2.1 Minimal knowledge and negation as failure (MKNF)

MKNF is a nonmonotonic logic formulated by Lifschitz (Reference Lifschitz1991). MKNF formulas extend first-order formulas with two modal operators, $\mathbf{K}$ for minimal knowledge, and $\mathbf{not}$ for negation as failure. We define a first-order interpretation $I$ as usual and denote the universe of $I$ by $|I|$ . An MKNF structure is a triple $(I,M,N)$ , where $M$ and $N$ are sets of first-order interpretations within the universe $|I|$ . The language of MKNF formulas contains a constant for each element of $\left |I\right |$ , which we call a name. We define the satisfaction relation between an MKNF structure $(I,M,N)$ and an MKNF formula as follows:

\begin{align*} &(I,M,N) \models \phi \text{ ($\phi $ is a first-order atom) if } \phi \text{ is true in } I, \\ &(I,M,N) \models \neg \phi \text{ if } (I,M,N) \not \models \phi, \\ &(I,M,N) \models \phi _1 \wedge \phi _2 \text{ if } (I,M,N) \models \phi _1 \text{ and } (I,M,N) \models \phi _2, \\ &(I,M,N) \models \exists x \phi \text{ if } (I,M,N) \models \phi [a\setminus x] \text{ for some } a, \\ &(I,M,N) \models \mathbf{K} \phi \text{ if } (J,M,N) \models \phi \text{ for all } J\in M, \\ &(I,M,N) \models \mathbf{not}\, \phi \text{ if } (J,M,N) \not \models \phi \text{ for some } J \in N. \end{align*}

The symbols $\top, \bot, \vee, \forall, \text{ and } \supset$ are interpreted as usual.

An MKNF interpretation $M$ is a nonempty set of first-order interpretations. Throughout this work, we employ the standard name assumption to avoid unintended behaviors (Motik and Rosati Reference Motik and Rosati2010). This assumes all interpretations are Herbrand interpretations with a countably infinite number of additional constants, and that the predicate $\approx$ is a congruence relation. Thus, we do not explicitly mention the universe associated with interpretations.

An MKNF interpretation $M$ satisfies an MKNF formula $\phi$ , written $M \models _{\text{MKNF}} \phi$ , if $(I,M,M) \models \phi$ for each $I \in M$ .

2.2 Hybrid MKNF knowledge bases (HMKNF-KBs)

Motik and Rosati (Reference Motik and Rosati2010) identify a subset of MKNF formulas as Hybrid MKNF. In this new language, an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ consists of a finite set of rules termed a rule base $\mathcal{P}$ , and an ontology $\mathcal{O}$ translatable to first-order logic as $\pi (\mathcal{O})$ . A rule $r$ is of the form

\begin{align*} h_0, \ldots, h_m\leftarrow p_0, \ldots, p_j, \neg n_0, \ldots, \neg n_k \end{align*}

where $h_i$ , $p_i$ , and $n_i$ are function-free first-order atoms. We denote $body^+(r)=\{p_0,\ldots, p_j\}$ , $body^-(r)=\{n_0,\ldots, n_k\}$ , $Body(r) = \bigwedge body^+(r) \wedge \neg \bigvee body^-(r)$ , and $head(r)=\{h_0,\ldots, h_m\}$ . A rule $r$ ’s semantics is governed by the following MKNF formula.

\begin{align*} \pi (r)= \forall \overrightarrow{x}: (\mathbf{K} h_0\vee \cdots \vee \mathbf{K} h_z)\subset (\mathbf{K} p_0\wedge \cdots \wedge \mathbf{K} p_j \wedge \mathbf{not}\ n_0 \wedge \cdots \wedge \mathbf{not}\ n_k) \end{align*}

where $\overrightarrow{x}$ is the vector of free variables in $r$ . Naturally, a rule base $\mathcal{P}$ translates to an MKNF formula as ${\pi (\mathcal{P})} = \bigcup _{r \in \mathcal{P}} \pi (r)$ . We say that an MKNF interpretation $M$ is an MKNF model of the HMKNF-KB $\mathcal{K}$ if it is an MKNF model of the MKNF formula $\pi ({\mathcal{K}}) = \mathbf{K}\pi (\mathcal{O}) \wedge{\pi (\mathcal{P})}$ .

A program $\mathcal{P}$ is ground if no rule $r \in{\mathcal{P}}$ has variables. Motik and Rosati (Reference Motik and Rosati2010) show that under the assumption of DL-safety,Footnote ² a first-order rule base is semantically equivalent to a finite ground rule base and that decidability is guaranteed for HMKNF-KBs with decidable ontologies. In this work, we assume rule bases are ground.

