2 Background: overview of forks

In this section, we revisit the basic definitions for the fork operators and their denotational semantics. This semantics is based in its turn on Equilibrium Logic Pearce (Reference Pearce1996) and its monotonic basis, the logic of Here-and-There (HT) Heyting (Reference Heyting1930), which is introduced in the first place. Then, we recall the definition of forks and some previous results that will be used later on for the proofs of correspondence with the other approaches. Finally, we conclude the section providing a new theorem (Th. 2) to be used later, that proves that the replacement of a disjunction by a fork in any arbitrary disjunctive logic programme always produces a superset of stable models.

2.1 Here-and-there and equilibrium logic

Let ${\mathcal AT}$ be a finite set of atoms called the alphabet or vocabulary. A (propositional) formula $\varphi$ is defined using the grammar:

\begin{equation*} \varphi \ ::= \ \ \bot \mid p \mid \varphi \wedge \varphi \mid \varphi \vee \varphi \mid \varphi \to \varphi \end{equation*}

where $p$ is an atom $p \in {{\mathcal AT}}$ . We use Greek letters $\varphi , \psi , \gamma$ and their variants to stand for formulas. We also define the derived operators $(\psi \leftarrow \varphi ) \stackrel {\scriptscriptstyle \mathrm{def}}{=} (\varphi \to \psi )$ , $\neg \varphi \stackrel {\scriptscriptstyle \mathrm{def}}{=} (\varphi \to \bot )$ and $\top \stackrel {\scriptscriptstyle \mathrm{def}}{=} \neg \bot$ . Given a formula $\varphi$ , by ${{\mathcal AT}}(\varphi ) \subseteq {{\mathcal AT}}$ we denote the set of atoms occurring in $\varphi$ . A theory $\Gamma$ is a finiteFootnote ³ set of formulas that can be also understood as their conjunction. When a theory consists of a single formula $\Gamma =\{\varphi \}$ we will frequently omit the braces. An extended disjunctive rule $r$ is an implication of the form:

(2) \begin{multline} \ p_{1} \vee \ldots \vee p_{m} \leftarrow p_{m+1}\wedge \ldots \wedge p_n \wedge \neg p_{n+1} \wedge \ldots \wedge \neg p_h \wedge \neg \neg p_{h+1} \wedge \ldots \wedge \neg \neg p_k \end{multline}

where all $p_i$ above are atoms in ${\mathcal AT}$ and $0 \leq m \leq n \leq h \leq k$ . The disjunction in the consequent is called the head of $r$ and denoted as $\mathit{Head}(r)$ , whereas the conjunction in the antecedent receives the name of body of $r$ and is denoted by $\mathit{Body}(r)$ . We define the sets of atoms $h(r)\stackrel {\scriptscriptstyle \mathrm{def}}{=} \{p_{1},\ldots ,p_m\}$ , $b^{+}(r)\stackrel {\scriptscriptstyle \mathrm{def}}{=} \{p_{m+1},\ldots ,p_n\}$ , $b^{-}(r)\stackrel {\scriptscriptstyle \mathrm{def}}{=} \{p_{n+1},\ldots ,p_h\}$ , $b^{-\,-}(r)\stackrel {\scriptscriptstyle \mathrm{def}}{=}\{p_{h+1},\ldots ,p_k\}$ and $b(r)\stackrel {\scriptscriptstyle \mathrm{def}}{=} b^{+}(r) \cup b^{-}(r) \cup b^{-\,-}(r)$ . We say that $r$ is an extended normal rule if $|h(r)| \leq 1$ . We drop the adjective “extended” when the rule does not have double negation. That is, when $k=h$ we simply talk about a disjunctive rule and further call it normal rule, if it satisfies $|h(r)| \leq 1$ . An empty head $h(r)=\emptyset$ represents falsum $\bot$ and, when this happens, the rule is called a constraint. An empty body $b(r)=\emptyset$ is assumed to represent $\top$ and, when this happens, we usually omit $\leftarrow \top$ simply writing the rule head. A rule with $b(r)=\emptyset$ and $|h(r)|=1$ is called a fact. A programme $P$ is a set of rules and, when the programme is finite, we will also understand it as their conjunction. We say that programme $P$ belongs to a syntactic category if all its rules belong to that category.

