2.1 Basics on relational structures

Fix three pairwise disjoint lexicographically enumerable infinite sets $\mathsf{C}$ of constants, $\mathsf{N}$ of nulls ( $\varphi$ , $\varphi_0$ , $\varphi_1$ , …), and $\mathsf{V}$ of variables (x, y, z, and variations thereof). Their union is denoted by $\mathsf{T}$ and its elements are called terms. For any integer $k \geq 0$ , we may write [k] for the set $ \{1,..., k \}$ ; in particular, as usual, if $k = 0$ , then $[k] = \emptyset$ .

An atom $\underline{a}$ is an expression of the form $P(\textbf{t})$ , where $\mathit{preds}(\underline{a})=P$ is a (relational) predicate, $\textbf{t}=t_1,..., t_k$ is a tuple of terms $\mathit{arity}(\underline{a}) = \mathit{arity}(P)=k \geq 0$ is the arity of both $\underline{a}$ and P, and $\underline{a}[i]$ denotes the i-th term $\textbf{t}[i] = t_i$ of $\underline{a}$ , for each $i \in [k]$ . In particular, if $k = 0$ , then $\textbf{t}$ is the empty tuple and $\underline{a} = P()$ . By $\mathit{consts}(\underline{a})$ and $\mathit{vars}(\underline{a})$ we denote, respectively, the set of constants and variables occurring in $\underline{a}$ . A fact is an atom that contains only constants.

A (relational) schema $\mathbf{S}$ is a finite set of predicates, each with its own arity. The set of positions of $\mathbf{S}$ , denoted by $\mathit{pos}(\mathbf{S})$ , is defined as the set $\{P[i] \ | \ P \in \mathbf{S}~\wedge~ 1 \leq i \leq \mathit{arity}(P)\}$ , where each P[i] denotes the i-th position of P. A (relational) structure over $\mathbf{S}$ is any (possibly infinite) set of atoms using only predicates from $\mathbf{S}$ . The domain of a structure S, denoted by $\mathit{dom}(S)$ , is the set of all the terms forming the atoms of S. An instance over $\mathbf{S}$ is any structure I over $\mathbf{S}$ such that $\mathit{dom}(I) \subseteq \mathsf{C} \cup \mathsf{N}$ . A database over $\mathbf{S}$ is any finite instance over $\mathbf{S}$ containing only facts. The active domain of an instance I, denoted by $\mathit{dom}(I)$ , is the set of all the terms occurring in I, whereas the Herbrand Base of I, denoted by ${{\mathit{HB}}}(I)$ , is the set of all the atoms that can be formed using the predicate symbols of $\mathbf{S}$ and terms of $\mathit{dom}(I)$ .

Consider two sets of terms $T_1$ and $T_2$ and a map $\mu : T_1 \rightarrow T_2$ . Given a set T of terms, the restriction of $\mu$ with respect to T is the map $\mu|_{T} = \{t \mapsto \mu(t):t \in T_1 \cap T\}$ . An extension of $\mu$ is any map $\mu'$ between terms, denoted by $\mu' \supseteq \mu$ , such that $\mu'|_{T_1} = \mu$ . A homomorphism from a structure $S_1$ to a structure $S_2$ is any map $h: \mathit{dom}(S_1) \rightarrow \mathit{dom}(S_2)$ such that both the following hold: (i) if $t \in \mathsf{C} \cap \mathit{dom}(S_1)$ , then $h(t) = t$ ; and (ii) $h(S_1) = \{P(h(\mathbf{t})) : P(\mathbf{t}) \in S_1\} \subseteq S_2$ .

2.2 Conjunctive queries

A conjunctive query (CQ) q over a schema $\mathbf{S}$ is a (first-order) formula of the form

(1) \begin{equation} \langle \mathbf{x} \rangle \leftarrow \exists \ \mathbf{y} \ \Phi(\mathbf{x,y}),\end{equation}

where $\mathbf{x}$ and $\mathbf{y}$ are tuples (often seen as sets) of variables such that $\mathbf{x} \cap \mathbf{y} = \emptyset$ , and $\Phi(\mathbf{x,y})$ is a conjunction (often seen as a set) of atoms using only predicates from $\mathbf{S}$ . In particular,

• – $\mathit{dom}(\Phi) \subseteq \mathbf{x} \cup \mathbf{y} \cup \mathsf{C}$ ,

• – whenever a variable z belongs to $\mathbf{x} \cup \mathbf{y}$ , then z occurs also in $\Phi$ ,

• – $\mathbf{x}$ are the output variables of q, and

• – $\mathbf{y}$ are the existential variables of q.

To highlight the output variables, we may write $q(\mathbf{x})$ instead of q. The evaluation of q over an instance I is the set q(I) of every tuple $\mathbf{t}$ of constants admitting a homomorphism $h_{\mathbf{t}}$ from $\Phi(\mathbf{x,y})$ to I such that $h_{\mathbf{t}}(\mathbf{x}) = \mathbf{t}$ .

A Boolean conjunctive query (BCQ) is a CQ with no output variable, namely an expression of the form $\langle \rangle \leftarrow \exists \ \mathbf{y} \ \Phi(\mathbf{y})$ . An instance I satisfies a BCQ q, denoted $I \models q$ , if q(I) is nonempty, namely q(I) contains only the empty tuple $\langle \rangle $ .

2.3 Tuple-generating dependencies

A tuple-generating dependency (TGD) $\sigma$ – also known as (existential) rule – over a schema $\mathbf{S}$ is a (first-order) formula of the form

(2) \begin{equation} \Phi (\mathbf{x}, \mathbf{y}) \rightarrow \ \exists \ \mathbf{z} \ \Psi(\mathbf{x}, \mathbf{z}),\end{equation}

where $\mathbf{x}$ , $\mathbf{y}$ , and $\mathbf{z}$ are pairwise disjoint tuples of variables, and both $\Phi(\mathbf{x,y})$ and $\Psi(\mathbf{x,z})$ are conjunctions (often seen as a sets) of atoms using only predicates from $\mathbf{S}$ . In particular,

• – $\Phi$ (resp., $\Psi$ ) contains all and only the variables in $\mathbf{x} \cup \mathbf{y}$ (resp., $\mathbf{x} \cup \mathbf{z}$ ),

• – constants (but not nulls) may also occur in $\sigma$ ,

• – $\mathbf{x} \cup \mathbf{y}$ are the universal variables of $\sigma$ denoted by $\mathit{vars_\forall}(\sigma)$ ,

• – $\mathbf{z}$ are the existential variables of $\sigma$ denoted by $\mathit{vars_\exists}(\sigma)$ , and

• – $\mathbf{x}$ are the frontier variables of $\sigma$ denoted by $\mathsf{vars_\curvearrowright}(\sigma)$ .

