 Tractable Reasoning for Ontologies

Across this section we discuss how to employ the chase to reason over DL ontologies and then, using some of the results from the previous section, we define tractable acyclicity, an acyclicity condition tailored for DL ontologies which ensures tractability of BCQ entailment.

We consider the $\mathcal {SRI}$ fragment of the description logic $\mathcal {SROIQ}$, which is the logical basis of OWL 2 DL. We present this DL using a normal form close to that of [8]. Note that, in such a normal form, occurrences of the negation logical constructor are normalized into axioms of the form (3) in Fig. 4. Moreover, we do not consider number restrictions nor nominals in our definition of DL, as the use of these logical constructors would require equality reasoning. There are well-known techniques to axiomatize the meaning of equality—e.g., singularization [10, 21]—but these are not our focus.

Fig. 4.

Mapping $\Psi$. In the above, $C_{(i)}, D \in \mathbf C$, $R, S_{(i)} \in \mathbf R ^-$, and $m, n \ge 1$. Moreover, for every $R \in \mathbf R$, $R^-\langle t, u\rangle = R(u, t)$ and $R\langle t, u\rangle = R(t, u)$.

Let C, R, and I be some infinite countable and pairwise disjoint sets of concepts, roles, and individuals, respectively. Moreover, let $\mathbf R ^- = \mathbf R \cup \{R^- \mid R \in \mathbf R \}$; and, for every $R \in \mathbf R ^-$, $R^{--} = R$. A TBox axiom is a formula of one of the forms given on the left hand side of Fig. 4. An ABox axiom or assertion is a formula of the form A(a) or R(a, b) with $A \in \mathbf C$, $R \in \mathbf R$ and $a, b \in \mathbf I$. A ontology is a tuple $\langle {\mathcal {T}, \mathcal {A}}\rangle$ with $\mathcal {T}$ a set of TBox axioms and $\mathcal {A}$ a set of assertions.

We do not consider any structural restrictions, such as role regularity [15], in our definition of ontologies. These restrictions are unnecessary for preserving correctness when using the chase and hence, we ignore them.

The semantics of ontologies are given by means of a mapping into programs.

Fact 17

An ontology $\mathcal {O}$ entails some query $Q$ iff $\langle {\Psi (\mathcal {R}), \mathcal {I}}\rangle \models Q$ with $\Psi$ the function mapping axioms to rules defined in Fig. 4.

Due to the close correspondence between DL axioms and rules highlighted by the previous result, we identify an axiom $\alpha$ with the rule $\Psi (\alpha )$, a TBox $\mathcal {T}$ with the set of rules $\Psi (\mathcal {R})$, and an ontology $\langle \mathcal {T}, \mathcal {A} \rangle$ with the program $\langle {\Psi (\mathcal {R}), \mathcal {A}}\rangle$.

By definition, every TBox $\mathcal {T}$ satisfies restrictions (f) and (w) and hence, we only need to determine whether $\mathcal {T}$ satisfies (a) and (b) to guarantee tractability of reasoning over a deterministic ontology $\langle \mathcal {T}, \mathcal {A} \rangle$. Unfortunately, the dependency graph of a TBox—which needs to be checked in order to verify (a) and (b)—cannot be computed in polynomial time.

Lemma 18

Given a TBox $\mathcal {T}$, the computation of $G(\mathcal {T})$ is ExpTime-hard.

Proof

The lemma follows from the fact that entailment of concept subsumptions by a TBox (which is ExpTime-hard) can be decided by computing the dependency graph of another TBox $\mathcal {T} '$.

Consider a TBox $\mathcal {T}$ and two concepts C and D. Moreover, let $\mathcal {T} ' = \mathcal {T} \cup \{\alpha _1 = C \sqsubseteq \exists R_Y. C \sqcap Y, \alpha _2 = Y \sqcap D \sqsubseteq \exists R_Z.Z\}$ with $R_Y$ and $R_Z$, and Y and Z some fresh roles and concepts, respectively. Then, $\mathcal {T} \models C \sqsubseteq D$ iff $y \rightarrow z \in G(\mathcal {T} ')$ with y and z the variables occurring in the rules $\Psi (\alpha _1)$ and $\Psi (\alpha _2)$.    $\square$

Since the computation of the dependency graph of a TBox is rather expensive, we define an over-approximation of this graph based on the definition of model-summarizing acyclicity ($\text {MSA}$) [10] which can be computed more efficiently.

Definition 19

Given a set of rules $\mathcal {R}$, let $\mathcal {R} _S$ be the set of rules that results from replacing every rule $\rho \in \mathcal {R}$ of the form (1) by the following rule.

$$\begin{aligned} B \rightarrow \bigwedge _{1 \le i \le n} \Big ( H _i \wedge \bigwedge _{x \in {\mathbf {x}}} \bigwedge _{v \in \mathbf {v} _i}{} \textit{Scc}(x, v)\Big ) \theta \end{aligned}$$

(3)

In the above, $\textit{Scc}$ is a fresh binary predicate and $\theta$ is the substitution mapping every variable in $v \in \mathbf {v} _i$ to a fresh constant $c_v$ (which, by ($\dagger$) and the definition of a rule, is also specific to the i-th disjunct in the head of the rule $\rho$).