A classical interpretation $T$ is a set of atoms $T \subseteq {{\mathcal AT}}$ . By $T \models \varphi$ we mean the usual classical satisfaction of a formula $\varphi$ . Moreover, we write $M(\varphi )$ to stand for the set of classical models of $\varphi$ . An HT-interpretation is a pair $\langle \, H,T \, \rangle$ (respectively called “here” and “there”) of sets of atoms $H \subseteq T \subseteq {{\mathcal AT}}$ ; it is said to be total when $H=T$ . Intuitively, an atom $p$ is considered false, when $p \not \in T$ , or true when $p \in T$ , but the latter has two cases: it may be certainly true when $p \in H$ or just assumed true when $p \in T \setminus H$ . An interpretation $\langle \, H,T \, \rangle$ satisfies a formula $\varphi$ , written $\langle \, H,T \, \rangle \models \varphi$ , when the following recursive rules hold:

• • $\langle \, H,T \, \rangle \not \models \bot$

• • $\langle \, H,T \, \rangle \models p$ if $p \in H$

• • $\langle \, H,T \, \rangle \models \varphi \wedge \psi$ if $\langle \, H,T \, \rangle \models \varphi$ and $\langle \, H,T \, \rangle \models \psi$

• • $\langle \, H,T \, \rangle \models \varphi \vee \psi$ if $\langle \, H,T \, \rangle \models \varphi$ or $\langle \, H,T \, \rangle \models \psi$

• • $\langle \, H,T \, \rangle \models \varphi \to \psi$ if both (i) $T \models \varphi \to \psi$ , and (ii) $\langle \, H,T \, \rangle \not \models \varphi$ or $\langle \, H,T \, \rangle \models \psi$

An HT-interpretation $\langle \, H,T \, \rangle$ is a model of a theory $\Gamma$ if $\langle \, H,T \, \rangle \models \varphi$ for all $\varphi \in \Gamma$ . Two formulas (or theories) $\varphi$ and $\psi$ are HT-equivalent, written $\varphi \equiv \psi$ , if they have the same HT-models.

A total interpretation $\langle \, T,T \, \rangle$ is an equilibrium model of a formula $\varphi$ iff $\langle \, T,T \, \rangle \models \varphi$ and there is no $H\subset T$ such that $\langle \, H,T \, \rangle \models \varphi$ . If so, we say that $T$ is a stable model of $\varphi$ and we write $\textit {SM}(\varphi )$ to stand for the set of stable models of $\varphi$ .

2.2 Forks

A fork $F$ is defined by the following grammar:

\begin{equation*} F ::= \bot \mid p \mid (F \mid F)\mid F \wedge F \mid \varphi \vee \varphi \mid \varphi \to F \end{equation*}

where $\varphi$ is a propositional formula over ${\mathcal AT}$ and $p \in {{\mathcal AT}}$ is an atom. It can be proved, by structural induction, that any propositional formula $\varphi$ is a fork. Note that a fork is not allowed as an argument of a disjunction nor as the antecedent of an implication. The intuition of this new connective “ $\mid$ ” is that the stable models of a fork such as $(\varphi _1 \mid \ldots \mid \varphi _n)$ – in fact, all forks are reducible to this form – will be the union of stable models of each $\varphi _i$ . The formal semantics of forks is based on the idea of denotations (sets of models) we define next in several steps.

Given a set of atoms $T\subseteq {{\mathcal AT}}$ , a $T$ -support $\mathcal{H} \subseteq 2^{T}$ is a set of subsets of $T$ so that, if $\mathcal{H} \neq \emptyset$ , then $T \in \mathcal{H}$ . Given a propositional formula $\varphi$ , the set of sets of atoms $\{H \subseteq T \, \mid \, \langle \, H,T \, \rangle \models \varphi \}$ forms a $T$ -support we denote as $[\![ \, \varphi \, ]\!]^T$ . For readability sake, we directly write a $T$ -support as a sequence of sets between square braces: for instance, some possible supports for $T=\{a,b\}$ are $[\{a,b\} \ \{a\}]$ , $[\{a,b\}\ \{b\}\ \emptyset ]$ or the empty support $[ \ ]$ . Given two $T$ -supports, $\mathcal{H}$ and $\mathcal{H}'$ , we define the order relation $\mathcal{H} \preceq \mathcal{H}'$ iff either $\mathcal{H}=[ \ ]$ or $[ \ ] \neq \mathcal{H}' \subseteq \mathcal{H}$ , read as $\mathcal{H}$ is “less supported” than $\mathcal{H}'$ . Intuitively, this means that $\mathcal{H}'$ is closer to make $T$ a stable model than $\mathcal{H}$ . Given a $T$ -support $\mathcal{H}$ , we define its complementary support $\overline {\mathcal{H}}$ as:

\begin{eqnarray*} \overline {\mathcal{H}} \stackrel {\scriptscriptstyle \mathrm{def}}{=} \left \{ \begin{array}{c@{\ \ }l} [\ ] & \text{if } \mathcal{H}=2^T\\ {[\ T\ ]} \cup \{H \subseteq T \mid H \notin \mathcal{H}\} & \text{otherwise.} \end{array} \right . \end{eqnarray*}