We refer to $\mathit{body}(\sigma) = \Phi$ and $\mathit{head}(\sigma) = \Psi$ as the body and head of $\sigma$ , respectively. If $|\mathit{head}(\sigma)| = 1$ , the TGD is called single-head, otherwise it called multi-head. If $\mathit{vars_\exists}(\sigma) = \emptyset$ and $|\mathit{head}(\sigma)|=1$ , then $\sigma$ is called $\mathit{datalog}$ rule. With $ \mathit{h}\textrm{-}\mathit{preds}(\sigma) $ (resp., $\mathit{b}\textrm{-}\mathit{preds}(\sigma)$ ) we denote the set of predicates in $\mathit{head}(\sigma)$ (resp., $\mathit{body}(\sigma)$ ). An instance I satisfies $\sigma$ , written $I \models \sigma$ , if the existence of a homomorphism h from $\Phi$ to I implies the existence of a homomorphism $h' \supseteq h_{| \mathbf{x}}$ from $\Psi$ to I.

An ontology $\Sigma$ is a set of rules. An instance I satisfies $\Sigma$ , written $I \models \Sigma$ , if $I \models \sigma$ for each $ \sigma \in \Sigma $ . Without loss of generality, we assume that $\mathit{vars}(\sigma_1) \cap \mathit{vars}(\sigma_2)~=~\emptyset$ , for each pair $\sigma_1, \sigma_2$ of rules in $\Sigma$ . Operators $\mathit{vars_\exists}$ , $\mathit{h}\textrm{-}\mathit{preds}$ , and $\mathit{b}\textrm{-}\mathit{preds}$ naturally extend on ontologies.

A class $\mathcal{C}$ of ontologies is any (typically infinite) set of TGDs fulfilling some syntactic or semantic conditions (see, e.g., the classes shown in Table 1, some of which will be formally defined in the subsequent sections). In particular, $\mathsf{Datalog}$ is the class of ontologies containing only datalog rules.

Finally, the schema of an ontology $\Sigma$ , denoted $\mathit{sch}(\Sigma)$ , is the subset of $ \mathbf{S}$ containing all and only the predicates occurring in $\Sigma$ , whereas $ \mathit{arity}(\Sigma) = \max_{P \in \mathit{sch}(\Sigma)} \mathit{arity}(P)$ . For simplicity of exposition, we write $\mathit{pos}(\Sigma)$ instead of $\mathit{pos}(\mathit{sch}(\Sigma))$ .

2.4 Ontological query answering

Consider a database D and a set $\Sigma$ of TGDs. A model of D and $\Sigma$ is an instance I such that $I \supseteq D$ and $I \models \Sigma$ . Let $\mathit{mods}(D, \Sigma)$ be the set of all models of D and $\Sigma$ . The certain answers to a CQ q w.r.t. D and $\Sigma$ are defined as the set of tuples $ \mathit{cert}(q, D, \Sigma) = \bigcap_{M \in \mathit{mods}(D,\Sigma)} q(M).$ Accordingly, for any fixed schema $\mathbf{S}$ , two ontologies $\Sigma_1$ and $\Sigma_2$ over $\mathbf{S}$ are said to be $\mathbf{S}$ -equivalent (in symbols $\Sigma_1 \equiv_\mathbf{S} \Sigma_2$ ) if, for each D and q over $\mathbf{S}$ , it holds that $\mathit{cert}(q,D,\Sigma_1) = \mathit{cert}(q,D,\Sigma_2).$ The pair D and $\Sigma$ satisfies a BCQ q, written $D \cup \Sigma \models q$ , if $\mathit{cert}(q, D, \Sigma) = \langle \rangle$ , namely $M \models q$ for each $M \in \mathit{mods}(D,\Sigma)$ . Fix a class $\mathcal{C}$ of ontologies. The computational problem studied in this work – called cert-eval ${{[{{\mathcal{C}}}]}}$ – can be schematized as follows:

In what follows, with a slight abuse of terminology, whenever we say that ${{\mathcal{C}}}$ is decidable, we mean that cert-eval ${{[{{\mathcal{C}}}]}}$ is decidable. Note that $\mathbf{c} \in \mathit{cert}(q, D, \Sigma)$ if, and only if, $D \cup \Sigma \models q(\mathbf{c})$ , where $q(\mathbf{c})$ is the BCQ obtained from $q(\mathbf{x})$ by replacing, for each $i \in \{1,...,|\mathbf{x}|\}$ , every occurrence of the variable $\mathbf{x}[i]$ with the constant $\mathbf{c}[i]$ . Actually, the former problem is ${{{ \textbf{AC}_0}}}$ reducible to the latter.

While considering the computational complexity of cert-eval ${{[{{\mathcal{C}}}]}}$ , we recall the following convention: (i) combined complexity means that D, $\Sigma$ , q, and $\mathbf{c}$ are given in input; and (ii) data complexity means that only D and $\mathbf{c}$ are given in input, whereas $\Sigma$ and q are considered fixed. Accordingly, we point out that complexity results reported in Table 1 refer to cert-eval ${{[{{\mathcal{C}}}]}}$ under this convention.

2.5 The chase procedure

The chase procedure (Deutsch et al. 2008) is a tool exploited for reasoning with TGDs. Consider a database D and a set $\Sigma$ of TGDs. Given an instance $I \supseteq D$ , a trigger for I is any pair $\langle \sigma, h\rangle$ , where $\sigma \in \Sigma$ is a rule as in equation (2) and h is a homomorphism from $\mathit{body}(\sigma)$ to I. Let $I' = I \cup h'(\mathit{head}(\sigma))$ , where $h' \supseteq h|_{\mathbf{x}}$ maps each $z \in \mathit{vars_\exists}(\sigma)$ to a “fresh” null h’(z) not occurring in I such that $z_1 \neq z_2$ in $\mathit{vars_\exists}(\sigma)$ implies $h'(z_1) \neq h'(z_2)$ . Such an operation which constructs I’ from I is called chase step and denoted $\langle \sigma, h \rangle (I) = I'$ .