We define $\mathbf{KA} (\mathcal{K})$ be the set containing every atom $\phi$ that occurs in $\mathcal{P}$ (either as $\mathbf{K} \phi$ or $\mathbf{not}\ \phi$ ) and we use $\mathbf{KA} (\mathcal{\mathcal{O}})$ to denote the maximal subset of $\mathbf{KA} (\mathcal{K})$ s.t. for each $p(t_1,\ldots, t_m) \in \mathbf{KA} (\mathcal{\mathcal{O}})$ , the predicate $p$ also appears in $\pi (\mathcal{O})$ .

We define the objective knowledge of an HMKNF-KB $\mathcal{K}$ w.r.t. a set $S \subseteq \mathbf{KA} (\mathcal{K})$ as the set $OB_{\mathcal{O}, S}= \{\pi (\mathcal{O})\}\cup \{\phi \ | \ \phi \in S\}$ . Intuitively, $OB_{\mathcal{O}, S}$ is a first-order formula that considers $\mathcal{O}$ with the assumption that $\mathbf{K} \phi$ holds for each $\phi \in S$ . In this paper, we prefer polynomial ontologies, that is for any $S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}})$ and $a \in \mathbf{KA} (\mathcal{\mathcal{O}})$ , the relation $OB_{\mathcal{O}, S} \models a$ can be checked in polynomial time.

The following notion of a $\mathbf{K}$ -interpretation allows for a simplified representation of an MKNF interpretation as a single set of atoms. A central focus of this work is to show which $\mathbf{K}$ -interpretation s are induced by MKNF models. Such a relationship is not completely straightforward, due to the quite different nature of the two representations.

Above, we use the notation $\hat{I}$ to express that as an interpretation in an entailment relation, $\hat{I} \models \phi$ , any atom $a \in \mathbf{KA} (\mathcal{K})$ but not in $\hat{I}$ is assigned to false.

2.3 Assignments and nogoods

Nogoods (Gebser et al. Reference Gebser, Kaufmann and Schaub2012) act as canonical representations of Boolean constraints, reflecting partial assignments, which cannot be extended to a solution. A nogood $\{\sigma _1, \ldots, \sigma _n\}$ , is a set of literals $\sigma _i$ , of the form ${\mathbf{T}} v_i$ or ${\mathbf{F}} v_i$ for $1\leq i \leq n$ , where $v_i$ is a propositional variable. The complement of a literal, is referred to by $\overline{{\mathbf{T}} v} ={\mathbf{F}} v$ and $\overline{{\mathbf{F}} v} ={\mathbf{T}} v$ . For any set $\delta$ of literals, $\delta ^T = \{v \ |{\mathbf{T}} v \in \delta \}$ and $\delta ^F = \{v \ |{\mathbf{F}} v \in \delta \}$ . The set of variables occurring within a set of nogoods $\Delta$ , is denoted $var(\Delta )=\bigcup _{\delta \in \Delta } (\delta ^{\mathbf{T}} \cup \delta ^{\mathbf{F}})$ . An assignment $A$ for $\Delta$ is any subset of $\{{\mathbf{T}} v,{\mathbf{F}} v \ | \ v \in var(\Delta )\}$ such that $A^{\mathbf{T}} \cap A^{\mathbf{F}} = \emptyset$ . A solution for $\Delta$ is an assignment $A$ for $\Delta$ , such that $A^{\mathbf{T}} \cup A^{\mathbf{F}} = var(\Delta )$ and $\delta \not \subseteq A$ for all $\delta \in \Delta$ . A nogood $\delta$ is unit-resulting for an assignment $A$ if $|\delta \setminus A| = 1$ . For a literal ${\mathbf{T}} \sigma$ in an assignment, $\sigma$ is true within the assignment, whereas for ${\mathbf{F}} \sigma$ , $\sigma$ is false.

4.1 Completion

Lee and Lifschitz (Reference Lee and Lifschitz2003) show that an interpretation of a tight disjunctive logic program is an answer set if and only if it satisfies a set of formulas termed the completion. A program’s completion is composed of a rule completion, clauses that ensure atoms whose truth is implied must be included in an answer set, and a support completion, clauses that ensure true atoms within an interpretation are supported by some rule.

The rule completion of Lee and Lifschitz (Reference Lee and Lifschitz2003) can be easily adopted as follows.

The formula $\mathcal{P}_{rule}$ guarantees that for any satisfying $\mathbf{K}$ -interpretation $\hat{I}$ , atoms implied by the rules of $\mathcal{P}$ under $\hat{I}$ are contained in $\hat{I}$ . We can construct an analogous formula for an ontology $\mathcal{O}$ that requires all atoms entailed by $\mathcal{O}$ given $\hat{I}$ to be within $\hat{I}$ . We define this notion as saturation and formulate what we call saturation completion.