The ideal of $\mathcal{H}$ is defined as $\,\downarrow \!\mathcal{H} = \{\mathcal{H}' \mid \mathcal{H}' \preceq \mathcal{H} \} \setminus \{ \ [\ ] \ \}$ . Note that, the empty support $[\ ]$ is not included in the ideal, so $\,\downarrow \![ \ ]=\emptyset$ . If $\Delta$ is any set of supports, we use its $\preceq$ -closure:

\begin{equation*}{}^\downarrow \!\Delta \stackrel {\scriptscriptstyle \mathrm{def}}{=} \bigcup _{\mathcal{H} \in \Delta } \,\downarrow \!\mathcal{H} = \bigcup _{\mathcal{H} \in \Delta } \{\, \mathcal{H}' \preceq \mathcal{H} \mid \mathcal{H}' \neq [ \ ]\,\}.\end{equation*}

We define a $T$ -view $\Delta$ as any $\preceq$ -closed set of $T$ -supports, that is, ${}^\downarrow \!\Delta = \Delta$ . Given a $T$ -view $\Delta$ , we write $\mathcal{H} \mathbin {\hat {\in }} \Delta$ iff $\mathcal{H} \in \Delta$ or both $\mathcal{H} = [\ ]$ and $\Delta = \emptyset$ .

Definition 1 ( $T$ -denotation). Let ${\mathcal AT}$ be a propositional signature and $T \subseteq {{\mathcal AT}}$ a set of atoms. The $T$ -denotation of a fork or a propositional formula $F$ , written $\langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T$ , is a $T$ -view recursively defined as follows:

\begin{equation*} \begin{array}{ccl} \langle \hspace {-2pt}\langle \, \bot \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& \emptyset \\[5pt] \langle \hspace {-2pt}\langle \, p \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& \,\downarrow \![\![ \, p \, ]\!]^T \quad \text{for any atom } p \\[5pt] \langle \hspace {-2pt}\langle \, F \wedge G \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& {}^\downarrow \!\{\,\mathcal{H} \cap \mathcal{H}'\mid \mathcal{H} \in \langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T \text{ and } \mathcal{H}' \in \langle \hspace {-2pt}\langle \, G \, \rangle \hspace {-2pt}\rangle ^T\,\} \\[5pt] \langle \hspace {-2pt}\langle \, \varphi \vee \psi \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& {}^\downarrow \!\{\,\mathcal{H} \cup \mathcal{H}'\mid \mathcal{H} \mathbin {\hat {\in }} \langle \hspace {-2pt}\langle \, \varphi \, \rangle \hspace {-2pt}\rangle ^T \text{ and } \mathcal{H}' \mathbin {\hat {\in }} \langle \hspace {-2pt}\langle \, \psi \, \rangle \hspace {-2pt}\rangle ^T\,\} \\[5pt] \langle \hspace {-2pt}\langle \, \varphi \to F \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& \left \{ \begin{array}{cl} \{ 2^T \} &\text{if } [\![ \, \varphi \, ]\!]^T = [ \ ] \\ {}^\downarrow \!\{\,\overline {[\![ \, \varphi \, ]\!]^T } \cup \mathcal{H}\mid \mathcal{H} \in \langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T\,\} &\text{otherwise} \end{array} \right . \\[9pt] \langle \hspace {-2pt}\langle \, F \mid G \, \rangle \hspace {-2pt}\rangle ^T &\stackrel {\scriptscriptstyle \mathrm{def}}{=}& \langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T \cup \langle \hspace {-2pt}\langle \, G \, \rangle \hspace {-2pt}\rangle ^T \end{array} \end{equation*}

where $F$ , $G$ denote forks or propositional formulas.

We say that $T$ is a stable model of a fork $F$ when $\langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T=\,\downarrow \![\, T\, ]$ or, equivalently, when $[\, T \, ] \in \langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T$ . The set $\textit {SM}(F)$ collects all the stable models of $F$ .

Definition 2 (Strong Entailment/Equivalence of forks). We say that fork $F$ strongly entails fork $G$ , in symbols $F \mathbin {|\kern -.17em\sim } G$ , if ${SM}(F \wedge L) \subseteq {SM}(G \wedge L)$ , for any fork $L$ . We further say that $F$ and $G$ are strongly equivalent, in symbols $F \cong G$ if both $F \mathbin {|\kern -.17em\sim } G$ and $G \mathbin {|\kern -.17em\sim } F$ , that is, ${SM}(F \wedge L) ={SM}(G \wedge L)$ , for any fork $L$ .