Without loss of generality, we assume that nulls introduced at each trigger functionally depend on the pair $ \langle \sigma, h \rangle $ that is involved in the trigger. For example, given a rule $\sigma$ as in equation (2) and a homomorphism h, it is sufficient to pick $ \varphi_{\langle \mathbf{z},h(\mathbf{x},\mathbf{y}) \rangle} $ as the fresh null replacing $\mathbf{z}$ when the chase produces the trigger $\langle \sigma, h \rangle$ . Accordingly, the processing order of rules and triggers does not change the result of the chase, and hence $\mathit{chase}(D, \Sigma$ ) can be considered unique. The chase procedure of $D \cup \Sigma$ is an exhaustive application of chase steps, starting from D, which produce a sequence $I_0 = D \subset I_1 \subset I_2 \subset \dots \subset I_m \subset \dots$ of instances in such a way that: (i) for each $i\geq 0$ , $I_{i+1} = \langle \sigma, h \rangle (I_{i})$ is a chase step obtained via some trigger $\langle \sigma, h \rangle$ for $I_{i}$ ; (ii) for each $i \geq 0$ , if there exists a trigger $\langle \sigma, h \rangle$ for $I_i$ , then there exists some $j>i$ such that $I_{j} = \langle \sigma, h\rangle (I_{j-1})$ is a chase step; and (iii) any trigger $\langle \sigma, h\rangle$ is used only once. We define $\mathit{chase}(D,\Sigma) = \cup_{i\geq0} I_i$ .

The chase bottom is the finite set of all null-free atoms in $\mathit{chase}(D,\Sigma)$ and is defined as $\mathit{chase}^{\bot} (D, \Sigma) = \mathit{chase}(D, \Sigma) \cap {{\mathit{HB}}}(D)$ .

It is well know that $\mathit{chase}(D,\Sigma)$ is a universal model of $D \cup \Sigma$ , that is, for each $M \in \mathit{mods}(D, \Sigma)$ there is a homomorphism from $\mathit{chase}(D,\Sigma)$ to M. Hence, given a BCQ q it holds that $\mathit{chase}(D,\Sigma) \models q \Leftrightarrow D \cup \Sigma \models q$ .

We recall that $\mathit{chase}(D, \Sigma)$ can be decomposed into levels (Calì et al. Reference Calì, Gottlob and Pieris2010): each atom of D has level $\gamma = 0$ ; an atom of $\mathit{chase}(D,\Sigma)$ has level $\gamma + 1$ if, during its generation, the exploited trigger $\langle \sigma, h \rangle$ maps the body of $\sigma$ via h to atoms whose maximum level is $\gamma$ . We refer to the part of the chase up to level $\gamma$ as $\mathit{chase}^\gamma(D, \Sigma)$ . Clearly, $\mathit{chase}(D,\Sigma) = \cup_{\gamma \geq 0} \mathit{chase}^\gamma(D, \Sigma)$ . Finally, a trigger involved at a certain level j if it gives rise to an atom of level j.

3.1 Preliminary notions

We start fixing some basics notions. We have chosen to provide a uniform notation for the key existing notions of affected and invaded positions, such as attacked, protected, harmless, harmful, and dangerous variables (Leone et al. Reference Leone, Manna, Terracina and Veltri2019; Calì et al. Reference Calì, Gottlob and Kifer2013; Krötzsch and Rudolph 2011; Berger et al. Reference Berger, Gottlob, Pieris and Sallinger2022; Gottlob et al. 2022). Basically, these notions serve to separate positions in which the chase can introduce only constants from those where nulls might appear.

Note that for every position $ \pi $ there exists a unique set S such that $\pi$ is S-affected. We write $\mathit{aff}(\pi)$ for this set S. Moreover, $ \mathit{aff}(\Sigma) = \{\pi \in \mathit{pos}(\Sigma) \ | \ \mathit{aff}(\pi) \neq \emptyset \}, $ and $ \mathit{nonaff}(\Sigma) = \mathit{pos}(\Sigma) \setminus \mathit{aff}(\Sigma).$

We point out that the notion above presented is a refined version of the classical notion of affected position (Calì et al. Reference Calì, Gottlob and Kifer2008). In particular, it holds that if a position $\pi$ is S-affected, then $\pi$ is also affected; whereas if $\pi$ is affected, then $\pi$ may not be S-affected. Moreover, the S-affected notion coincides with the one of attacked positions by a variable (Leone et al. Reference Leone, Manna, Terracina and Veltri2019; Krötzsch and Rudolph 2011). We highlight that its key nature and properties are not modified by the notion of S-affected position introduced above. Hence, for simplicity of exposition, we give only this refined definition. In the same spirit, we classify variables occurring in a conjunction of atoms.

Given a variable x that is S-dangerous, we write $ \mathit{dang}(x) $ for the set S . Hereinafter, for simplicity of exposition, the prefix S- is omitted when it is not necessary. Consider an ontology $\Sigma$ . Given a rule $\sigma \in \Sigma$ , we denote by $\mathit{dang}(\sigma)$ (resp., $\mathit{harmless}(\sigma)$ and $\mathit{harmful}(\sigma)$ ) the dangerous (resp., harmless and harmful) variables in $\sigma$ . These sets of variables naturally extend to the whole $\Sigma$ by taking, for each of them, the union over all the rules of $\Sigma$ .

3.2 Decidable classes of existential rules

We now survey the 15 concrete classes reported in Table 1 as well as the known abstract classes based on semantic conditions. On the one side, we report some specific syntactic conditions whenever these are useful for the rest of the presentation; on the other side, for all of them (both concrete and abstract), we recall their containment relationships. For the rest of the section, fix a ${{\textsf{Datalog}^{_\exists}}}$ ontology $\Sigma$ .