In the following lemma, we provide alternative characterizations of saturation to provide further insight into its relevance and establish a relationship between $\mathbf{K}$ -interpretation s and MKNF interpretations. For instance, characterization 3 below implies that all $\mathbf{K}$ -interpretation s induced by MKNF models are saturated.

The relationship between saturated $\mathbf{K}$ -interpretation s which satisfy the rule completion and MKNF interpretations is described by the following proposition.

Unlike the rule completion, the support completion of Lee and Lifschitz (Reference Lee and Lifschitz2003) is not immediately adaptable to HMKNF-KBs. This is because their notion of support is based on the idea that atoms must be implied by rules, but applying this idea directly to HMKNF-KBS would be misguided as they combine both rules-based and ontological reasoning. We instead define two different notions of support for an atom $a$ occurring within a $\mathbf{K}$ -interpretation $\hat{I}$ of an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ . Firstly, $a$ is supported by a rule $r \in \mathcal{P}$ , if $\hat{I} \models r$ while $\hat{I} \setminus \{a\} \not \models r$ . Secondly, $a$ is supported via the ontology $\mathcal{O}$ if $OB_{\mathcal{O}, \hat{I} \setminus \{a\}} \models a$ .

We can now specify a formula which ensures that all atoms within a satisfying $\mathbf{K}$ -interpretation are supported by either a rule or the ontology.

Definition 7 The support completion of an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ , w.r.t. a $\mathbf{K}$ -interpretation $\hat{I}$ , is the set of formulas ${\mathcal{K}}_{sup}(\hat{I}) = \{\phi (a) \ | \ a \in \mathbf{KA} (\mathcal{K}), OB_{\mathcal{O}, \hat{I}\setminus \{a\}} \not \models a\}$ , where

\begin{align*} \phi (a) = a \supset \bigvee _{\substack{ r \in \mathcal{P} \,\&\, a \in head(r) }} \big ( Body(r) \wedge \bigwedge _{p \in head(r) \setminus \{a\}} \neg p \big ). \end{align*}

When we combine the rule completion, saturation completion, and support completion, we obtain a formula which determines whether a $\mathbf{K}$ -interpretation of a tight HMKNF-KB is induced by some MKNF model.

4.2 Loop formulas

The loop formulas of Lee and Lifschitz (Reference Lee and Lifschitz2003) generalize the support completion and allow for the assumption of tightness to be dropped by ensuring that each set of atoms forming a loop has support. We generalize our earlier notion of support to any subset $L$ , of a $\mathbf{K}$ -interpretation $\hat{I}$ for an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ as follows. A rule $r \in \mathcal{P}$ supports $L$ if $\hat{I} \models r$ but $\hat{I} \setminus L \not \models r$ , whereas $L$ is supported by the ontology $\mathcal{O}$ if $OB_{\mathcal{O}, \hat{I} \setminus L} \models \bigvee L$ . Thus, the conversion of loop formulas to the case of HMKNF-KBs, is similar to that for the support completion, instead of requiring that each atom within a $\mathbf{K}$ -interpretation is supported, we require that each subset forming a loop within $G({\mathcal{K}})$ is supported by either a rule or the ontology.

Definition 9 The loop formulas of an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ with respect to a $\mathbf{K}$ -interpretation $\hat{I}$ , is the set of formulas ${\mathcal{K}}_{loop} (\hat{I}) = \{ \psi (L) \ | \ L \in Loops({\mathcal{K}}), OB_{\mathcal{O}, \hat{I} \setminus L} \not \models \bigvee L \}$ , where

\begin{align*} \psi (L) = \bigvee L \supset \bigg ( \bigvee _{\substack{r \in \mathcal{P} \\ head(r)\cap L \ne \emptyset \\ body^+(r)\cap L = \emptyset }} \Big ( Body(r)\wedge \bigwedge _{a\in head(r) \setminus L} \neg a \Big ) \bigg ). \end{align*}

We combine the completion and loop formulas to determine whether a $\mathbf{K}$ -interpretation of an arbitrary HMKNF-KB is induced by any MKNF model.

Example 1 Consider the following HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ , representing whether a person $p$ , is a good candidate for a blood-pressure medication:

\begin{align*} \mathcal{P} = \left \{ \begin{array}{ll} r_1 = goodCand(p) \leftarrow (cand(p), \neg highRisk(p)). & r_2 =highBP(p). \\ r_3 = highRisk(p) \leftarrow (riskFactor(p), \neg risksTreated(p)). & \\ \end{array} \right \} \end{align*}

and $\pi (\mathcal{O}) = \{\forall x, (highBP(x) \supset cand(x)) \wedge (highRisk(x) \supset riskFactor(x))\}$ .