Interestingly, Aguado et al. (Reference Aguado, Cabalar, Fandinno, Pearce, Pérez and Vidal2019) (Prop. 11) proved that $F \mathbin {|\kern -.17em\sim } G$ is equivalent to $\langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T \subseteq \langle \hspace {-2pt}\langle \, G \, \rangle \hspace {-2pt}\rangle ^T$ , for every set of atoms $T \subseteq {{\mathcal AT}}$ and, thus, $F \cong G$ amounts to $\langle \hspace {-2pt}\langle \, F \, \rangle \hspace {-2pt}\rangle ^T = \langle \hspace {-2pt}\langle \, G \, \rangle \hspace {-2pt}\rangle ^T$ , for every $T$ . Other properties proved by Aguado et al. (Reference Aguado, Cabalar, Fandinno, Pearce, Pérez and Vidal2019) we will use below are:

(3) \begin{eqnarray} (F \mid G) \mid L & \cong & F \mid (G \mid L) \end{eqnarray}

(4) \begin{eqnarray}\quad\quad\,\, (F \mid G) \wedge L & \cong & (F \wedge L) \mid (G \wedge L) \end{eqnarray}

(5) \begin{eqnarray}\quad\quad\,\, \textit {SM}(F \mid G) & = & \textit {SM}(F)\cup \textit {SM}(G) \end{eqnarray}

Example 1. Consider the fork:

(6) \begin{eqnarray} (a \mid b) \wedge (a \mid c) \end{eqnarray}

We can apply distributivity (4) and associativity (3) to conclude that (6) is actually strongly equivalent to:

\begin{eqnarray*} a \wedge a \mid a \wedge c \mid b \wedge a \mid b \wedge c \end{eqnarray*}

which is a fork built with 4 propositional formulas. By (5), the stable models of this fork are the union of stable models of these 4 formulas, namely, $\{a\}$ , $\{a,c\}$ , $\{a,b\}$ and $\{b,c\}$ .

We conclude this section introducing the polynomial reduction of any fork $F$ into a propositional formula $\mathit{pf}(F)$ by Aguado et al. (Reference Aguado, Cabalar, Fandinno, Pearce, Pérez and Vidal2022) that may help for a better understanding of the behaviour of forks, and is used in the proof of Theorem4 later on. For simplicity, we constrained here $\mathit{pf}(F)$ to the case in which $F$ has the form $P^{\mid }$ for some extended disjunctive programme $P$ , using less definitions and getting $\mathit{pf}(F)$ in the form of a disjunctive logic programme.

Definition 3. Let $P$ be some extended disjunctive logic programme. For each $r \in P$ we define $\mathit{pf}(r^{\mid })$ as: $\mathit{pf}(r^{\mid })\stackrel {\scriptscriptstyle \mathrm{def}}{=} r$ if $r$ is an extended normal rule, that is $|h(r)|\leq 1$ ; otherwise, given $h(r)=\{p_1,\ldots ,p_m\}$ :

\begin{equation*} \mathit{pf}(r^{\mid }) \stackrel {\scriptscriptstyle \mathrm{def}}{=} (x_1 \vee \ldots \vee x_m \leftarrow \mathit{Body}(r) )\ \wedge \ \bigwedge _{i=1}^m (p_i \leftarrow x_i) \end{equation*}

for a set of fresh propositional atoms $x_1, \ldots , x_m$ .

We also define $\mathit{pf}(P^{\mid })\stackrel {\scriptscriptstyle \mathrm{def}}{=} \bigwedge _{r \in P} \mathit{pf}(r^{\mid })$ . For example, $\mathit{pf}(P^{\mid }_{(1)})$ is the conjunction of:

\begin{eqnarray*} x_1 \vee x_2 \hspace {30pt} a \leftarrow x_1 \hspace {30pt} b \leftarrow x_2 \hspace {30pt} y_1 \vee y_2 \hspace {30pt} a \leftarrow y_1 \hspace {30pt} c \leftarrow y_2 \end{eqnarray*}

Theorem 1 (From Main Theorem Aguado et al. (Reference Aguado, Cabalar, Fandinno, Pearce, Pérez and Vidal2022)). Let $P$ be an extended logic programme. $P^{\mid }$ and $\mathit{pf}(P^{\mid })$ are strongly equivalent, modulo alphabet ${{\mathcal AT}}(P)$ .