The class ${{\textsf{FES}}}$ (Baget et al. Reference Baget, Leclère, Mugnier and Salvat2009) stands for finite expansions sets, which intuitively are sets of TGDs which ensure the termination of the chase procedure. The class ${{\textsf{BTS}}}$ (Baget et al. Reference Baget, Leclère, Mugnier and Salvat2009) stands for bounded treewidth sets, which intuitively are sets of TGDs which guarantee that the (possibly infinite) instance constructed by the chase procedure has bounded treewidth. The class ${{\textsf{FUS}}}$ (Baget et al. Reference Baget, Leclère, Mugnier and Salvat2011) stands for finite unification sets, which intuitively are sets of TGDs which guarantee the termination of (resolution-based) backward chaining procedures. The class ${{\textsf{SPS}}}$ (Leone et al. Reference Leone, Manna, Terracina and Veltri2019) stands for strongly parsimonious sets, which intuitively are sets of TGDs which guarantee that the parsimonious chase procedure can be reapplied a number of times that is linear in the size of the query.

The class ${{\textsf{Weakly-Acyclic}}}$ (Fagin et al. Reference Fagin, Kolaitis, Miller and Popa2005) is based on the acyclicity condition. To define the latter, we recall the label graph of $\Sigma$ , $G(\Sigma) = \langle N,A\rangle$ , defined as follows: (i) $N = \cup_{P \in \mathit{sch}(\Sigma)} \mathit{pos}(P)$ ; (ii) $(\pi_1, \pi_2, \forall) \in A$ if there are $\sigma \in \Sigma$ and $x \in \mathsf{vars_\curvearrowright}(\sigma)$ such that x occurs both in $\mathit{body}(\sigma)$ at position $\pi_1$ and in $\mathit{head}(\sigma)$ at position $\pi_2$ ; and (iii) $(\pi_1, \pi_2, \exists) \in A$ if there are $\sigma \in \Sigma$ , $x \in \mathsf{vars_\curvearrowright}(\sigma)$ , and $y \in \mathit{vars_\exists}(\sigma)$ such that both x occurs in $\mathit{body}(\sigma)$ at position $\pi_1$ and y occurs in $\mathit{head}(\sigma)$ at position $\pi_2$ . The existential graph of $\Sigma$ is $G_\exists(\Sigma) = \langle N,A\rangle$ , where: (i) $N = \cup_{\sigma \in \Sigma}\mathit{vars_\exists}(\sigma)$ ; and (ii) $(X,Y) \in A$ if the rule $\sigma$ where y occurs contains a universal variable x-affected and occurring in $\mathit{head}(\sigma)$ . Therefore, $\Sigma$ belongs to ${{\textsf{Weakly-Acyclic}}}$ (resp., ${{\textsf{Jointly-Acyclic}}}$ ) if $G(\Sigma)$ (resp., $G_\exists(\Sigma)$ ) has no cycle going through an $\exists$ -arc (resp., is acyclic).

We now recall the notion of marked variable, in order to define the class ${{\textsf{Sticky}}}$ (Calì et al. Reference Calì, Gottlob and Pieris2012b). A variable x of $\Sigma$ is marked if (i) there is $\sigma \in \Sigma$ such that x occurs in $\mathit{body}(\sigma)$ but not in $\mathit{head}(\sigma$ ); or (ii) there is $\sigma \in \Sigma$ such that x occurs in $\mathit{head}(\sigma)$ at position $\pi$ together with some $\sigma' \in \Sigma$ having a marked variable in its body at position $\pi$ . Accordingly, the stickiness condition states that $\Sigma$ is ${{\textsf{Sticky}}}$ if, for each $\sigma \in \Sigma$ , x occurs multiple times in $\mathit{body}(\sigma)$ implies x is not marked.

The class ${{\textsf{Linear}}}$ (Calì et al. Reference Calì, Gottlob and Lukasiewicz2012a) is based on the linearity condition: an ontology $\Sigma$ belongs to ${{\textsf{Linear}}}$ if each rule contains at most one body atoms. This class generalize the class ${{\textsf{Inclusion-Dependencies}}}$ (Abiteboul et al. Reference Abiteboul, Hull and Vianu1995; Johnson and Klug Reference Johnson and Klug1984) in which rules contain only one body atom and one head atom and the repetition of variables is not allowed neither in the body nor in the head.

The class ${{\textsf{Guarded}}}$ (Calì et al. Reference Calì, Gottlob and Kifer2013) is based on the guardedness condition: an ontology $\Sigma$ belongs to ${{\textsf{Guarded}}}$ if for each rule $\sigma \in \Sigma$ there is $\underline{a}$ in $\mathit{body}(\sigma)$ such that $\mathit{vars_\forall}(\sigma) = \mathit{vars}(\underline{a})$ . In similar fashion, $\Sigma$ belongs to ${{\textsf{Weakly-Guarded}}}$ if, for each $\sigma \in \Sigma$ , there is an atom of $\mathit{body}(\sigma)$ containing all the affected variables of $\sigma$ .

We recall the shyness condition underlying the class ${{\textsf{Shy}}}$ (Leone et al. Reference Leone, Manna, Terracina and Veltri2019). An ontology $\Sigma$ is ${{\textsf{Shy}}}$ if, for each $\sigma \in \Sigma$ the following conditions both hold: (i) if a variable x occurs in more than one body atom, then x is harmless; (ii) for every pair of distinct dangerous variable z and w in different atoms, $ \mathit{dang}(z) \cap \mathit{dang}(w) = \emptyset$ .

The class ${{\textsf{Ward}}}$ (Gottlob and Pieris 2015) is based on the wardedness condition: $\Sigma \in {{\textsf{Ward}}}$ if, for each $\sigma \in \Sigma$ , there are no dangerous variables in $\mathit{body}(\sigma)$ , or there exists an atom $\underline{a} \in \mathit{body}(\sigma)$ , called a ward, such that (i) all the dangerous variables in $\mathit{body}(\sigma)$ occur in $\underline{a}$ , and (ii) each variable of $\mathit{vars}(\underline{a}) \cap \mathit{vars}(\mathit{body}(\sigma) \setminus \{\underline{a}\})$ is harmless.

