2.1 Disjunctive logic programming

In what follows we consider a propositional ^{Footnote 2} language ${\mathfrak L}$ , whose atomic formulas are denoted by p,q,r (possibly indexed), and that contains the propositional constants ${\sf T}$ (representing truth), ${\sf F}$ (falsity), ${\sf U}$ (unknown), and ${\sf C}$ (contradictory information). The connectives in ${\mathfrak L}$ include negation $\neg$ , conjunction $\wedge$ , disjunction $\vee$ , and implication $\leftarrow$ . Formulas are denoted by $\phi$ , $\psi$ , $\delta$ (again, possibly indexed). Logic programs in ${\mathfrak L}$ may be divided to different kinds as follows: a (propositional) disjunctive logic program ${\cal P}$ in ${\mathfrak L}$ (a dlp in short) is a finite set of rules of the form $\bigvee_{i=1}^n p_i~\leftarrow~\psi$ , where the head $\bigvee_{i=1}^n p_i$ is a non-empty disjunction of atoms, and the body $\psi$ is a formula not containing $\leftarrow$ . A rule is called normal (nlp), if its body is a conjunction of literals (i.e., atomic formulas or negated atoms), and its head is atomic. A rule is disjunctively normal if its body is a conjunction of literals and its head is a non-empty disjunction of atoms. We will use these denominations for programs if all rules in the program satisfy the denomination, for example, a program is normal if all its rules are normal. The set of atoms occurring in $\mathcal{P}$ is denoted ${\cal A}_{\cal P}$ .

The semantics of dlps are given in terms of four-valued interpretations. A four-valued interpretation of a program ${\cal P}$ is a pair (x,y), where $x \subseteq {\cal A}_{\cal P}$ is the set of the atoms that are assigned a value in $\{{\sf T},{\sf C}\}$ and $y \subseteq {\cal A}_{\cal P}$ is the set of atoms assigned a value in $\{{\sf T},{\sf U}\}$ . We define $-{\sf T}={\sf F}$ , $-{\sf F}={\sf T}$ and ${\sf X}=-{\sf X}$ for ${\sf X}={\sf C},{\sf U}$ . Truth assignments to complex formulas are as follows:

A four-valued interpretation of the form (x,x) may be associated with a two-valued (or total) interpretation x. (x,y) is a three-valued (or consistent) interpretation, if $x \subseteq y$ . Interpretations are compared by two-order relations which form a pointwise extension of the structure ${\cal FOUR}$ consisting of ${\sf T}, {\sf F},{\sf C}$ , and ${\sf U}$ with ${\sf U}<_i {\sf F},{\sf T}<_i {\sf C}$ and ${\sf F}<_t {\sf C},{\sf U}<_t {\sf T}$ . The pointwise extension of these orders corresponds to the information order, which is equivalently defined as $(x,y)\leq_i (w,z)$ iff $x\subseteq w$ and $z\subseteq y$ , and the truth order, where $(x,y)\leq_t (w,z)$ iff $x\subseteq w$ and $y\subseteq z$ .

The immediate consequence operator for normal programs (van Emden and Kowalski Reference van Emden and Kowalski1976) is extended to dlps as follows:

Thus, $\mathit{IC}_{\cal P}(x)$ consists of sets of atoms that occur in activated rule heads, each set contains at least one representative from every disjuncts of a rule in ${\cal P}$ whose body is x-satisfied. Denoting by $\wp({\cal S})$ , the powerset of ${\cal S}$ , $\mathit{IC}_{\cal P}$ is an operator on the lattice ${\langle \wp({\cal A}_{\cal P}),\subseteq \rangle}$ .^{Footnote 3}