2.3 Replacing disjunctions by forks

As expected, the definition of stable models for forks is a proper extension of stable models for propositional theories (or if preferred, equilibrium models Pearce (Reference Pearce1996)) and so, in its turn, it also applies to the more restricted syntax of logic programmes with disjunction Gelfond and Lifschitz (Reference Gelfond and Lifschitz1991). This means that disjunction " $\vee$ " in logic programmes respects the principle of minimality. For instance, under this definition we still have the same two stable models for programme $P_{(1)}$ , namely, $\textit {SM}(P_{(1)})=\{\{a\},\{b,c\}\}$ . However, minimality is lost if we replace " $\vee$ " by " $\mid$ ," as illustrated next.

For any disjunctive rule $r$ , let us denote by $r^{\mid }$ the fork obtained by substituting in $h(r)$ the operator " $\vee$ " by " $\mid$ " and let $P^{\mid }\stackrel {\scriptscriptstyle \mathrm{def}}{=} \bigwedge _{r \in P} r^{\mid }$ for any programme $P$ as expected. For instance, the fork $P_{(1)}^{\mid }$ would correspond to (6) in Example1 whose stable models were $\{a\}$ , $\{b,c\}$ , $\{a,b\}$ and $\{a,c\}$ – the last two are not minimal whereas the first two coincide with $\textit {SM}(P_{(1)})$ . The main result in this section proves that the replacement of regular disjunctions by forks in any rule $r$ always produces a superset of stable models, even if that rule is included in a larger arbitrary context. Namely, we have the strong entailment relation $r \mathbin {|\kern -.17em\sim } r^{\mid }$ .

Theorem 2. Let $\varphi$ and $\alpha _1, \ldots , \alpha _n$ be propositional formulas with $n\geq 1$ . Then:

\begin{equation*} \varphi \rightarrow (\alpha _1 \vee \cdots \vee \alpha _n) \; \mathbin {|\kern -.17em\sim } \; \varphi \rightarrow (\alpha _1 \mid \cdots \mid \alpha _n)\end{equation*}

Since strong entailment allows us to proceed rule by rule, we conclude:

Corollary 1. Let $P$ be any extended disjunctive logic programme, then $P \mathbin {|\kern -.17em\sim } P^{\mid }$ .

As a result, a disjunctive programme that has no stable models may restore coherence (existence of stable model) if we replace disjunctions by forks. Take the following example (adapted from Ex. 1 by Shen and Eiter (Reference Shen and Eiter2019), ).

Example 2. Consider the programme $P_{(7)}$ consisting of the three rules:

(7) \begin{eqnarray} a \vee b \hspace {60pt} a \hspace {60pt} b \leftarrow \neg b \end{eqnarray}

Disjunction $a\vee b$ is redundant and can be removed, because it is an HT-consequence of $a$ . But once $a \vee b$ disappears, it is clear that $b \leftarrow \neg b$ prevents obtaining any stable model. Yet, if we change the disjunction in $a \vee b$ by a fork, we can restore coherence. The fork $P_{(7)}^{\mid }$ corresponds to:

\begin{equation*} \begin{array}{cl@{\hspace {10pt}}l} & (a \mid b) \wedge a \wedge (\neg b \to b)\\ \cong & (a \mid b) \wedge a \wedge \neg \neg b & \text{HT-equivalence}\\ \cong & (a \wedge a \wedge \neg \neg b) \mid (b \wedge a \wedge \neg \neg b) & \text{by distributivity } (4)\\ \cong & (a \wedge \neg \neg b) \mid (a \wedge b) & \text{HT-equivalence} \end{array} \end{equation*}

and then $\textit {SM}(P_{(7)})=\textit {SM}(a \wedge \neg \neg b) \cup \textit {SM}(a \wedge b) = \emptyset \cup \{\{a,b\}\} = \{\{a,b\}\}$ , so $P^{\mid }_{(7)}$ has a unique stable model $\{a,b\}$ .

3 Justified models

We proceed now to compare the forks semantics with justified models Cabalar and Muñiz (Reference Cabalar and Muñiz2024). This approach was originally introduced to provide a definition of explanations for the stable models of a logic programme. Such explanations have the form of graphs built with rule labels and reflect the derivation of atoms in the model. A classical model of a logic programme is said to be justified if it admits at least one of these explanation graphs. In the case of normal logic programmes, justified and stable models coincide, but Cabalar and Muñiz (Reference Cabalar and Muñiz2024) observed that, when the programme is disjunctive, it may have more justified models than stable models. In other words, although every stable model of a disjunctive programme admits an explanation, we may have classical models of the programme that admit an explanation but are not stable, breaking the principle of minimality in many cases. In this way, justified models provide a weaker semantics for disjunctive programmes that, as we will see, actually coincides with the behaviour of fork-based disjunction. Let us start recalling some basic definitions, examples and results by Cabalar and Muñiz (Reference Cabalar and Muñiz2024).