Having finished with syntactic and semantic conditions, we close the section with a proposition stating their containment relationships (Baget et al. Reference Baget, Leclère, Mugnier and Salvat2011; Krötzsch and Rudolph 2011; Leone et al. Reference Leone, Manna, Terracina and Veltri2019; Baldazzi et al. Reference Baldazzi, Bellomarini, Favorito and Sallinger2022).

Proposition 1

The following classes are pairwise uncomparable, except for:

• – ${{\textsf{Inclusion-Dependencies}}} \subset {{\textsf{Joinless}}}$ , ${{\textsf{Inclusion-Dependencies}}} \subset {{\textsf{Linear}}}$ ;

• – ${{\textsf{Joinless}}} \subset {{\textsf{Sticky}}} \subset {{\textsf{Sticky}}}$ - $\mathsf{Join} \subset {{\textsf{FUS}}}$ ;

• – ${{\textsf{Linear}}} \subset {{\textsf{Guarded}}}$ , ${{\textsf{Linear}}} \subset \mathsf{Protected}$ ;

• – ${{\textsf{Guarded}}} \subset \mathsf{Weakly}$ - ${{\textsf{Guarded}}}$ , ${{\textsf{Guarded}}} \subset \mathsf{Fr}$ -Guarded;

• – $\mathsf{Weakly}$ - ${{\textsf{Guarded}}} \subset \mathsf{Weakly}$ - $\mathsf{Fr}$ -Guarded $\subset {{\textsf{BTS}}}$ ;

• – $\mathsf{Fr}$ -Guarded $\subset \mathsf{Weakly}$ - $\mathsf{Fr}$ -Guarded;

• – ${{\textsf{Datalog}}} \subset \mathsf{Weakly}$ - ${{\textsf{Guarded}}}$ , ${{\textsf{Datalog}}} \subset \mathsf{Protected}$ , ${{\textsf{Datalog}}} \subset {{\textsf{Weakly-Acyclic}}}$ ;

• – $\mathsf{Protected} \subset {{\textsf{Ward}}}$ , $\mathsf{Protected} \subset {{\textsf{Shy}}} \subset {{\textsf{SPS}}}$ ;

• – ${{\textsf{Weakly-Acyclic}}} \subset {{\textsf{Jointly-Acyclic}}} \subset {{\textsf{FES}}}$ .

Throughout the remainder of the paper, let $\mathbb{E}_{syn}$ denote the set of all 15 decidable syntactic classes reported in Table 1. Analogously, let $\mathbb{E}_{sem}$ denote the set of known decidable abstract classes considered in this paper, namely ${{\textsf{FES}}}$ , ${{\textsf{FUS}}}$ , ${{\textsf{BTS}}}$ , and ${{\textsf{SPS}}}$ .

3.3 Autonomous full inclusion dependencies

The aim of this section is to introduce a very simple new class of existential rules called ${{\textsf{Af-Inds}}}$ . Additionally, we characterize the main properties of this class.

Now, we show that any class ${{\mathcal{C}}}$ of TGDs in $\mathbb{E}_{syn} \cup \mathbb{E}_{sem}$ includes the class just defined. Formally, it holds the following.

We conclude the section by providing the complexity of the class Af-Inds.

4.1 Formal definition

We start introducing the concept of head-ground set of rules, being roughly “non-recursive” rules in which nulls are neither created nor propagated.

The following example is given to better understand the above definition.

Example 1 Consider the next set of rules:

$$\begin{array}{rrcl} \sigma_1: & R(x_1, y_1), {S(y_1,u_1), T(u_1,v_1)} & \rightarrow & \exists \ z_1, w_1 \ Q(z_1, w_1)\\ \sigma_2: & C(y_2), R(x_2, z_2) & \rightarrow & S(y_2, z_2) \\ \sigma_3: & D(y_3,z_3), R(x_3, w_3) & \rightarrow & T(x_3, y_3)\end{array}$$

$$\begin{array}{rrcl} \sigma_4: & Q(x_4, y_4) & \rightarrow & \exists \ z_4 A(x_4,z_4) \\ \sigma_5: & A(x_5,z_5), D(y_5,z_5) & \rightarrow & Q(x_5,y_5)\end{array}$$

A subset of head-ground rule w.r.t. $\Sigma$ is given by $ \Sigma_\mathrm{HG} = \{ \sigma_2, \sigma_3 \}$ . In fact, $\mathit{harmless}(\Sigma)$ is the set $\{x_1, y_1, y_2, x_2, z_2, x_3,y_3, z_3, y_5, z_5\}$ ; hence, according to Definition 4, it is easy to check that (i) $\sigma_2$ and $\sigma_3$ are datalog rules; (ii) the head atoms of $\sigma_2$ and $\sigma_3$ contain only harmless variables; (iii) both predicates S and T do not occur in the body of any rule in $\Sigma_\mathrm{HG}$ , and (iv) both predicates S and T do not occur in the head of any rule in $\{\sigma_1, \sigma_4, \sigma_5\}$ . On the contrary, none of the rules in $\{\sigma_1, \sigma_4, \sigma_5\}$ can be part of any head-ground subset of $\Sigma$ . Indeed, according to Definition 4, both $\sigma_1$ and $\sigma_2$ violate properties (1) and (2), whereas $\sigma_5$ violates property (2). Hence, we observe that the set $\Sigma_\mathrm{HG}$ is also maximal.

Having in mind the notion of head-ground set of rules, we can now formally define what is a dyadic pair.

Whenever the above definition applies, we also say, for short, that $\Pi$ is a ${{\mathcal{C}}}$ -dyadic pair. Consider the following example to more easily understand the concept of dyadic pair.