The ontology states that any person with high blood-pressure is a candidate, and that anyone who is high-risk for the drug has a risk factor. The rule base states that $p$ is a good candidate for the drug if they are a non-high-risk candidate, that they are high risk if they have a risk factor they have not been treated for, and that they have high blood-pressure. Below is the dependency graph $G({\mathcal{K}})$ .

The following are selected $\mathbf{K}$ -interpretation s for $\mathcal{K}$ , of which only $\hat{I}_4$ is induced by any MKNF model: $\hat{I}_1 = \{highBP(p)\}$ , $\hat{I}_2 = \{highBP(p),$ $cand(p), goodCand(p),$ $risksTreated(p)\}$ , $\hat{I}_3 = \{highBP(p),$ $cand(p),$ $highRisk(p),$ $riskFactor(p)\}$ , and $\hat{I}_4 = \{goodCand(p),$ $cand(p),$ $highBP(p)\}$ .

Clearly, $cand(p)$ is entailed by the ontology given $\hat{I}_1$ , so $\hat{I}_1$ fails to satisfy $\mathcal{O}_{satr}(\hat{I}_1)$ . For $\hat{I}_2$ , it is clear that $risksTreated(p)$ has no form of support, therefore it fails to satisfy ${\mathcal{K}}_{sup}(\hat{I}_2)$ . Finally, from examination of $G({\mathcal{K}})$ , $\{highRisk(p), riskFactor(p)\}$ forms a loop $L \in Loops({\mathcal{K}})$ . As each atom is only supported by the other $\hat{I}_3$ does not satisfy ${\mathcal{K}}_{loop}(\hat{I}_3)$ .

In contrast to the other $\mathbf{K}$ -interpretation s, $\hat{I}_4$ satisfies ${\mathcal{K}}_{comp}(\hat{I}_4)$ and ${\mathcal{K}}_{loop}(\hat{I}_4)$ . It avoids the problem with $\hat{I}_1$ as $\hat{I}_4 \models cand(p)$ , the one with $\hat{I}_2$ as $\hat{I}_4 \not \models risksTreated(p)$ , and the one with $\hat{I}_3$ as $\hat{I}_4 \not \models \bigvee L$ . This is consistent with the fact that it is the only $\mathbf{K}$ -interpretation induced by an MKNF model.

5.1 Completion nogoods

5.1.3 Support nogoods

To reflect whether an atom $p$ is supported by a rule $r$ , we use literal sets of the form $\beta _{\mathcal{P}}(r,p) = \{{\mathbf{T}} \beta (r)\} \cup \{{\mathbf{F}} q \ | \ q \in head(r) \setminus \{p\}\}$ , as is done in Gebser et al. (Reference Gebser, Kaufmann and Schaub2013).Footnote ³ The truth of literals based on these sets are also enforced by conjunction nogoods.

We introduce a novel set of support nogoods corresponding to the support completion, to prevent cases where there is no support from a rule or the ontology for some true atom within an assignment.

Definition 12 The support nogood for any atom $p \in \mathbf{KA} (\mathcal{K})$ , is defined as

\begin{align*} \psi _{\mathcal{K}} (p) = \{{\mathbf{T}} p\} \cup \{{\mathbf{F}} \beta _{\mathcal{P}} (r,p) \ | \ r \in \mathcal{P}, p \in head(r)\} \cup \{{\mathbf{F}} \beta _{\mathcal{O}} (p) \,|\, \textrm{if } p \in \mathbf{KA} (\mathcal{\mathcal{O}})\}. \end{align*}

The support nogoods of $\mathcal{K}$ , are $\Psi _{\mathcal{K}} = \{\psi _{\mathcal{K}}(p) \ | \ p \in \mathbf{KA} (\mathcal{K})\}$ .

5.1.5 Entailment nogoods

To ensure the literals of the form ${\mathbf{T}} \beta _{\mathcal{O}} (p)$ or ${\mathbf{F}} \beta _{\mathcal{O}} (p)$ appear within assignments in accordance with whether $p$ is supported via the ontology, we introduce entailment nogoods.