Given a dlp ${\cal P}$ a consistent interpretation (x,y) is a (three-valued) model of ${\cal P}$ , if for every $\phi\leftarrow \psi \in {\cal P}$ , $(x,y)(\phi)\geq_t (x,y)(\psi)$ . The GL-transformation $\frac{\cal P}{(x,y)}$ of a disjunctively normal dlp ${\cal P}$ with respect to a consistent (x,y) is the positive program obtained by replacing in every rule in ${\cal P}$ of the form $p_1\lor\ldots\lor p_n \leftarrow \bigwedge_{i=1}^m q_i\land \bigwedge_{j=1}^n \lnot r_j$ a negated literal $\lnot r_i$ ( $1\leq i\leq k$ ) by $(x,y)(\lnot r_i)$ . (x,y) is a three-valued stable model of ${\cal P}$ iff it is a $\leq_t$ -minimal model of $\frac{{\cal P}}{(x,y)}$ . ^{Footnote 4}

2.2 Approximation fixpoint theory

We now recall basic notions from AFT, as described by Denecker et al. (2000). We restrict ourselves here to the necessary formal details and refer to more detailed introductions by Denecker et al. (2000) and Bogaerts (Reference Bogaerts2015) for more informal details. AFT introduces constructive techniques for approximating the fixpoints of an operator O over a lattice $L= {\langle {\cal L},\leq \rangle}$ .^{Footnote 5} Approximations are pairs of elements (x,y). Thus, given a lattice $L = {\langle {\cal L},\leq \rangle}$ , the induced bilattice is the structure $L^2 ={\langle {\cal L}^2,\leq_i,\leq_t \rangle}$ , in which ${\cal L}^2 = {\cal L} \times {\cal L}$ , and for every $x_1,y_1,x_2,y_2 \in {\cal L }$ , $(x_1,y_1) \leq_i (x_2,y_2)$ if $x_1 \leq x_2$ and $y_1 \geq y_2$ , and $(x_1,y_1) \leq_t (x_2,y_2)$ if $x_1 \leq x_2$ and $y_1 \leq y_2$ . ^{Footnote 6}

An approximating operator ${\cal O}:{\cal L}^2\rightarrow {\cal L}^2$ of an operator $O:{\cal L}\rightarrow {\cal L}$ is an operator that maps every approximation (x,y) of an element z to an approximation (x’,y’) of another element O(z), thus approximating the behavior of the approximated operator O. In more details, an operator ${\cal O}:{\cal L}^2\rightarrow {\cal L}^2$ is $\leq_i$ -monotonic, if when $(x_1,y_1)\leq_i(x_2,y_2)$ , also ${\cal O}(x_1,y_1)\leq_i {\cal O}(x_2,y_2)$ ; ${\cal O}$ is approximating, if it is $\leq_i$ -monotonic and for any $x\in {\cal L}$ , ${\cal O}_l(x,x) = {\cal O}_u(x,x)$ .^{Footnote 7} ${\cal O}$ approximates of $O:{\cal L}\rightarrow {\cal L}$ , if it is $\leq_i$ -monotonic and ${\cal O}(x,x) = (O(x), O(x))$ (for every $x\in {\cal L}$ ). Finally, for a complete lattice L, let ${\cal O}:{\cal L}^2 \rightarrow {\cal L}^2$ be an approximating operator. We denote ${\cal O}_{l}(\cdot,y) = \lambda x.{\cal O}_{l}(x,y)$ and similarly for ${\cal O}_{u}$ . The stable operator for ${\cal O}$ is then defined as $S({\cal O})(x,y)=(lfp({\cal O}_l(.,y)),lfp({\cal O}_u(x,.))$ , where lfp(O) denotes the least fixpoint of an operator O.

Approximating operators induce a family of fixpoint semantics. Given a complete lattice $L = {\langle {\cal L},\leq \rangle}$ and an approximating operator ${\cal O}:{\cal L}^2 \rightarrow {\cal L}^2$ , (x,y) is a Kripke–Kleene fixpoint of ${\cal O}$ if $(x,y) =lfp_{\leq_i}({\cal O}(x,y))$ ; (x,y) is a three-valued stable fixpoint of ${\cal O}$ if $(x,y)= S({\cal O})(x,y) $ ; (x,y) is a two-valued stable fixpoints of ${\cal O}$ if $x=y$ and $(x,x)= S({\cal O})(x,x)$ ; (x,y) is the well-founded fixpoint of ${\cal O}$ if it is the $\leq_i$ -minimal (three-valued) stable fixpoint of ${\cal O}$ .