If $r$ is a labelled rule, we keep the definitions of the formulas $\mathit{Body}(r)$ and $\mathit{Head}(r)$ and sets of atoms $h(r)$ , $b(r)$ , $b^+(r)$ , $b^- (r)$ and $b^{-\,-}(r)$ as before, that is, ignoring the additional label. Similarly, if $P$ is a labelled logic programme, $P^{\mid }$ denotes the fork that results from removing the labels and, as before, replacing disjunctions $\vee$ by $\mid$ . A set of atoms $I$ is a classical model of a labelled rule $r$ iff $I \models \mathit{Body}(r) \to \mathit{Head}(r)$ in classical logic. Given a labelled logic programme $P$ , by $\mathit{Lb}(P)$ we denote the set of labels of the programme $\mathit{Lb}(P) \stackrel {\scriptscriptstyle \mathrm{def}}{=} \{\mathit{Lb}(r)\mid r \in P\}$ . Note that no label is repeated, but $P$ can contain two rules $r,r'$ with the same body and head and different labels $\mathit{Lb}(r) \neq \mathit{Lb}(r')$ . For instance, we could have two repeated facts with different labels $\ell _1:p$ and $\ell _2:p$ possibly representing two different and simultaneously applicable sources of information.

The fact that $\lambda$ is injective guarantees that there are no repeated labels in the graph. Additionally, the definition tells us that if an atom $p$ is labelled with $\lambda (p)=\ell$ then $\ell$ must be the label of some rule $r$ where (1) $p$ occurs in the head, (2) the body of the rule is satisfied by $I$ and (3) the incoming edges for $p$ are formed from the atoms in the positive body of $r$ . Since an explanation $G=\langle \, I,E,\lambda \, \rangle$ for a model $I$ is uniquely determined by its atom labelling $\lambda$ , we can abbreviate $G$ as a set of pairs $p \mapsto \lambda (p)$ for $p \in I$ .

We can also define $\mathit{SPM}(P)$ and $\mathit{JM}(P)$ for any non-labelled programme $P$ by assuming we previously label each rule in $P$ with a unique arbitrary identifier. Note that different labels produce different explanations, but the definition of justified/supported model is not affected by that.

However, if we allow disjunction, we may have justified models that are not stable models, as illustrated below.

Example 3. Let $P_{(8)}$ be the following labelled version of $P_{(1)}$ :

(8) \begin{eqnarray} \ell _1: a \vee b \hspace {70pt} \ell _2: a \vee c \end{eqnarray}

The classical models of $P_{(8)}$ are $\{a\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}$ . The last one, $\{a,b,c\}$ , is not justified, since we would need three different labels and we only have two rules. Each model $\{a,c\}$ , $\{a,b\}$ , $\{b,c\}$ has a unique explanation corresponding to the atom labellings $\{a \mapsto \ell _1, c \mapsto \ell _2\}$ , $\{b \mapsto \ell _1, a \mapsto \ell _2\}$ and $\{b \mapsto \ell _1, c \mapsto \ell _2\}$ , respectively. On the other hand, model $\{a\}$ has two possible explanations, corresponding to $\{ a \mapsto \ell _1\}$ and $\{a \mapsto \ell _2\}$ . To sum up, we get four justified models, $\{a,c\}$ , $\{a,b\}$ , $\{b,c\}$ and $\{a\}$ but only two of them are stable, $\{a\}$ and $\{b,c\}$ .

In other words, the justified models of $P_{(8)}$ coincide with the stable models of its fork version $P_{(8)}^{\mid }=P_{(1)}^{\mid }=(6)$ seen before. This is in fact, a general property that constitutes the main result of this section.

