2 Approximation fixpoint theory

Here, we recall the basics of AFT from the work by Denecker et al. (Reference Denecker, Marek and Truszczyński2000). In many non-monotonic languages, a theory defines a semantic lattice ^{Footnote 1} operator $O: L\rightarrow L$ . If O is monotone, its least fixpoint is often taken as the semantics of the theory. Otherwise, AFT can be applied. The first step is to associate the approximation space $L^2$ to L. A pair $(x,y)\in L^2$ is an approximation of any $z\in[x,y]$ . With $x \leq y$ , the interval is non-empty and the pair is consistent. $L^c$ is the subspace of consistent pairs. $L^2$ and $L^c$ possess (i) a precision order, $(x,y)\leq_p (u,v)$ if $x\leq u$ and $y\geq v$ and (ii) the embedding of L, namely the set of exact pairs (x,x) which approximate only x. The least precise point is $(\bot,\top)$ , with $\bot=glb(L),\top=lub(L)$ .

The second step is to assign an approximating operator A to O: a $\leq_p$ -monotone operator on $L^2$ or $L^c$ such that if (x,y) approximates z then A(x,y) approximates O(z); an approximator on $L^2$ also has to be symmetric: $A(x,y)=(u,v)$ iff $A(y,x)=(v,u)$ . Thus, increasing the precision of the input to an approximator A increases the precision of the output. With an approximator A, several types of fixpoints are definable. The least fixpoint (lfp) construction $(\bot,\top)$ , $A(\bot,\top)$ ,, $A^\alpha(\bot,\top)$ , produces the KK fixpoint $\textit{KK}(A)=\textit{lfp}(A)$ . The KK fixpoint is a pair $(x,y) \in L^2$ . If exact, $x (= y)$ is the only fixpoint of O. Otherwise, it approximates all fixpoints of O, including “self-supported” ones that are often not minimal in L. Stable and WF fixpoint definitions contain mechanisms to reduce self-support. ^{Footnote 2} A stable fixpoint $x\in L$ is one such that $x=\textit{lfp}(\lambda z: A(z,x)_1)$ , where $A(z,x)_1$ is the first component of the pair A(z,x). The WF fixpoint $\textit{WF}(A)$ is the least precise pair (x,y) with $x=\textit{lfp}(\lambda z: A(z,y)_1)$ and $y=\textit{lfp}(\lambda z: A(x,z)_2)$ . It is the least precise fixpoint of the monotone operator $(x,y)\mapsto (\textit{lfp}(\lambda z:A(z,y)_1),\textit{lfp}(\lambda z:A(x,z)_2))$ . We have that $\textit{KK}(A)$ and $\textit{WF}(A)$ are consistent and for each stable fixpoint x, $\textit{KK}(A)\leq_p \textit{WF}(A)\leq_p x$ and x is a minimal fixpoint of O. ^{Footnote 3}

AFT induces relationships between fixpoints of different approximators of O. If approximator A is pointwise less precise than B, then $\textit{KK}(A)\leq_p \textit{KK}(B)$ , $\textit{WF}(A)\leq_p \textit{WF}(B)$ and any stable fixpoint of A is a stable fixpoint of B. Thus, with increasing precision of the approximator, KK and WF fixpoints increase in precision, and the set of stable fixpoints grows. There exists a most precise approximator $\textit{Ult}_O$ of O, with the most precise KK and WF fixpoint and the largest set of stable fixpoints. For consistent pairs (x,y), $\textit{Ult}_O(x,y)$ is the most precise pair approximating O([x,y]), that is, $(\textit{glb}(O([x,y]),\textit{lub}(O([x,y])))$ . It follows, perhaps surprisingly, that if an even moderately precise approximator A has a stable fixpoint x, then x is approximated by the most precise WF fixpoint $\textit{WF}(\textit{Ult}_O)$ associated with O (but keep in mind that unprecise approximators are unlikely to have stable fixpoints). This counter-intuitive fact tells us that in all cases, if stable fixpoints exist, they are in the proximity of the most precise WF fixpoint that can be associated to O. This may explain the good quality of stable semantics in capturing intuitions. But while KK and WF are constructive, stable semantics is not really. It only has a constructive test: testing if x is stable is by testing if x is the limit of the $\textit{lfp}$ construction of $\lambda z: A(z,x)_1$ .