Example 2 Consider the following pair $\Pi = (\Sigma_\mathrm{HG}, \Sigma_\mathcal{C}) $ of TGDs, where $\Sigma_\mathrm{HG}$ is:

$\begin{array}{rcl} P(x_1) & \rightarrow & H_1(x_1)\\ P(x_2) & \rightarrow & H_2(x_2)\\ Q(x_3) & \rightarrow & H_3(x_3).\\ \end{array}$

and $\Sigma_\mathcal{C}$ is:

In particular, $\Pi$ is a dyadic pair with respect to any ${{\mathcal{C}}} \in \{{{\textsf{Guarded}}}, {{\textsf{Shy}}}, {{\textsf{Ward}}}.\}$ . To this aim, let $\Sigma = \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}$ . It easy computable that $\mathit{aff}(\Sigma) = \{R[1],R[2],Q[1],S[1]\}$ , where $\mathit{aff}(R[1])=\{y_1\}$ , $\mathit{aff}(R[2])=\{z_1\}$ , $\mathit{aff}(Q[1])=\{y_2\}$ , and $\mathit{aff}(S[1])=\{y_1\}$ . Accordingly, $\mathit{harmless}(\Sigma) = \{x_1,x_2,x_3\}$ , $\mathit{harmful}(\Sigma) = \{y_3\}$ and $ \mathit{dang}(\Sigma) = \{y_3\}$ . To prove that $\Pi$ is a dyadic pair, we have first to show that $\Sigma_\mathrm{HG}$ is an head-ground set of rules with respect to $\Sigma$ . Clearly, $\Sigma_\mathrm{HG} \in {{\textsf{Datalog}}}$ and each head atom contains only harmless variables; moreover, the head predicates do not appear neither in body atoms of $\Sigma_\mathrm{HG}$ nor in head atoms of $\Sigma_\mathcal{C}$ . Hence, $\Sigma_\mathrm{HG}$ is head-ground with respect to $\Sigma$ . It remains to show that $\Sigma_\mathcal{C} \in {{\mathcal{C}}}$ . We focus on the last rule of $\Sigma_\mathcal{C}$ , since the first two rules are linear rules, and hence are trivially guarded, shy and ward rules. The last rule belongs to ${{\textsf{Guarded}}}$ since the atom $R(y_3,x_3)$ contains all the universal variables of the rule (guardedness condition); it belong to ${{\textsf{Shy}}}$ since the variable $x_3$ that occurs in two body atoms is harmless (shyness condition); finally, it belongs to ${{\textsf{Ward}}}$ since atom $R(y_3,x_3)$ is the ward that contains the dangerous variables $(y_3)$ and shares with the rest of the body only harmless variables $(x_3)$ (wardedness condition).

The next step is to extend the query answering problem – classically defined over an ontology – over a dyadic pair. Therefore, we extend both notions of chase and certain answers for a dyadic pair. Accordingly, given a dyadic pair $\Pi = (\Sigma_\mathrm{HG}, \Sigma_\mathcal{C})$ , we define

(3) \begin{equation}{{\mathit{dp}\textrm{-}\mathit{chase}}}(D,\Pi) = \mathit{chase}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})\end{equation}

and

(4) \begin{equation}{{\mathit{dp}\textrm{-}\mathit{cert}}}(q,D,\Pi) = \mathit{cert}(q,D,\Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}).\end{equation}

Now we can fix the problem studied in the rest of the paper.

4.2 Computational properties

For the rest of the section, fix a decidable class ${{\mathcal{C}}}$ of TGDs. Given a database D and a ${{\mathcal{C}}}$ -dyadic pair $\Pi$ of TGDs, we define the following set of ground atoms:

(5) \begin{equation} {{\mathit{gra}}}(D,\Pi) = \{\underline{a} \in \mathit{chase}(D, \Pi)~|~ \Pi = (\Sigma_\mathrm{HG},\Sigma_\mathcal{C}) \ \wedge \ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \}.\end{equation}

Our idea is to reduce query answering over a dyadic pair $\Pi$ to query answering over $\mathcal{C}$ , the latter being decidable by assumption.

Proof. Consider $\Pi = (\Sigma_\mathrm{HG},\Sigma_\mathcal{C})$ . By equation (4), ${{\mathit{dp}\textrm{-}\mathit{cert}}}(q,D,\Pi) = \mathit{cert}(q,D,\Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})$ . Moreover, by fixing any arbitrary $|{\bf x}|$ -ary tuple c of constants, it holds that ${\bf c} \in \mathit{cert}(q,D,\Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \Leftrightarrow D \cup \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C} \models q({\bf c})$ and ${\bf c} \in \mathit{cert}(q,D^+,\Sigma_\mathcal{C}) \Leftrightarrow D^+ \cup \Sigma_\mathcal{C} \models q({\bf c})$ . Let $q' = q({\bf c})$ . Accordingly, the thesis boils down to showing that

$$D \cup \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C} \models q' \Leftrightarrow D^+ \cup \Sigma_\mathcal{C} \models q'.$$

$[\Rightarrow]$ Assume that $D \cup \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C} \models q'$ holds. Hence, $\mathit{chase}(D , \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \models q'$ . Given that ${{\mathit{gra}}}(D,\Pi) \subseteq \mathit{chase}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})$ , it holds that $\mathit{chase}(D \cup {{\mathit{gra}}}(D,\Pi) , \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \models q'$ . Moreover, since ${{\mathit{gra}}}(D,\Pi)$ contains all the auxiliary ground consequences of $\Sigma_\mathrm{HG}$ , the latter becomes equivalent to $\mathit{chase}(D \cup {{\mathit{gra}}}(D,\Pi) , \Sigma_\mathcal{C}) \models q'$ . Hence, $D \cup {{\mathit{gra}}}(D,\Pi) \cup \Sigma_\mathcal{C} \models q'$ , that is $D^+ \cup \Sigma_\mathcal{C} \models q'$ .

$[\Leftarrow]$ Assume that $D^+ \cup \Sigma_\mathcal{C} \models q'$ , hence $\mathit{chase}(D^+ , \Sigma_\mathcal{C}) \models q'$ . Since $\Sigma_\mathcal{C} \subseteq \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}$ , it holds that $\mathit{chase}(D^+, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \models q'$ . By hypothesis, ${{\mathit{gra}}}(D,\Pi) \subseteq \mathit{chase}(D , \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})$ ; hence $\mathit{chase}(D , \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \models q'$ , that is $D \cup \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C} \models q'$ .

According to Theorem 1, to solve dp-cert-eval $[\mathcal{C}]$ one can first “complete” D and then performing classical query evaluation. To this aim we design Algorithms 1 and 2. The correctness of the latter will be proved in Proposition 4. Consequently, the correctness of the former is guaranteed by Theorem 1.