Definition 14 Let $\beta _{\mathcal{O}}(p)$ be a variable associated with an atom $p \in \mathbf{KA} (\mathcal{\mathcal{O}})$ . A positive entailment nogood, of an atom $p\in \mathbf{KA} (\mathcal{\mathcal{O}})$ and set $S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}})$ , is defined as $\gamma _{\mathcal{O}}^+ (p, S) = \{{\mathbf{F}} \beta _{\mathcal{O}}(p)\} \cup \{{\mathbf{T}} s \ | \ s \in S\}$ , whereas a negative entailment nogood, is defined as $\gamma _{\mathcal{O}}^- (p, S) = \{{\mathbf{T}} \beta _{\mathcal{O}}(p)\} \cup \{{\mathbf{F}} s \ | \ s \in S\}.$ The entailment nogoods of $\mathcal{K}$ are

\begin{align*} \Gamma _{\mathcal{O}} = & \{\gamma _{\mathcal{O}}^+ (p, S) \ | \ p \in \mathbf{KA} (\mathcal{\mathcal{O}}) \cup \{\bot \},S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}}), OB_{\mathcal{O}, S \setminus \{p\}} \models p\} \\ &\cup \{\gamma _{\mathcal{O}}^- (p, S) \ | \ p \in \mathbf{KA} (\mathcal{\mathcal{O}}),S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}}), OB_{\mathcal{O}, \mathbf{KA} (\mathcal{\mathcal{O}}) \setminus (S \cup \{p\})} \not \models p \}. \end{align*}

For some atom $p \in \mathbf{KA} (\mathcal{\mathcal{O}})$ and set of atoms $S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}})$ , the purpose of a positive entailment nogood $\gamma _{\mathcal{O}}^+(p,S)$ is to indicate that $p$ is supported via $\mathcal{O}$ by true atoms within an assignment, given it has all atoms in $S$ as true. Conversely, the purpose of a negative entailment nogood $\gamma _{\mathcal{O}}^-(p,S)$ is to indicate that $p$ has no way of being supported via $\mathcal{O}$ by true atoms within an assignment, given it has all atoms in $S$ as false.

5.1.6 Relation to MKNF models

So far we have discussed the relationship between assignments and $\mathbf{K}$ -interpretation s using general language. The exact correspondence is given by the following definition.

Definition 15 Let $\hat{I}$ a $\mathbf{K}$ -interpretation for $\mathcal{K}$ . The induced assignment of $\hat{I}$ for $\mathcal{K}$ is

\begin{align*} &A_{\mathcal{K}}^{\hat{I}} = \{{\mathbf{T}} p \ | \ p \in \hat{I}\} \cup \{{\mathbf{F}} p \ | \ p \in \mathbf{KA} (\mathcal{K}) \setminus \hat{I}\} \cup \{{\mathbf{F}} \bot \} \\ &\cup \{{\mathbf{T}} \beta (r) \ | \ r \in \mathcal{P}, \hat{I} \models Body(r)\} \cup \{{\mathbf{F}} \beta (r) \ | \ r \in \mathcal{P}, \hat{I} \not \models Body(r)\} \\ &\cup \{{\mathbf{T}} \beta _{\mathcal{P}} (r,p) \ | \ r \in \mathcal{P}, \hat{I} \models Body(r), head(r) \cap (\hat{I} \cup \{p\}) = \{p\}\} \\ &\cup \{{\mathbf{F}} \beta _{\mathcal{P}} (r,p) \ | \ r \in \mathcal{P}, \hat{I} \not \models Body(r), p \in head(r)\} \\ &\cup \{{\mathbf{F}} \beta _{\mathcal{P}} (r,p) \ | \ r \in \mathcal{P}, p \in head(r), head(r) \cap \hat{I} \not \subseteq \{p\} \} \\ &\cup \{{\mathbf{T}} \beta _{\mathcal{O}} (p) \ | \ p \in \mathbf{KA} (\mathcal{\mathcal{O}}) \cup \{\bot \}, OB_{\mathcal{O}, \hat{I} \setminus \{p\}} \models p \} \\ &\cup \{{\mathbf{F}} \beta _{\mathcal{O}} (p) \ | \ p \in \mathbf{KA} (\mathcal{\mathcal{O}}) \cup \{\bot \}, OB_{\mathcal{O}, \hat{I} \setminus \{p\}} \not \models p\}. \end{align*}

The nogoods we have defined thus far make up our completion nogoods.

Definition 16 The completion nogoods of $\pi ({\mathcal{K}})$ are

\begin{align*} \Delta _{\mathcal{K}} = \Phi _{\mathcal{P}} \cup \Phi _{\mathcal{O}} \cup \Psi _{\mathcal{K}} \cup \Gamma _{\mathcal{P}} \cup \Gamma _{\mathcal{O}}. \end{align*}

We have designed the induced assignment of any $\mathbf{K}$ -interpretation to be a total assignment for the completion nogoods, and the nogoods themselves to occur in and only in assignments induced by $\mathbf{K}$ -interpretation s which do not satisfy their completion formulas. In showing this to be the case, we obtain the following result.

Theorem 3 Let $\hat{I}$ be a $\mathbf{K}$ -interpretation for an HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ , then we have that $\hat{I} \models{\mathcal{K}}_{comp}(\hat{I})$ if and only if $A^{\hat{I}}_{{\mathcal{K}}}$ is a solution to $\Delta _{\mathcal{K}} \cup \{{\mathbf{T}} \bot \}$ .