The summarizing dependency graph $G_S(\mathcal {R})$ of a rule set $\mathcal {R}$ is the smallest graph containing an edge $y \rightarrow z$ if $\langle {\mathcal {R} _S, \mathcal {I} ^\star _{\mathcal {R}}}\rangle \models \textit{Scc}(c_y, c_z)$ where $\mathcal {I} ^\star _{\mathcal {R}}$ is the critical instance of $\mathcal {R}$; i.e., the set of all facts that can be constructed using the predicates in $\mathcal {R}$ and the special constant $\star$.

Lemma 20

Consider a rule set $\mathcal {R}$. Then, the summarizing dependency graph of $\mathcal {R}$ is a superset of the dependency graph of $\mathcal {R}$.

Proof

Consider some chase tree T of a program $\langle \mathcal {R}, \mathcal {I} \rangle$; and a function h mapping every constant to $\star$, and every skolem term of the form $f_y({\mathbf {t}})$ to the constant $c_y$. Then, for every set of facts $\mathcal {F}$ associated to some node $\alpha$ in T, $h(\mathcal {F})$ is contained in the result of the chase of $\langle {\mathcal {R} _S, \mathcal {I} ^\star _{\mathcal {R}}}\rangle$. The previous claim can be verified by induction on the path from the root of T to $\alpha$.

Let us assume that there is some edge $y \rightarrow z \in G(\mathcal {R})$. Then, by the definition of the dependency graph, there must be some terms $f_z({\mathbf {t}})$ and $f_y({\mathbf {s}})$ with $f_y({\mathbf {s}}) \in {\mathbf {t}}$ occurring in some set of facts $\mathcal {F}$ in some chase tree of a program $\langle \mathcal {R}, \mathcal {I} \rangle$. Let $B [{\mathbf {x}},\mathbf {y} ] \rightarrow \bigvee _{i=1}^n \exists \mathbf {v} _i.H _i[{\mathbf {x}},\mathbf {v} _i]$ be the only rule in $\mathcal {R}$ containing z in some disjunct in the head. Then, $B[{\mathbf {x}}/ {\mathbf {t}} ] \subseteq \mathcal {F}$, and hence, $h(B[{\mathbf {x}}/ {\mathbf {t}} ])$ is contained in the result of the chase of $\langle {\text {MSA} (\mathcal {R}), \mathcal {I} ^\star _{\mathcal {R}}}\rangle$. Since $B \rightarrow \bigwedge _{i=1}^n (H _i' \wedge \bigwedge _{x \in {\mathbf {x}}} \bigwedge _{v \in \mathbf {v} _i}{} \textit{Scc}(x, v))\theta \in \text {MSA} (\mathcal {R})$, then $\textit{Scc}(c_y, c_z)$ is also in the result of the chase of $\langle {\text {MSA} (\mathcal {R}), \mathcal {I} ^\star _{\mathcal {R}}}\rangle$. In turn, this implies that $y \rightarrow z \in G_S(\mathcal {R})$.    $\square$

We proceed with the definition of tractable acyclicity, and thereafter establish the complexity of checking this condition and reasoning over such ontologies

Definition 21

A TBox $\mathcal {T}$ is k-tractable acyclic ($\text {TA}_{k}$) if its summarizing graph is acyclic and the length of every braid in this graph is at most k.

Theorem 22

Deciding $\text {TA}_{k}$ membership of a TBox $\mathcal {T}$ is P-complete.

Proof

To verify membership, we propose a polynomial procedure to determine if $G_S(\mathcal {T})$ is acyclic and then compute the length of the longest braid in $G_S(\mathcal {T})$. Let $\mathcal {P} = \langle {\mathcal {R}, \mathcal {I}}\rangle$ be the program where $\mathcal {I}$ is the instance containing $E(c_y, c_z)$ for every $y \rightarrow z \in G_S(\mathcal {T})$, and $\textit{Neq}(c_y, c_z)$ for every pair of nodes y and z in $G_S(\mathcal {T})$ with $y \ne z$; and $\mathcal {R} = \{\rightarrow \textit{P}(x, x), \textit{E}(x, y) \rightarrow \textit{P}(x, y), \textit{P}(x, y) \wedge \textit{P}(y, z) \rightarrow \textit{P}(x, z), \textit{P}(x, y) \wedge \textit{P}(y, z) \rightarrow \textit{P}(x, z), \textit{P}(x, y) \wedge \textit{P}(x, z) \wedge \textit{Neq}(y, z) \wedge \textit{E}(y, w) \wedge \textit{E}(z, w) \rightarrow \textit{B}(x, w), \textit{B}(x, y) \wedge \textit{B}(y, z) \rightarrow \textit{B}(x, z)\}$. Then, there is a braid starting in y and ending in z in $G_S(\mathcal {R})$ if and only if $\mathcal {P} \models \textit{B}(c_{y}, c_{z})$. Thus, to determine the maximum length of a braid in $G_S(\mathcal {R})$, we simply have to look for the largest path over the binary predicate B in the result of the chase of $\mathcal {P}$. Moreover, $G_S(\mathcal {R})$ is acyclic if and only if $\mathcal {P}$ does not entail the query $\exists x . \textit{P}(x, x)$. Note that, the program $\mathcal {P}$ can be constructed in polynomial time since the computation of $G_S(\mathcal {T})$ is tractable. Moreover, as the number of variables per rule in $\mathcal {R}$ is at most 4 and the maximum arity of a predicate is 2, the chase of such a program can be computed in polynomial time.