We now sketch how to use AFT to define constructive semantics for programs based on some logic ${\mathcal{L}}$ . ^{Footnote 4} We assume ${\mathcal{L}}$ is a first-order logic with a (Herbrand) model semantics defined using a 2-valued truth function $\mathcal{H}^2$ , or equivalently, a satisfaction relation $\models_2$ (where $I\models_2 \phi$ iff $\mathcal{H}_{I}^2(\phi)={\mathbf{t}}$ ). An ${\mathcal{L}}$ -program is then defined as follows:

Importantly, $\leftarrow$ is not a connective of ${\mathcal{L}}$ but AFT defines its meaning as a construction operator. LP and (non-disjunctive) ASP are instances of this where ${\mathcal{L}}$ is simply the logic of conjunctions of literals p or $\neg p$ under standard interpretation. Given an ${\mathcal{L}}$ -program P, the corresponding lattice $L,\leq$ is the set of P’s Herbrand interpretations ordered by the truth order ( ${\mathbf{f}}<{\mathbf{t}}$ ). The sets $L^c$ and $L^2$ correspond to 3- and 4-valued interpretations. Any 3- or 4-valued interpretation ${\mathcal{I}}$ can be split into a pair (I,J) by splitting truth values as in ${\mathbf{t}}\sim({\mathbf{t}},{\mathbf{t}}), {\mathbf{f}}\sim({\mathbf{f}},{\mathbf{f}}), {\mathbf{u}}\sim({\mathbf{f}},{\mathbf{t}})$ and $\mathbf{i}\sim({\mathbf{t}},{\mathbf{f}})$ . If ${\mathcal{I}}$ is 3-valued, then $I\leq J$ , and I is a lower bound, J an upper bound of ${\mathcal{I}}$ . The 3- and 4-valued structures are equipped with a truth order, which isomorphically corresponds to the product order $\leq$ of $L^2$ and $L^c$ , and with a precision order $\leq_p$ ( ${\mathbf{u}} <_p {\mathbf{t}} <_p \mathbf{i}, {\mathbf{u}}<_p{\mathbf{f}}<_p\mathbf{i}$ which corresponds to the precision order of $L^2$ and $L^c$ . Any truth or precision monotone operator $\Gamma$ on 2-, 3- or 4-valued structures corresponds to a truth or precision monotone operator on $L, L^c, L^2$ .

An ${\mathcal{L}}$ -program P is characterised by the immediate consequence operator ${T_{P}}:L\rightarrow L$ . This operator fulfills the role of O, the approximated operator. The immediate consequence operator ${T_{P}}:L\rightarrow L$ for a program P is such that ${T_{P}}(I)=J$ if for every ground atom p, $\mathcal{H}_{J}^2(p) = \textit{lub}_\leq(\{\mathcal{H}_{I}^2(\psi) | (p \leftarrow \psi) \in P\})$ . There are two ways to define AFT semantics using an approximator $A (= A_P)$ for ${T_{P}}$ .

One way is to use $A_P = \textit{Ult}({T_{P}})$ , the most precise approximator of ${T_{P}}$ in $L^c$ leading to ultimate versions of the family of AFT semantics as described by Denecker et al. (Reference Denecker, Marek and Truszczyński2004) and used by Denecker et al. (Reference Denecker, Pelov and Bruynooghe2001) for defining semantics of Aggregate LP. This is the most precise approach, but computationally costly. The other way is to extend ${\mathcal{L}}$ ’s truth assignment to 3- or 4-valued interpretations and by interpreting ${T_{P}}$ ’s definition in this broader context. For example, a 3-valued truth assignment $\mathcal{H}^3$ induces a 3-valued immediate consequence operator ${\Phi_{P}}$ where ${\Phi_{P}}({\mathcal{I}})={\mathcal{I}}'$ if for every atom p, $\mathcal{H}_{{\mathcal{I}}'}^3(p) = \textit{lub}_{\leq}\{\mathcal{H}_{{\mathcal{I}}}^3(\psi)|(p\leftarrow \psi) \in P\}$ . The operator ${\Phi_{P}}$ corresponds isomorphically to an $L^c$ approximator $A_P$ on pairs $I<J$ of 2-valued interpretations. But for $A_P$ to be an approximator of ${T_{P}}$ , the 3-valued truth assignment $\mathcal{H}^3$ should satisfy a condition introduced for 3-valued logic by Kleene (Reference Kleene1952):