In particular, given a database D and a dyadic pair $\Pi = (\Sigma_\mathrm{HG}, \Sigma_\mathcal{C})$ , Algorithm 2 iteratively constructs the set $D^+ = D \cup {{\mathit{gra}}}(D,\Pi)$ , with ${{\mathit{gra}}}(D,\Pi)$ being the set defined by equation (5). Roughly speaking, the first two instructions are required, respectively, to add D to $D^+$ and to initialize a temporary set $\tilde{D}$ used to store ground consequences derived from $\Sigma_\mathrm{HG}$ . The rest of the algorithm is an iterative procedure that computes the certain answers (instruction 5) to the queries constructed from the rules of $\Sigma_\mathrm{HG}$ (instruction 4) and completes the initial database D (instruction 7) until no more auxiliary ground atoms can be produced (instruction 6). We point out that, in general, $\tilde{D} \subseteq {{\mathit{gra}}}(D,\Pi)$ holds; in particular, $\tilde{D} = {{\mathit{gra}}}(D,\Pi)$ holds in the last execution of instruction 7 or, equivalently, when the condition $D \cup \tilde{D} \supset D^+$ examined at instruction 6 is false, since all the auxiliary ground atoms have been added to $D^+$ .

Algorithm 1: DpCertEval[$\mathcal{C}$](q, D, Π, c)

Before we prove that Algorithm 2 always terminates and correctly constructs $D^+$ , we show the following preliminary lemma.

Algorithm 2: Complete[$_\mathcal{C}$](D, Π)

Base case: $i = 1$ . We want to prove that $\mathit{chase}^1(D \cup X, \Sigma') \subseteq \mathit{chase}(D, \Sigma)$ . Let $\underline{a}$ be an atom of $\mathit{chase}^1(D \cup X, \Sigma')$ generated exactly at level $i = 1$ . By definition, $\underline{a}$ is obtained due to some trigger $\langle \sigma, h \rangle$ such that h maps $\mathit{body}(\sigma)$ to $D \cup X$ . If $\underline{a} \in \mathit{chase}^p(D, \Sigma)$ , then the claim holds trivially. Otherwise, we can show that $\underline{a} \in \mathit{chase}^{p+1}(D, \Sigma)$ . Indeed, since h maps $\mathit{body}(\sigma)$ to $D \cup X$ , then h is also a trigger involved at level $p+1$ since it maps $\mathit{body}(\sigma)$ to $\mathit{chase}^{p}(D, \Sigma)$ . In particular, h maps at least one atom of $\mathit{body}(\sigma)$ to some atom $\underline{a}_k \in X$ such that $\mathit{lev}(\underline{a}_k) = p$ . Since, by definition, nulls introduced during the chase functionally depend on the involved triggers, then $\underline{a}$ necessarily belongs to $\mathit{chase}^{p+1}(D, \Sigma)$ .

Induction step: $i = \ell$ . Given that for every level $i \leq \ell -1$ , $\mathit{chase}^{i}(D \cup X, \Sigma') \subseteq \mathit{chase}(D,\Sigma)$ (induction hypothesis), we prove that $ \mathit{chase}^{\ell}(D \cup X, \Sigma') \subseteq \mathit{chase}(D,\Sigma) $ holds, too. Let $\beta$ be an atom of $ \mathit{chase}^{\ell}(D \cup X, \Sigma') $ generated exactly at level $i=\ell$ . By definition, $\beta$ is obtained via some trigger $ \langle \sigma', h' \rangle $ such that h’ maps $\mathit{body}(\sigma')$ to atoms with level at most $\ell-1$ . Accordingly, by induction hypothesis, h’ maps $\mathit{body}(\sigma')$ also to $\mathit{chase}(D,\Sigma)$ . Hence, since the processing order of rules and triggers does not change the result of the chase and nulls functionally depend on the involved triggers, it follow that also $\beta \in \mathit{chase}(D,\Sigma)$ .

With the next proposition, we prove that Algorithm 2 always terminates and correctly constructs $D^+$ .

Termination. To prove the termination of Algorithm 2, it suffices to show that each instruction alone always terminates and that the overall procedure never falls into an infinite loop. First, observe that $|{{\mathit{gra}(D,\Pi)}}| \leq |\mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG})|\cdot d^\mu$ , where $d=|\mathit{consts}(D)|$ and $ \mu = \max_{P \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG})}{\mathit{arity}(P)}$ . Instructions 1, 2, 4, 8, and 9 trivially terminate. Instructions 6 and 7 both terminate, since $\tilde{D} \subseteq {{\mathit{gra}(D,\Pi)}}$ always holds (see correctness below). Each time instruction 3 is reached, the for-loop simply scans the set $\Sigma_\mathrm{HG}$ , which is finite by definition. Concerning instruction 5, it suffices to observe that its termination relies on the termination of cert-eval ${{[{{\mathcal{C}}}]}}$ – which is true by hypothesis – and on the fact that, for each query q, to construct the set $\{ \mathtt{H}_i(\mathbf{t}) \ | \ \mathbf{t} \in \mathit{cert}(q,D^+,\Sigma_\mathcal{C}) \}$ , the problem cert-eval ${{[{{\mathcal{C}}}]}}$ must be solved at most $d^\mu$ times, where $d^\mu$ is the maximum number of tuples t for which the check $\mathbf{t} \in \mathit{cert}(q,D^+,\Sigma_\mathcal{C})$ has to be performed. Since each instruction alone terminates, it remains to analyze the overall procedure. It contains two loops. The first, namely the for-loop at instruction 3, is not problematic; indeed, we shown that it locally terminates. The second one, namely the go to-loop, depends on the evaluation of the if-instruction, which can be executed at most $|{{\mathit{gra}(D,\Pi)}}|$ times. Thus, also the go to-loop does the same.

Correctness. We now claim that Algorithm 2 correctly completes the database. Let $D^+$ be the output of Algorithm 2. Our claim is that $D^+ = D \cup {{\mathit{gra}(D,\Pi)}}$ .