2.3 Non-deterministic AFT

AFT was generalized to non-deterministic operators, that is, operators which map elements of a lattice to a set of elements of that lattice (like the operator $\mathit{IC}_{\cal P}$ for dlps) by Heyninck et al. (Reference Heyninck, Arieli and Bogaerts2022). We recall the necessary details, referring to the original paper for more details and explanations.

A non-deterministic operator on ${\cal L}$ is a function $O : {\cal L}\rightarrow \wp({\cal L}) \setminus \{\emptyset\}$ . For example, the operator $\mathit{IC}_{\cal P}$ from Definition 1 is a non-deterministic operator on the lattice ${\langle \wp({\cal A}_{\cal P}),\subseteq \rangle}$ .

As the ranges of non-deterministic operators are sets of lattice elements, one needs a way to compare them, such as the Smyth order and the Hoare order. Let $L = {\langle {\cal L},\leq \rangle}$ be a lattice, and let $X,Y \in \wp({\cal L})$ . Then, $X \preceq^S_L Y$ if for every $y\in Y$ there is an $x\in X$ such that $x\leq y$ ; and $X \preceq^H_L Y$ if for every $x\in X$ there is a $y\in Y$ such that $x\leq y$ . Given some $X_1,X_2,Y_1,Y_2\subseteq {\cal L}$ , $X_1\times Y_1 \preceq^A_i X_2\times Y_2$ iff $X_1\preceq^S_L X_2$ and $Y_2\preceq^H_L Y_1$ . Let $L={\langle {\cal L},\leq \rangle}$ be a lattice. Given an operator ${\cal O}:{\cal L}^2\rightarrow {\cal L}^2$ , we denote by ${\cal O}_l$ the operator defined by ${\cal O}_l(x,y)={\cal O}(x,y)_1$ , and similarly for ${\cal O}_u(x,y)={\cal O}(x,y)_2$ . An operator ${\cal O}:{\cal L}^2\rightarrow \wp({\cal L}){\setminus\emptyset}\times \wp({\cal L}){\setminus\emptyset}$ is called a non-deterministic approximating operator (ndao, for short), if it is $\preceq^A_i$ -monotonic (i.e., $(x_1,y_1)\leq_i (x_2,y_2)$ implies ${\cal O}(x_1,y_1)\preceq^A_i {\cal O}(x_2,y_2)$ ) and is exact (i.e., for every $x\in {\cal L}$ , ${\cal O}(x,x)={\cal O}_l(x,x)\times {\cal O}_{l}(x,x)$ ). We restrict ourselves to ndaos ranging over consistent pairs (x,y).

We finally define the stable operator (given an ndao ${\cal O}$ ) as follows. The complete lower stable operator is defined by (for any $y\in {\cal L}$ ) $C({\cal O}_l)(y) \ = \{x \in {\cal L}\mid x\in {\cal O}_l(x,y) \mbox{ and }\lnot \exists x'< x: x'\in {\cal O}_l(x',y)\}$ . The complete upper stable operator is defined by (for any $x\in {\cal L}$ ) $C({\cal O}_u)(x) \ =\ \{y \in {\cal L}\mid y\in {\cal O}_u(x,y) \mbox{ and }\lnot \exists y'<y: y' \in {\cal O}_u(x,y')\}$ . The stable operator: $S({\cal O})(x,y)=C({\cal O}_l)(y)\times C({\cal O}_u)(x)$ . (x,y) is a stable fixpoint of ${\cal O}$ if $(x,y)\in S({\cal O})(x,y)$ .^{Footnote 8}

Other semantics, for example, the well-founded state and the Kripke–Kleene fixpoints and state are defined by Heyninck et al. (Reference Heyninck, Arieli and Bogaerts2022) and can be immediately obtained once an ndao is formulated.