The above theorem allows us to conclude that the $\mathbf{K}$ -interpretation s that induce solutions to the completion nogoods are the ones which satisfy their completion formulas. It naturally follows from Theorem 1 , that for tight HMKNF-KBs these are also the $\mathbf{K}$ -interpretation s induced by MKNF models.

5.2 Loop nogoods

Loop nogoods are intended to parallel the loop formulas of Section 4.2 in ensuring non-circular support. Directly following Gebser et al. (Reference Gebser, Kaufmann and Schaub2013), we denote the external program supports of a set of atoms $L$ , by $\mathcal{E}_{\mathcal{P}} (L) = \{r \in \mathcal{P} \ | \ head(r) \cap L = \emptyset, body^+(r) \cap L \neq \emptyset \}$ . Moreover, $\rho (r, L) = \{{\mathbf{F}} \beta (r)\}\cup \{{\mathbf{T}} p \ | \ p \in head(r) \setminus L\}$ , collects all literals satisfying a rule $r$ , regardless of whether any atom from $L$ is true.

Definition 17 The loop nogoods for any $L \subseteq \mathbf{KA} (\mathcal{K})$ and $S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}})$ are defined as

\begin{align*} \lambda _{\mathcal{K}} (L, S) = \{& \{{\mathbf{T}} p, \sigma _1,\ldots, \sigma _k,{\mathbf{F}} s_1, \ldots, {\mathbf{F}} s_n \} \ | \ p \in L, \mathcal{E}_{\mathcal{P}}(L)= \{r_1,\ldots, r_k\}, \\ &\sigma _1 \in \rho (r_1,L),\ldots, \sigma _k \in \rho (r_k, L), S = \{s_1, \ldots, s_n\} \}. \end{align*}

The loop nogoods of $\mathcal{K}$ are

\begin{align*} \Lambda _{\mathcal{K}} = \bigcup _{\substack{ L \in Loops({\mathcal{K}}), S \subseteq \mathbf{KA} (\mathcal{\mathcal{O}}), \\ L\cap S = \emptyset, OB_{\mathcal{O}, \mathbf{KA} (\mathcal{\mathcal{O}}) \setminus (S \cup L) \not \models \bigvee L} }} \lambda _{\mathcal{K}} (L, S). \end{align*}

Above, $\Lambda _{\mathcal{K}}$ includes a nogood for each loop, $L$ , and subset of $\mathbf{KA} (\mathcal{\mathcal{O}})$ , $S$ , for which $L$ cannot be supported via $\mathcal{O}$ whenever all atoms in $S$ are false. Each nogood $\lambda (L,S)$ requires that the loop $L$ must be supported by a rule if all atoms in $S$ are false. We can show that loop nogoods occur only within assignments induced by $\mathbf{K}$ -interpretation s which do not satisfy their loop formulas, to obtain the following result.

This theorem, together with Theorem2, implies that, for any HMKNF-KB, the K-interpretations that induce solutions to the completion and loop nogoods are precisely those that are induced by MKNF models.

Example 2 Consider the following HMKNF-KB ${\mathcal{K}} = (\mathcal{P}, \mathcal{O})$ where only the $\mathbf{K}$ -interpretation $\hat{I} = \{a, b\}$ is induced by any MKNF model.

\begin{align*} \mathcal{P} = \left \{ \begin{array}{ll} r_1 = a. & r_2 = a \vee d. \\ r_3 = f \leftarrow d. & r_4 = e \leftarrow f. \end{array} \right \} \end{align*}

and $\pi (\mathcal{O}) = \{(a \supset b) \wedge (c \supset d) \wedge (c \supset e) \wedge (e \supset f)\}$ . $G({\mathcal{K}})$ is shown below.

We will show that $A^{\hat{I}}_{{\mathcal{K}}}$ , the assignment $\hat{I}$ induces, is the only solution to our nogoods. To do so we take $A$ to be an arbitrary solution and prove that it contains every literal within $A^{\hat{I}}_{{\mathcal{K}}}$ . Some steps are omitted for conciseness, for instance we leave it as an exercise to the reader to show that:

\begin{align*} A^{\hat{I}}_{{\mathcal{K}}} = \{&{\mathbf{T}} a,{\mathbf{T}} b,{\mathbf{F}} c,{\mathbf{F}} d,{\mathbf{F}} e,{\mathbf{F}} f, \\ &{\mathbf{T}} \beta (r_1) ={\mathbf{T}} \beta (r_2) ={\mathbf{T}} \emptyset, {\mathbf{F}} \beta (r_3) ={\mathbf{F}} \{d\},{\mathbf{F}} \beta (r_4) ={\mathbf{F}} \{f\}, \\ &{\mathbf{T}} \beta (r_1, a) ={\mathbf{T}} \{{\mathbf{T}} \beta (r_1)\},{\mathbf{T}} \beta (r_2, a) ={\mathbf{F}} \{{\mathbf{T}} \beta (r_2),{\mathbf{F}} d\}, \\ &{\mathbf{F}} \beta (r_2, d) ={\mathbf{F}} \{{\mathbf{T}} \beta (r_2),{\mathbf{F}} a\},{\mathbf{F}} \beta (r_3, f) ={\mathbf{F}} \{{\mathbf{T}} \beta (r_3)\},{\mathbf{F}} \beta (r_4, f), ={\mathbf{F}} \{{\mathbf{T}} \beta (r_4)\},\\ &{\mathbf{F}} \beta _{\mathcal{O}} (a),{\mathbf{T}} \beta _{\mathcal{O}} (b),{\mathbf{F}} \beta _{\mathcal{O}} (c),{\mathbf{F}} \beta _{\mathcal{O}} (d),{\mathbf{F}} \beta _{\mathcal{O}} (e),{\mathbf{F}} \beta _{\mathcal{O}} (f) \} \end{align*}

Firstly, we show that ${\mathbf{T}} a$ and ${\mathbf{T}} b$ must be in $A$ . Consider the conjunction nogood $\gamma (r_1) = \{\{{\mathbf{F}} \beta (r_1)\} \cup \beta (r_1) \} \cup \{\{{\mathbf{T}} \beta (r_1), \overline{\sigma }\} \ | \ \sigma \in \beta (r_1)\} \in \Gamma _{\mathcal{P}}$ . As the body of $r_1$ is empty $\beta (r_1) = \emptyset$ , thus $\gamma (r_1) = \{{\mathbf{F}} \emptyset \}$ , and so ${\mathbf{T}} \emptyset \in A$ . Combined with the rule nogood $\phi _{\mathcal{P}}(r_1) = \{{\mathbf{F}} a,{\mathbf{T}} \beta (r_1)\} \in \Phi _{\mathcal{P}}$ , this also means that ${\mathbf{T}} a \in A$ . Due to the fact that $OB_{\mathcal{O}, a} \models b$ , the positive entailment nogood $\gamma _{\mathcal{O}}^+(b, \{a\})$ occurs within $\Gamma _{\mathcal{O}}$ . As $\gamma _{\mathcal{O}}^+(b, \{a\}) = \{{\mathbf{F}} \beta _{\mathcal{O}} (b),{\mathbf{T}} a\}$ it follows that ${\mathbf{T}} \beta _{\mathcal{O}}(b) \in A$ . Consequently, due to the saturation nogood $\phi _{\mathcal{O}}(b) = \{{\mathbf{F}} b,{\mathbf{T}} \beta _{\mathcal{O}}(b)\} \in \Phi _{\mathcal{O}}$ , ${\mathbf{T}} b \in A$ as well.

It is simple to show that ${\mathbf{F}} c \in A$ , as such we focus on the more interesting case of ${\mathbf{F}} d$ . Turn your attention to the conjunction nogoods $\gamma _{\mathcal{P}} (\beta _{\mathcal{P}} (d, r_2)) \subseteq \Gamma _{\mathcal{P}}$ , which are $\{{\mathbf{F}} \beta _{\mathcal{P}} (d, r_2),{\mathbf{T}} \beta (r_2),{\mathbf{F}} a\}$ , $\{{\mathbf{T}} \beta _{\mathcal{P}}(d,r_2),{\mathbf{F}} \beta (r_2)\}$ , and $\{{\mathbf{T}} \beta _{\mathcal{P}}(d,r_2),{\mathbf{T}} a\}$ . The last one shows that ${\mathbf{F}} \beta _{\mathcal{P}}(d,r_2) \in A$ . Also note that unless $c$ is true $d$ cannot be supported by the ontology – $OB_{\mathcal{O}, \mathbf{KA} (\mathcal{\mathcal{O}}) \setminus \{c, d\}} \not \models d$ – therefore the negative entailment nogood $\gamma _{\mathcal{O}}^- (d, \{c\}) = \{{\mathbf{T}} \beta _{\mathcal{O}} (d),{\mathbf{F}} c\}$ is in $\Gamma _{\mathcal{K}}$ and thereby ${\mathbf{F}} \beta _{\mathcal{O}} (d) \in A$ . Now consider the support nogood $\psi _{\mathcal{K}}(d) = \{{\mathbf{T}} d,{\mathbf{F}} \beta _{\mathcal{P}}(d, r_2),{\mathbf{F}} \beta _{\mathcal{O}}(d)\} \in \Psi _{\mathcal{K}}$ , to see that ${\mathbf{F}} d \in A$ .