Inclusion 1 ( $D^+ \supseteq D \cup {{\mathit{gra}(D,\Pi)}}$ ). Assume, by contradiction, that $D \cup {{\mathit{gra}(D,\Pi)}}$ contains some atom that does not belong to $D^+$ . This means that there exists some $j>0$ such that both $\bar{D} = ((D \cup {{\mathit{gra}(D,\Pi)}}) \cap \mathit{chase}^{j-1}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})) \subseteq D^+$ and $((D \cup {{\mathit{gra}(D,\Pi)}}) \cap \mathit{chase}^j(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})) \setminus D^+ \neq \emptyset$ hold. Thus, there exists some $\underline{a} \in \mathit{chase}^j(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})$ whose level is exactly j and that does not belong to $D^+$ . Let $\langle \sigma, h\rangle$ be the trigger used by the chase to generate $\underline{a}$ , where $\sigma$ is of the form $\Phi(\mathbf{x,y}) \rightarrow \mathtt{H}(\mathbf{x})$ . Clearly, h maps $\Phi(\mathbf{x},\mathbf{y})$ to $\mathit{chase}^{j-1}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C})$ , and we also have that $\underline{a} = \mathtt{H}(h(\mathbf{x}))$ . Consider now the query $q =\langle \mathbf{x} \rangle \leftarrow \Phi(\mathbf{x,y})$ constructed from $\sigma$ by Algorithm 2 at instruction 4. Thus, $\mathit{chase}^{j-1}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) \models q(h({\bf x}))$ holds. Since $\bar{D} \subseteq D^+$ , we have that $\mathit{chase}^{j-1}(D, \Sigma_\mathrm{HG}\cup\Sigma_\mathcal{C}) \subseteq\mathit{chase}(\bar{D}, \Sigma_\mathcal{C})\subseteq\mathit{chase}(D^+, \Sigma_\mathcal{C})$ . Hence, $\mathit{chase}(D^+, \Sigma_\mathcal{C}) \models q(h({\bf x}))$ , namely $h({\bf x}) \in \mathit{cert}(q,D^+,\Sigma_\mathcal{C})$ and, thus, $\underline{a} \in D^+$ , which is a contradiction.

Inclusion 2 ( $D^+ \subseteq D \cup {{\mathit{gra}(D,\Pi)}}$ ). Let $D^+$ be the set produced by Algorithm 2. Let $\ell$ be the number of time instruction 7 of Algorithm 2 is executed. At each execution $i \in [\ell]$ of instruction 7, the algorithm computes the set $\tilde{D}_i$ containing only auxiliary ground atoms, and produces the set $D^+_i = D \cup \tilde{D}_i$ . By construction, $D^+ = D \cup \tilde{D}_\ell$ . Let $\tilde{D}_0 = \emptyset$ , $D^+_0 = D \cup \tilde{D}_0$ , and $I_i = \tilde{D}_i \setminus \tilde{D}_{i-1}$ , for each $i \in [\ell]$ . We show that $D \cup \tilde{D}_\ell \subseteq D \cup {{\mathit{gra}(D,\Pi)}}$ , that is $\tilde{D}_\ell \subseteq {{\mathit{gra}(D,\Pi)}}$ . We proceed by induction on the number $\ell$ of iterations.

Base case: Let $i = 1$ . We claim that $ \tilde{D}_1 \subseteq {{\mathit{gra}(D,\Pi)}} $ . By construction, the set $\tilde{D}_1 = \{ \mathtt{H}_i(\mathbf{t}) ~|~ i \in [|\Sigma_\mathrm{HG}|] ~\wedge~ \mathbf{t} \in \mathit{cert}(q, D^+_0, \Sigma_\mathcal{C}) \}$ . Since $D^+_0 = D$ and the component $\Sigma_\mathcal{C}$ does not produce any atom in the first iteration of the algorithm, the latter is equal to $\{ \mathtt{H}(\mathbf{t}) ~|~ i \in [|\Sigma_\mathrm{HG}|] ~\wedge~ \mathbf{t} \in \mathit{cert}(q, D, \emptyset) \}= \{ \underline{a} \in \mathit{chase}^1(D, \Sigma_\mathrm{HG}) ~|~ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \}\subseteq \{ \underline{a} \in \mathit{chase}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) ~|~ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \} = {{\mathit{gra}(D,\Pi)}}$ .

Induction step: Given that, for $i = \ell-1, \ \tilde{D}_{\ell -1} \subseteq {{\mathit{gra}(D,\Pi)}} $ (induction hypothesis), we prove that $ \tilde{D}_\ell \subseteq {{\mathit{gra}(D,\Pi)}} $ holds, too. By construction $\tilde{D}_\ell = \{ \mathtt{H}_i(\mathbf{t}) ~|~ i \in [|\Sigma_\mathrm{HG}|] ~\wedge~ \mathbf{t} \in \mathit{cert}(q, D^+_{\ell-1}, \Sigma_\mathcal{C}) \}= \{ \mathtt{H}_i(\mathbf{t}) ~|~ i \in [|\Sigma_\mathrm{HG}|] ~\wedge~ \mathbf{t} \in \mathit{cert}(q, D \cup \tilde{D}_{\ell-1}, \Sigma_\mathcal{C}) \}= \{ \underline{a} \in \mathit{chase}(D \cup \tilde{D}_{\ell-1}, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) ~|~ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \}$ . Since the set $ D \cup \tilde{D}_{\ell-1}$ already contains all the ground consequences of $\Sigma_\mathrm{HG}$ , the latter is equivalent to $\{ \underline{a} \in \mathit{chase}(D \cup \tilde{D}_{\ell-1}, \Sigma_\mathcal{C}) ~|~ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \}$ . Applying Lemma 1 together with the induction hypothesis, the last one is a subset of $ \{ \underline{a} \in \mathit{chase}(D, \Sigma_\mathrm{HG} \cup \Sigma_\mathcal{C}) ~|~ \mathit{preds}(\underline{a}) \in \mathit{h}\textrm{-}\mathit{preds}(\Sigma_\mathrm{HG}) \} = {{\mathit{gra}(D,\Pi)}}$ .

We conclude the section by proving the decidability of the problem dp-cert-eval ${{[{{\mathcal{C}}}]}}$ , under the assumption that cert-eval ${{[{{\mathcal{C}}}]}}$ is decidable.