For example, Kleene’s strong 3-valued truth assignment $\mathcal{H}^{SK}$ of FO (introduced by Kleene Reference Kleene1952) and Belnap’s 4-valued extension (introduced by Belnap Reference Belnap1977) are regular. They induce multi-valued extensions ${\Phi_{P}}$ of ${T_{P}}$ first introduced by Fitting (Reference Fitting1985). Later, ${\Phi_{P}}$ was found to correspond to an approximator $A_P$ of ${T_{P}}$ whose KK, WF and stable fixpoints correspond to the semantics of the same name.

In LP and ASP, it is often taken for granted that LP’s nonmonotonicity is due to its non-classical negation not. But it is evident in AFT-based semantics, that the main non-classical connective is the rule operator: its semantics is defined via operators and constructive processes while negation in bodies is treated like the other FO connectives, using three-valued logic only for approximation of the standard classical connectives.

The AFT road is quite unlike other semantic techniques in ASP. In the next section, we reformulate the framework in more accessible terms for the ASP community.

3 Ternary satisfaction relations

Ternary satisfaction relations were used originally by Liu et al. (Reference Liu, Pontelli, Son and Truszczyński2010) in the context of Aggregate ASP semantics where they were called sub-satisfiability relations.

We still assume a base logic ${\mathcal{L}}$ equipped with satisfaction relation $\models_2$ . Below, we restrict ourselves to 3-valued interpretations corresponding to pairs (I,J) of 2-valued Herbrand interpretations $I\subseteq J$ (but extension to non-Herbrand interpretations is possible).

For any pair of a lower-regular TSR $\models_3$ and an upper-regular TSR $\models_3^\uparrow$ , it always holds that $\models_3\subseteq\models_3^\uparrow$ since $(I,J)\models_3\psi$ implies $(I,I)\models_2\psi$ which implies $(I, J)\models_3^\uparrow \psi$ . Also, a three-valued truth-function $\mathcal{H}^3$ corresponds one to one to pairs $(\models_3,\models_3^\uparrow)$ of TSRs satisfying $\models_3 \subseteq \models_3^\uparrow$ . The correspondence is: (i) $(I, J) \models_3 \psi$ iff $\mathcal{H}_{(I, J)}^3(\psi) = {\mathbf{t}}$ and (ii) $(I, J) \models_3^\uparrow \psi$ iff $\mathcal{H}_{(I, J)}^3(\psi) \in \{{\mathbf{t}}, {\mathbf{u}}\}$ .

Taking ${\mathcal{L}}$ as FO and $\mathcal{H}^3$ as the strong Kleene truth assignment, it is a folk result that $(I,J)\models_3 \psi$ if $\psi$ evaluates to true when interpreting all positively occurring atoms in I and all negatively occurring ones in J. For $(I,J)\models_3^\uparrow \psi$ , exchange the roles of I and J.

For each ${\mathcal{L}}$ -program P, the lower- and upper-regular TSR induce two distinct operators on consistent pairs $I\subseteq J$ : $A_P^{\models_3}(I, J) = \{p| \exists (p \leftarrow \psi) \in P: (I, J)\models_3 \psi\}$ and $A_P^{\models_3^\uparrow}(I, J) = \{p| \exists (p \leftarrow \psi) \in P: (I, J)\models_3^\uparrow \psi \}$ . Now, we define $A_P(I, J) = (A_P^{\models_3}(I, J), A_P^{\models_3^\uparrow}(I, J))$ .