Supported models $\mathit{SPM}(P)$ correspond to the case in which we also accept cyclic explanation graphs. Obviously, $\mathit{JM}(P) \subseteq \mathit{SPM}(P)$ , because all acyclic explanations are still acceptable for $\mathit{SPM}(P)$ . Cabalar and Muñiz (Reference Cabalar and Muñiz2024) also proved that $\mathit{SPM}(P)$ generalise the standard notion of supported models – that is models of Clark’s completion Clark (Reference Clark1978) – to the disjunctive case. For instance, the programme $P_{(9)}$ consisting of the rule:

(9) \begin{eqnarray} \ell _1: p \leftarrow p \end{eqnarray}

has two supported models, $I=\emptyset$ (which is also stable and justified) and $I=\{p\}$ with a cyclic support graph where node $p$ is connected to itself. As a remark, notice that the definition of our “graph-based” supported models Cabalar and Muñiz (Reference Cabalar and Muñiz2023) does not correspond to the (also called) supported models obtained from the programme completion defined by Alviano and Dodaro (Reference Alviano, Dodaro and Kambhampati2016) for disjunctive programmes. The latter, we denote $AD(P)$ , impose a stronger condition: a rule $r$ supports an atom $p \in \mathit{Hd}(r)$ with respect to interpretation $I$ not only if $I \models \mathit{Body}(r)$ but also $I \not \models q$ for all $q \in \mathit{Hd}(r) \setminus {p}$ . To illustrate the difference, take programme $P_{(8)}$ : as it has no cyclic dependencies, graph-based supported and justified models coincide, that is, $\mathit{SPM}(P_{(8)})=\mathit{JM}(P_{(8)})=\{\{a\},\{a,b\},\{a,c\},\{b,c\}\}$ we saw before. However, $AD(P_{(8)})=\{\{a\},\{b,c\}\}$ that, in this case, coincide with the stable models of the programme. Model $\{a,b\}$ is supported (under Def, 6) because $a$ is justified by rule $\ell _1$ and $b$ by rule $\ell _2$ . However, under Alviano and Dodaro’s definition, rule $\ell _1$ is not a valid support for $a$ since we would further need $b$ (the other atom in the disjunction $\ell _1$ ) to be false. The situation for model $\{a,c\}$ is analogous. It is not hard to see that $\textit {SM}(P) \subsetneq AD(P) \subsetneq \mathit{SPM}(P)$ (the first inclusion proved by Alviano and Dodaro (Reference Alviano, Dodaro and Kambhampati2016)), so clearly, $AD(P)$ is more interesting for computation purposes when our goal is approximating $\textit {SM}(P)$ . However, $\mathit{SPM}(P)$ provides a more liberal generalisation of the definition of supported model from normal logic programmes: as in that case, $I$ is a supported model of $P$ if, for every atom $p \in I$ , there exists some rule $r$ with $p$ “in the head” and $I\models \mathit{Body}(r)$ . This definition has also a closer relation to $\mathit{JM}(P)$ and explanation generation or to the DI-semantics (as we see in Theorem8 in the next section).

4 Determining inference

The third approach we consider, DI Shen and Eiter (Reference Shen and Eiter2019), also introduces a new disjunction operator in rule heads, with the same syntax as forks “ $|$ ”. Besides, first-order formulas are allowed to play the role of atoms, and so, the syntax accepts regular disjunction “ $\vee$ ” too. However, in this paper (for the sake of comparison) we describe the DI-semantics directly on the syntax of extended disjunctive rules of the form (2) seen before,Footnote ⁴ using “ $\vee$ ” to play the role of the DI disjunctive operator.

The DI-semantics understands disjunction as a non-deterministic choice and is based on the definition of a head selection function. This function will tell us, beforehand, which head atom will be chosen if we have to apply a rule for derivation. We introduce next a slight generalisation of that definition.

Definition 7 (Open/Closed Head Selection Function). Let $P$ be an extended disjunctive logic programme and $I \subseteq {{\mathcal AT}}$ an interpretation. A head selection function $sel$ for $I$ and some $r \in P$ is a formula:

\begin{equation*} sel(\mathit{Head}(r),I) \stackrel {\scriptscriptstyle \mathrm{def}}{=} \left \{ \begin{array}{ll} \bot & \mbox{ if } h(r) \cap I = \emptyset \\ p_i & \mbox{ otherwise, for some } p_i \in h(r) \cap I \end{array} \right . \end{equation*}

We say that $sel$ is closed if $sel(\mathit{Head}(r),I)=sel(\mathit{Head}(r'),I)$ for any pair of rules $r,r'$ with the same head atoms $h(r)=h(r')$ . If this restriction does not apply, we just say that $sel$ is open.

The original definition by Shen and Eiter (Reference Shen and Eiter2019) (Def. 4) corresponds to what we call here closed selection function and forces the same choice when two rule heads are formed by the same set of atoms.

The reduct of a programme $P$ with respect to some interpretation $I$ and selection function $sel$ is defined as the logic programme $ P_{sel}^{I} \stackrel {\scriptscriptstyle \mathrm{def}}{=} \{\ sel(\mathit{Head}(r),I) \leftarrow \mathit{Body}(r) \mid I \models \mathit{Body}(r) \ \}$ . Note that $P^I_{sel}$ is an extended normal logic programme (possibly containing constraints) where we replaced each disjunction by the atom determined by the selection function $sel$ .