Since ${\cal IC}_{\cal P}^l(x,y)={\cal IC}_{\cal P}^u(y,x)$ (see Heyninck et al. Reference Heyninck, Arieli and Bogaerts2022, Lemma 1), ${\cal IC}_{\cal P}$ behaves as follows:

• • For any (x,y) with $q\not\in x$ and $q\not\in y$ , ${\cal IC}_{\cal P}(x,y)=\{\{p\},\{q\},\{p,q\}\}\times \{\{p\},\{q\},\{p,q\}\}$ ,

• • For any (x,y) with $q\not\in x$ and $q\in y$ , ${\cal IC}_{\cal P}(x,y)=\{\emptyset\}\times \{\{\{p\},\{q\},\{p,q\}\}$ ,

• • For any (x,y) with $q\in x$ and $q\not\in y$ , ${\cal IC}_{\cal P}(x,y)=\{\{p\},\{q\},\{p,q\}\}\times \{\emptyset\}$ , and

• • For any (x,y) with $q\in x$ and $q\in y$ , ${\cal IC}_{\cal P}(x,y)=\{(\emptyset,\emptyset)\}$ .

We see, for example, that $C({\cal IC}^l_{\cal P})(\{p\})=\{\{p\},\{q\}\}$ and thus $(\{p\},\{p\})$ is a stable fixpoint of ${\cal IC}_{\cal P}$ . $(\emptyset,\{q\})$ is the second stable fixpoint of ${\cal IC}_{\cal P}$ . $(\emptyset,\{p,q\})$ is a fixpoint of ${\cal IC}_{\cal P}$ that is not stable.

In general, (total) stable fixpoints of ${\cal IC}_{\cal P}$ correspond to (total) stable models of ${\cal P}$ , and weakly supported models of ${\cal IC}_{\cal P}$ correspond to fixpoints of ${\cal IC}_{\cal P}$ . (Heyninck et al. Reference Heyninck, Arieli and Bogaerts2022).

5.1 Preliminaries on aggregates

We survey the necessary preliminaries on aggregates and the corresponding programs, restricting ourselves to propositional aggregates and leaving aggregates with variables for future work.

A set term S is a set of pairs of the form $[ \overline{t}: \mathit{Conj}]$ with t a list of constants and $\mathit{Conj}$ a ground conjunction of standard atoms For example, $[1:p; 2:q; -1:r]$ intuitively assigns 1 to p, 2 to q and $-1$ to r. An aggregate function is of the form f(S) where S is a set term, and f is an aggregate function symbol (e.g., $\#{\tt Sum}$ , $\#{\tt Count}$ or $\#{\tt Max}$ ). An aggregate atom is an expression of the form $f(S)\ast w$ where f(S) is an aggregate function, $\ast\in \{<,\leq,\geq,>,=\}$ and w is a numerical constant. We denote by ${\sf At}(f(S)\ast w)$ the atoms occuring in S.

A disjunctively normal aggregate program consists of rules of the form (where $\Delta$ is a set of propositional atoms, and $\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_m$ are aggregate or propositional atoms):

\[ \bigvee \Delta\leftarrow \alpha_1,\ldots,\alpha_n,\lnot \beta_1,\ldots,\lnot \beta_m\]