Now, supported, KK, WF and stable models of the ${\mathcal{L}}$ -programs P can be defined in terms of $\models_3$ and $\models_3^\uparrow$ . Interestingly, J is a stable model of P iff J is the least fixpoint of $A_P^{\models_3}(I,J) =\lambda I\in [\bot,J]: \{p| \exists (p \leftarrow \psi) \in P: (I, J)\models_3 \psi\}$ . No need of $\models_3^\uparrow$ ! Clearly there exists an asymmetry between truth and falsity in stable models. While information about the truth of formulas, encoded by $\models_3$ , is essential to determine the stable fixpoints of a program, information about their falsity, given by $\models_3^\uparrow$ , is disregarded.

4 Generalizing the concept of answer set

Here, we generalize stable models of ${\mathcal{L}}$ -programs to answer sets. Now, ${\mathcal{L}}$ , the logic of the rule bodies, has, besides a satisfaction relation $\models_2$ also a TSR $\models_3$ .

In general, the (transfinite) fixpoint sequence may leave $[\emptyset,J]$ , or end up with a fixpoint $I \subseteq J$ . In case it is J, it is an answer set. Let $\models_3$ be lower-regular. Combining it with any upper-regular TSR $\models_3^\uparrow$ induces a regular truth assignment $\mathcal{H}^3$ , as well as an entire family of AFT fixpoints and models of ${\mathcal{L}}$ -programs: KK models, WF models and AFT-stable models approximated by the WF models. The AFT-stable models in this framework depend only on $\models_3$ and they correspond exactly to answer sets of $\models_3$ , since a lower-regular TSR is also lower-monotone.

To finish this section, we analyze the link with the original definition of answer set given by Gelfond and Lifschitz (Reference Gelfond and Lifschitz1988). There, ${\mathcal{L}}$ is the logic of ground sets/conjunctions of literals with FO’s standard satisfaction relation $\models_2$ . Let P be an ${\mathcal{L}}$ -program.

• all rules with a negative literal $\neg l$ such that $l \in J$ .

• all negative literals in the bodies of the remaining rules.

J is a GL-answer set of P if $J \models_2 P^J$ and there is no $I\subset J$ such that $I\models_2 P^J$ .

We now identify the lower-regular TSR $\models_{GL}$ such that answer sets of programs induced by $\models_{GL}$ , coincide with GL-answer sets. The TSR that is needed evaluates bodies $\psi$ in pairs (I,J) by interpreting atomic literals of $\psi$ in I and negative literals in J. But as explained in the previous section, that is how the lower-regular TSR $\models_{SK}$ of the strong Kleene truth assignment operates: $(I,J)\models_{SK} \psi$ iff atoms in $\psi$ hold in I and negative literals hold in J. Thus, $\models_{GL}$ is the restriction of $\models_{SK}$ to conjunctions of literals. With this in mind, the following proposition is straightforward.

Thus, this type of answer set fits in the AFT-landscape of KK, WF and stable models.

5 Aggregates programs in the AFT framework

Aggregate Programs For the remainder of the text, we will consider ${\mathcal{L}}$ -programs where ${\mathcal{L}}$ is the logic of conjunctions of literals and positive aggregate atoms. ^{Footnote 7}

An aggregate atom $a^\textit{Aggr}$ is of the form: $\textit{Agg}(\{a_1: \textit{cond}_1,..., a_n: \textit{cond}_n\})*w$ with aggregate symbol $\textit{Agg}$ (e.g., SUM), comparison connective $*$ (e.g., $\leq, =, \neq,\dots$ ), numerical value w and multiset $\{a_1: \textit{cond}_1,..., a_n: \textit{cond}_n\}$ wher each $cond_i$ is a literal and each $a_i$ is a weight. The $\models_2$ relation for ${\mathcal{L}}$ is naturally extended with a rule for evaluating aggregate atoms, so also ${T_{P}}$ is defined. This enables the first approach to apply AFT on this type of programs, using the 3-valued ultimate approximator $\textit{Ult}_{{T_{P}}}$ yielding the ultimate KK, WF and stable semantics. Up to the syntax, this is the semantics of Denecker et al. (Reference Denecker, Pelov and Bruynooghe2001). The second approach to apply AFT is based on defining a regular 3-valued truth assignment for aggregate atoms, leading to a three valued operator ${\Phi^{Aggr}_{P}}$ . Up to the syntax, this was the approach followed by Pelov et al. (Reference Pelov, Denecker and Bruynooghe2007).