Finally we show that ${\mathbf{F}} e$ and ${\mathbf{F}} f$ are in $A$ . Clearly the set of both atoms is a loop $\{e, f\} \in Loops({\mathcal{K}})$ , and since neither atom can be supported by the ontology unless $c$ , $e$ , or $f$ is true – $OB_{\mathcal{O}, \mathbf{KA} (\mathcal{\mathcal{O}}) \setminus \{c, e, f\}} \not \models \bigvee \{e, f\}$ – the loop nogoods $\lambda (\{e, f\}, \{c\})$ are in $\Lambda _{\mathcal{K}}$ . The external supports for $\{e, f\}$ are simply $\mathcal{E}(\{e,f\}) = \{r_3\}$ , and $\rho (r_3, \{e, f\}) = \{{\mathbf{F}} \beta (r_3)\}$ , therefore $\lambda (\{e, f\}, \{c\}) = \{\{{\mathbf{T}} e,{\mathbf{F}} \beta (r_3)\}, \{{\mathbf{T}} f,{\mathbf{F}} \beta (r_3)\}\}$ . From the fact that ${\mathbf{F}} d \in A$ , it is easy to show that ${\mathbf{F}} \{d\} ={\mathbf{F}} \beta (r_3) \in A$ . Consequently ${\mathbf{F}} e$ and ${\mathbf{F}} f$ both must be in $A$ .

We have that $\{{\mathbf{T}} a,{\mathbf{T}} b,{\mathbf{F}} c,{\mathbf{F}} d,{\mathbf{F}} e,{\mathbf{F}} f\} \subseteq A$ which is the most essential aspect of $A^{\hat{I}}_{{\mathcal{K}}}$ ’s relationship to $\hat{I}$ , the rest is left as an exercise.

6.1 Main procedures

Due to the similarity of their formulation to that of Gebser et al. (Reference Gebser, Kaufmann and Schaub2012) the details of the main conflict-driven procedures will be explained only at a high level, with some comments on specific differences.

Algorithm 1 CDNL is primarily responsible for keeping track of an assignment representing a partial candidate solution, through a tree-like search procedure. Additionally, it tracks a set of nogoods which are a subset of those introduced in Section 5, and a decision level indicating the current depth of the search tree. It operates by calling the reasoning procedure NogoodProp to deterministically extend the assignment, and track additional nogoods which prove helpful in doing so. Following this it takes one of three actions. If the current assignment is incomplete but compatible with the current nogood, an arbitrary decision literal is added to the assignment increasing the decision level. If the current assignment conflicts with the current nogoods, the search backtracks to a lower decision level or returns “no model” if already at the lowest decision level. Finally, if the assignment is total and compatible with all tracked nogoods, a model-checking oracle is consulted. If the assignment is verified as a solution the $\mathbf{K}$ -interpretation that induces it is returned. Otherwise, the trivial nogood – the full invalid assignment – is added to the tracked set enabling backtracking.

The significant changes from the main algorithm of Gebser et al. (Reference Gebser, Kaufmann and Schaub2012) are: the different initial set of nogoods, and the final model check ultimately required for the soundness of the algorithm. This check can only be avoided if we assume we can always determine a nogood whenever one is a subset of a total assignment, however this requires substantial procedures for determining loop nogoods.

Algorithm 1. CDNL

Algorithm 2. NogoodProp

Algorithm 2 NogoodProp is the core reasoning procedure. A unit-resulting nogood for an assignment contains one literal which is not within the assignment. We use the term unit-propagation to refer to the process of adding the complement of this literal to the assignment which the nogood is unit-resulting for. NogoodProp unit-propagates the current assignment based on the set of tracked nogoods, and adds additional nogoods whenever propagation reaches a fixpoint. The novel feature of this procedure is that its first resort after reaching a fixpoint is to call EntNogoods, which will attempt to add unit-resulting entailment nogoods to $\nabla$ . If no such nogoods can be found based on the current assignment, the procedure will instead attempt to generate a loop nogood, by calling the procedure UnfoundedSet. This procedure returns a set $U \in Loops({\mathcal{K}})$ Footnote ⁴ for which $U \subseteq A^T$ , and $S \subseteq A^F$ such that $OB_{\mathcal{O}, \mathbf{KA} (\mathcal{K}) \setminus (U \cup S)} \not \models \bigvee U$ . In principle, this can be a modification of existing methods for detecting unfounded sets. Eventually, it reaches a point where either a tracked nogood conflicts with the current assignment or no more unit-resulting nogoods can be discovered. At this point, the procedure returns.