An aggregate symbol is evaluated w.r.t. a set of atoms as follows. First, let x(S) denote the multiset $[t_1 \mid \langle t_1,\ldots,t_n:{Conj}\rangle\in S\mbox{ and }{Conj}\mbox{ is true w.r.t. }x]$ . x(f(S)) is then simply the result of the application of f on x(S). If the multiset x(S) is not in the domain of f, $x(f(s))=\curlywedge$ where $\curlywedge$ is a fixed symbol not occuring in ${\cal P}$ . An aggregate atom $f(S)\ast w$ is true w.r.t. x (in symbols, $x(f\ast w)={\sf T}$ ) if: (1) $x(f(S))\neq \curlywedge$ and (2) $x(f(S))\ast w$ holds; otherwise, $f(S)\ast w$ is false (in symbols, $x(f\ast w)={\sf F}$ ). $\lnot f(S)\ast w$ is true if: (1) $x(f(S))\neq \curlywedge$ and (2) $x(f(S))\ast w$ does not hold; otherwise, $\lnot f(S)\ast w$ is false. Evaluating a conjunction of aggregate atoms is done as usual. We can now straightforwardly generalize the immediate consequence operator for disjunctive logic programs to disjunctive aggregate programs by generalizing $\mathit{HD}_{\cal P}$ to take into account aggregate formulas as described above: $\mathit{HD}_{\cal P}(x)=\{ \Delta\mid \bigvee\Delta\leftarrow \phi\in {\cal P}, x(\phi)={\sf T}\}$ . $\mathit{IC}_{\cal P}$ from Definition 1 is then generalized straightforwardly by simply using the generalized $\mathit{HD}_{\cal P}$ . Thus, the only difference with the immediate consequence operator for dlps is that the set of activated heads $\mathit{HD}_{\cal P}$ now takes into account the truth of aggregates as well.

The first semantics we consider is the one formulated by Gelfond and Zhang (Reference Gelfond and Zhang2019) (defined there only for logic programs with aggregates occurring positively in the body of a rule):

We now move to the semantics by Denecker et al. (Reference Denecker, Marek and Truszczynski2002). They are defined only for non-disjunctive aggregate programs. They are defined on the basis of the ultimate (deterministic) approximator ${\cal IC}^{\sf DMT}_{\cal P}$ (Definition 2). In more detail, an interpretation (x,y) is ${\sf DMT^d}$ -stable if and only if $(x,y)\in S({\cal IC}^{\sf DMT^d}_{\cal P})(x,y)$ , that is, $x\in lfp({\cal IC}^{\sf DMT^d}_{\cal P}(.,y))$ and $y\in lfp({\cal IC}^{\sf DMT^d}_{\cal P}(x,.))$ .

We first explain why $\{p\}$ is not a GZ-stable model. First, we construct ${\cal P}^{\{p\}}_{\sf GZ}=\{ p\leftarrow p\}$ . Since $\{p\}$ is not a stable model of ${\cal P}^{\{p\}}_{\sf GZ}$ , we see that $\{p\}$ is not a GZ-stable model. Likewise, since ${\cal P}^{\emptyset}_{\sf GZ}=\{p\leftarrow \emptyset\}$ , we see that $\emptyset$ is not a stable model of ${\cal P}^{\emptyset}_{\sf GZ}$ and therefore not GZ-stable.

To see $\{p\}$ is a ${\sf DMT^d}$ -stable model, observe that ${\cal IC}_{\cal P}^{{\sf DMT^d},l}(\emptyset,\{p\})={\cal IC}_{\cal P}^{{\sf DMT^d},l}(\{p\},\{p\})=\{p\}$ . Thus, $lfp({\cal IC}_{\cal P}^{{\sf DMT^d},l}(.,\{p\})=\{p\}$ , that is, $(\{p\},\{p\})=S({\cal IC}^{\sf DMT^d}_{\cal P})(\{p\},\{p\})$ .

5.2 Non-deterministic approximation operators for disjunctive aggregate programs

We now proceed to define ndaos for disjunctive aggregate programs. The first ndao we consider generalizes the trivial operator (Pelov et al. Reference Pelov, Denecker and Bruynooghe2007), which maps two-valued interpretations to their immediate consequences, whereas three-valued interpretations are mapped to the least precise pair $(\emptyset,{\cal A}_{\cal P})$ (or, in the non-deterministic case, $\{\emptyset\}\times \{{\cal A}_{\cal P}\}$ ). We also study the ndao ${\cal IC}^{\sf DMT}_{\cal P}$ based on the deterministic ultimate approximation, and the ultimate ndao ${\cal IC}_{\cal P}^{\cal U}$ .