We now start the study of existing approaches for handling aggregates in ASP. Some of these use more extensive programs than the ones analysed here, however our analysis only considers the simpler ${\mathcal{L}}$ -programs. Proposition 3 shows that there is a constructive test for answer sets of non-disjunctive ${\mathcal{L}}$ -programs for semantics with lower-monotone TSR’s. A lower-regular TSR is lower-monotone by definition. Thus, this constructive test is applicable for all semantics that fit in the AFT-framework. However, not all semantics for ASP programs in the literature have lower-regular or even lower-monotone ternary satisfaction relations.

Before we start, we define the precision order on the TSRs analogous to the precision order on truth assignments defined by Pelov et al. (Reference Pelov, Denecker and Bruynooghe2007).

5.2 Other ternary satisfaction relations

In the AFT framework, semantics of ASP aggregate atoms are based on lower regular TSRs. As we show in this section, also other well-known semantics can be characterized using TSRs, however, they are not regular. This is no coincidence. The definition of answer set semantics in terms of TSRs strongly resembles another well-known semantic method of ASP, namely using the logic of here-and-there (HT) (for an overview of HT and its applications to ASP, see the work by Cabalar et al. Reference Cabalar, Pearce and Valverde2017). Due to very different points of view on answer sets, the two frameworks obtain different requirements for the TSRs. AFT treats answer sets as the result of constructive processes; the rule operator serves to produce them. A production is safe if the TSR is lower-regular. In contrast, HT takes a non-constructive take on answer sets. In HT, extensions build on the inherently non-regular TSRs derived from the three-valued logic G3 introduced by GÖdel (Reference GÖdel1932) where the rule operator is treated as HT-material implication.

Reference Marek and Remmel Marek and Remmel (2004) . They study Set Constraints (SC) Programming. It builds an NSS-reduct for an SC-program. Liu et al. (Reference Liu, Pontelli, Son and Truszczyński2010) prove that this semantics for set constraints is also obtained by the following satisfaction rule for a set constraint $\alpha$ : $(I, J)\models_{\textit{MR}} \alpha$ , if $J\models_2 \alpha$ and there exists an interpretation $Z \subseteq I$ such that $Z\models_2 \alpha$ . This leads to the TSR $\models_{\textit{MR}}$ :

Definition 12 ( $\models_{\textit{MR}}$ ) $\models_{\textit{MR}}$ extends $\models_{\textit{GL}}$ with: $(I, J) \models_{\textit{MR}} a^{\textit{Aggr}} $ iff $J\models_2 a^{\textit{Aggr}}$ and there exists an interpretation Z such that $Z \subseteq I$ and $Z \models_2 a^{\textit{Aggr}}$ .

Consider the aggregate atom $\mathit{SUM}(\{1:p, -1:q\}) \geq 0$ and two intervals, namely $(\emptyset, \{p, q, s\}) <_p (\emptyset, \{q\})$ . We have that $(\emptyset, \{p, q, s\}) \models_{\textit{MR}} \mathit{SUM}(\{1:p, -1:q\}) \geq 0$ while $(\{p\}, \{q\}) \not\models_{\textit{MR}} \mathit{SUM}(\{1:p, -1:q\}) \geq 0$ . Hence, $\models_\textit{MR}$ is not lower-regular.

Proposition 9 (i) $\models_{\textit{MR}}$ extends $\models_2$ , that is, $(I,I) \models_{\textit{MR}} \psi $ iff $I \models_2 \psi$ . (ii) $\models_{\textit{MR}}$ is lower-monotone, that is, if $I \subseteq I'$ and $(I, J) \models_{\textit{MR}} \psi$ , then $(I', J)\models_{\textit{MR}} \psi$ .