Hardness of the $\text {TA}_{k}$ membership check can be readily ascertained via reduction from propositional horn entailment.    $\square$

Theorem 23

Deciding BCQ entailment for $\text {TA}_{k}$ ontologies $\langle \mathcal {T}, \mathcal {A} \rangle$ is coNP-complete provided the size of the query is fixed. Moreover, if $\mathcal {T}$ is a deterministic TBox, then it is P-complete.

Proof

If $\mathcal {T}$ is $\text {TA}_{k}$, then $G_S(\mathcal {T})$ is acyclic and every braid in $G_S(\mathcal {T})$ is of length at most k. In turn, this implies that $G(\mathcal {T})$ is acyclic and every braid in $G(\mathcal {T})$ is of length at most k by Lemma 20. Since the TBox $\mathcal {T}$ satisfies restrictions (a), (b), (f), and (w), the theorem follows from Theorems 7 and 23.    $\square$

5 Evaluation

To assess the empirical generality of $\text {TA}_{k}$, we analyzed ontologies from MOWLCorp [22] and Oxford Ontology Library,² two large corpora of real-world OWL ontologies. These ontologies were transformed into the normal form defined in Fig. 4 using standard normalization techniques [8]. After this step, we disregarded ontologies with nominals and number restrictions; and also ontologies without any axiom of type (9), as these are trivially $\text {TA}_{0}$. Since the MOWLCorp is rather large, we only considered ontologies in this corpus with up to 1,000 axioms of type (9). The final set contained 1,576 TBoxes from MOWLCorp and 225 TBoxes from the Oxford Ontology Library.

To determine $\text {TA}_{k}$ membership, we first constructed the summarizing dependency graphs of the TBoxes. For this, we transformed axioms to rules using the mapping in Fig. 4 and derived the programs described in Definition 19, over which we reasoned using the RDFox [24] datalog rule engine. Out of the obtained graphs, we found 974 (61.8%) acyclic ones from MOWLCorp and 171 (76%) from Oxford Library. Then, we determined $TA_k$ membership of acyclic graphs by counting the length of their longest braid. We did this by constructing the program defined in Theorem 22, over which we reasoned using RDFox.

As our results show in Table 1, 78.3% of acyclic ontologies from MOWLCorp are $\text {TA}_{1}$, 90.8% are $\text {TA}_{2}$, 95.5% are $\text {TA}_{3}$, 98.8% are $\text {TA}_{4}$ and 99% are $\text {TA}_{5}$. In the Oxford Library, 51.4% of the acyclic ontologies are $\text {TA}_{1}$, 69.5% are $\text {TA}_{2}$, 81.2% are $\text {TA}_{3}$, 92.3% are $\text {TA}_{4}$, 97.6 % are $\text {TA}_{5}$ and 98.2 % are $\text {TA}_{6}$. There was only one ontology from the Oxford corpus (00477.owl), containing more than 150,000 rules of type (9), for which computing $\text {TA}_{k}$ membership did not terminate.

Our acyclicity notion is theoretically equivalent to MSA and as general as MFA with respect to the evaluated ontologies: In our test set, there are no MFA ontologies which were not MSA. This validates the claims from [7, 10], where it was observed that MFA (the most general known acyclicity criterion for the skolem chase) is not empirically more general than MSA. Moreover, our results show that almost all acyclic ontologies are $\text {TA}_{k}$ with a small k: $\text {TA}_{5}$ characterizes 97% of the ontologies in both corpora.

Table 1. Histogram of $\text {TA}_{k}$ on ontologies from MOWL and Oxford corpora, where for ${TA_k}$ we only count ontologies that do not also belong to $\text {TA}_{j}$ for all $j < k$

6 Conclusions and Future Work

To the best of our knowledge, this is the first systematic study of tractability of CQ answering with disjunctive existential rules. An important application is tractable query answering over OWL ontologies, a task which in general is known to be intractable [25]. We have shown that our restrictions do indeed apply, for small bounds of the related parameters, to many practical ontologies.

Our work therefore suggests a new approach to efficient reasoning that might be applicable to many realistic ontologies, and which might be natural to implement in existing reasoners such as HermiT [23], which use chase-like procedures already. The extension of our work with more general conditions for restricted chase termination, which was recently shown to work well with many OWL ontologies [7], may further help to extend the applicability of this approach.