Definition 5 Given a disjunctively normal aggregate program ${\cal P}$ and a (consistent) interpretation (x,y), let

The ndaos ${\cal IC}^{\sf DMT}_{\cal P}$ and ${\cal IC}^{\cal U}_{\cal P}$ are defined exactly the same as in Section 3 (recall that $\mathit{IC}_{\cal P}(x)$ was generalized for aggregates in Section 5.1). We illustrate these semantics with an example:

We now look at the ${\sf DMT}$ -semantics. For this, we first calculate $\mathit{HD}_{\cal P}$ and $\mathit{IC}_{\cal P}$ for all members of $\wp(\{r,q,s\})$ (with $\Delta_1=\{\{r\},\{q\},\{r,q\}\}$ and $\Delta_2=\{\{s,r\},\{s,q\},\{s,r,q\}\}$ ):

We then see that, for example, ${\cal IC}^{\sf DMT}_{\cal P}(\{r,s\},\{r,s\})=\{\{r,s\},\{q,s\},\{r,q,s\}\}\times \{\{r,s\},\{q,s\},\{r,q,s\}\}$ , whereas ${\cal IC}^{\sf DMT}_{\cal P}(\emptyset,\{r,s\})=\{\emptyset\}\times \{ \{r,s\},\{q,s\},\{r,q,s\}\}$ .

We see that ${\cal IC}^{\cal U}_{\cal P}(\{r,s\},\{r,s\})=\{\{r,s\},\{q,s\},\{r,q,s\}\}\times \{\{r,s\},\{q,s\},\{r,q,s\}\}$ , whereas ${\cal IC}^{\cal U}_{\cal P}(\emptyset,\{r,s\})=\wp(\{r,s,q\})\times \wp(\{r,s,q\})$ .

We now show that these operators are approximation operators with increasing orders of precision: ${\cal IC}^{\sf GZ}_{\cal P}$ is the least precise, ${\cal IC}^{\sf DMT}_{\cal P}$ holds a middle ground, and ${\cal IC}^{\cal U}$ is the most precise:

The following properties follow from the general properties shown by Heyninck et al. (Reference Heyninck, Arieli and Bogaerts2022):

The ndao ${\cal IC}_{\cal P}^{\sf GZ}$ only admits two-valued stable fixpoints, and these two-valued stable fixpoints generalize the GZ-semantics (Gelfond and Zhang Reference Gelfond and Zhang2019):

We finally show that stable semantics based on ${\cal IC}^{\sf DMT}_{\cal P}$ generalize those for non-disjunctive logic programs with aggregates by Denecker et al. (Reference Denecker, Marek and Truszczynski2002).

We have shown how semantics for disjunctive aggregate logic programs can be obtained using the framework of non-deterministic AFT, solving the open question (Alviano et al. Reference Alviano, Faber and Gebser2023) of how operator-based semantics for aggregate programs can be generalized to disjunctive programs. This means AFT can be unleashed upon disjunctive aggregate programs, as demonstrated in this paper, as demonstrated in this section. Other semantics, such as the weakly supported semantics, the well-founded state semantics (Heyninck et al. Reference Heyninck, Arieli and Bogaerts2022) and semi-equilibrium semantics (Section 4, as in the appendix of the full version of this article Heyninck and Bogaerts Reference Heyninck and Bogaerts2023) are obtained without any additional effort and while preserving desirable properties shown algebraically for ndaos. None of these semantics have, to the best of our knowledge, been investigated for dlps with aggregates. Other ndaos, left for future work, can likely be obtained straightforwardly on the basis of deterministic approximation operators for aggregate programs that we did not consider in this paper (e.g., the operator defined by Vanbesien et al. (Reference Vanbesien, Bruynooghe and Denecker2021) to characterize the semantics of Marek and Remmel (Reference Marek and Remmel2004) or the bounded ultimate operator introduced by Pelov and Truszczynski (Reference Pelov and Truszczynski2004).