To understand the difference between closed and open selection functions, take the following programme $P_{(10)}$ :

(10) \begin{eqnarray} p \qquad\qquad \bot \leftarrow c \qquad\qquad a \vee b \qquad\qquad b \vee a \leftarrow p \end{eqnarray}

The set $\mathit{CSM}(P_{(10)})$ consists of $\{p, a\}$ , $\{p, b\}$ and $\{p,a,b\}$ , but only the first two models are closed, since they make the same choice in both disjunctions $a \vee b$ and $b \vee a$ that have the same atoms. Note that this condition is rather syntax-dependent: if we replace $b \vee a \leftarrow p$ by the rule $b \vee a \vee c \leftarrow p$ , then open candidate stable models are not affected ( $c$ must be false due to constraint $\bot \leftarrow c$ ) but $\{p,a,b\}$ becomes now a closed candidate stable model, since the sets of atoms in $a \vee b$ and $b \vee a \vee c$ are different.

A DI-stable model $I$ of a programme $P$ is a model that is minimal among the closed candidate stable models (Def. 7 by Shen and Eiter (Reference Shen and Eiter2019)). Thus, DI-semantics actually imposes an additional minimality condition. However, if we focus on the previous step, $\mathit{CSM}(P)$ , we can prove that they coincide with $\textit {SM}(P^{\mid })$ and, by Theorem4, with $\mathit{JM}(P)$ too.

We conclude this section by proving that the decision problem $\mathit{CSM}(P)\neq \emptyset$ is NP-complete, recalling the following complexity result proved by Shen and Eiter (Reference Shen and Eiter2019)

As one last result in this section, we provide an alternative characterisation of the supported models from Def. 6 using DI-semantics. For normal logic programmes, $I$ is a supported model of $P$ if, for every atom $p \in I$ , there exists some rule $r$ with $p$ in the head and $I\models \mathit{Body}(r)$ . Alternatively, supported models can also be captured as the fixpoints of the immediate consequences van Emden and Kowalski (Reference Van Emden and Kowalski1976) operator $T_P\stackrel {\scriptscriptstyle \mathrm{def}}{=} \{p \mid (p \leftarrow B) \in P, I \models B\}$ , namely, $I$ is a supported model of $P$ iff $I=T_P(I)$ . We can extend this relation for disjunctive logic programmes as follows.

5 Strongly supported models

For our last comparison, we consider strongly supported models by Doherty and Szałas (Reference Doherty and Szałas2015):

Doherty and Szałas (Reference Doherty and Szałas2015) (Th. 1) proved that the stable models of $P$ , $\textit {SM}(P)$ , coincide with the minimal elements of $\mathit{SSM}(P)$ and, furthermore, $\textit {SM}(P)=\mathit{SSM}(P)$ when $P$ has no disjunction. However, in general, the $\mathit{SSM}$ semantics makes disjunction to behave classically. For instance, from Def. 6 above, we can easily observe that, if $P$ is a set of disjunctions of atoms, then $\mathit{SSM}(P)=M(P)$ . As a result, since (1) is a pair of disjunctions, $\mathit{SSM}(P_{(1)})=M(P_{(1)})$ that is the five classical models $\{a\}$ , $\{a,b\}$ , $\{a,c\}$ , $\{b,c\}$ and $\{a,b,c\}$ mentioned before. Note that $\mathit{CSM}$ did not accept $\{a,b,c\}$ , pointing our that it is a stronger semantics, as corroborated next:

To conclude this section, we observe that, despite their name similarity, supported $\mathit{SPM}(P)$ and strongly supported models $\mathit{SSM}(P)$ are unrelated. To prove $\mathit{SPM}(P) \not \subseteq \mathit{SSM}(P)$ , just take the programme $P_{(9)}$ with no disjunctions, so that $\mathit{SSM}(P)=\textit {SM}(P)=\{\emptyset \}$ . However, $\{p\} \in \mathit{SPM}(P)$ as we discussed before. To prove $\mathit{SSM}(P) \not \subseteq \mathit{SPM}(P)$ we already saw that $\{a,b,c\} \in \mathit{SSM}(P_{(1)}) \setminus \mathit{JM}(P_{(1)})$ . But, since $P_{(1)}$ has no implications, the support graphs contain no edges, so that acyclicity is irrelevant meaning $\mathit{JM}(P_{(1)})=\mathit{SPM}(P_{(1)})$ .