While non-regular, the ternary relation $\models_{\textit{MR}}$ still extends the satisfaction relation $\models_2$ and induces a monotone operator $\lambda I: A_{\models_{MR}}(I,J)$ . Thus, an answer set may still be defined as an interpretation J that is a least fixpoint of this operator. But, due to non-regularity, some unexpected answer sets are obtained.

Example 1 Take the program:

\begin{gather*} s \leftarrow \mathit{SUM}(\{1:p, -1:q\}) \geq 0.\\ q \leftarrow \mathit{SUM}(\{1:s\}) > 0. ~~~~~~~ p \leftarrow \mathit{SUM}(\{1:q\}) > 0. \end{gather*}

The bodies of these rules are equivalent to $p\lor\neg q$ , respectively s and q. Therefore, one expects its answer sets to be the same as those of the simplified program:

\begin{gather*}s \leftarrow p. ~~~~~~~~ s \leftarrow \neg q.~~~~~~~~q \leftarrow s. ~~~~~~ p \leftarrow q.\end{gather*}

To check that $J=\{p, q, s\}$ is an answer set, we observe that $(\emptyset, J)\models_{\textit{MR}} \mathit{SUM}(\{1:p, -1:q\})\geq 0$ , hence the first iteration derives the head s. Two more iterations reconstruct the fixpoint $\{p, q, s\}$ which therefore is an answer set according to these semantics.

On the other hand, the Gelfond-Lifschitz reduct of the simplified program with respect to $J = \{p, q, s\}$ is:

\begin{equation*}s \leftarrow p.~~~~~~q \leftarrow s. ~~~~~~p \leftarrow q.\end{equation*}

Since $\emptyset$ is a model of the reduct, $\{p,q,s\}$ is not an answer set of the simplified program. The culprit for “too many” answer sets of the aggregate program is the non-regularity of $\models_{MR}$ which leads to the unsafe derivation of aggregate atoms: in the first derivation, $(\emptyset,\{p, q, s\})\models_{MR}\mathit{SUM}(\{1:p, -1:q\})\geq 0$ which derived s, but this derivation was unsafe since this aggregate atom is not satisfied in the more precise $(\emptyset,\{q\})$ . However, things change when looking only at convex aggregate atoms.

Definition 13 (Convex aggregate atom) An aggregate atom $a^{\textit{Aggr}}$ is convex iff for all interpretations I, Z, J, such that $I \subseteq Z \subseteq J$ , it holds that if $I\models_2 a^{\textit{Aggr}}$ and $J\models_2 a^{\textit{Aggr}}$ , then $Z\models_2 a^{\textit{Aggr}}$ .

Proposition 10 For convex aggregate atoms, $\models_{\textit{MR}}$ behaves lower-regular and equivalent with $\models_{\textit{ult}}$ .

Reference Faber, Pfeifer and Leone Faber et al. (2011) . They provide semantics for a broader class of ASP programs including negated aggregate atoms. This approach constructs a reduct $P^J$ for a program P with respect to an interpretation J by deleting all rules of which the body is not satisfied in J. In general, the immediate consequence operator $T_{P^J}$ is not monotone, so answer sets cannot be defined using the $\textit{lfp}$ construction of this operator. Nevertheless, the $\textit{FPL}$ -answer sets are the answer sets of the following TSR:

Definition 14 ( $\models_{\textit{FPL}}$ ) We define $\models_{\textit{FPL}}$ as follows: $(I, J)\models_{\textit{FPL}} \psi$ iff $I \models_2 \psi$ and $J \models_2 \psi$ .

Proposition 11 $\models_{\textit{FPL}}$ extends $\models_2$ , that is, $(I,I) \models_{\textit{FPL}} \psi $ iff $I \models_2 \psi$ . For conjunctions of aggregate free literals, $\models_{\textit{FPL}}$ coincides with $\models_{\textit{GL}}$ .

This very simple TSR is neither lower-regular nor lower-monotone, thus the constructive test is inapplicable.

As a consequence, discrepancies between some aggregate programs and their aggregate-free simplification are also present. Using again Example 1, $(\emptyset, \{p, q, s\})\models_{\textit{FPL}} \mathit{SUM}(\{1:p, -1:q\})\geq 0$ . But again, $(\emptyset, \{q\})\not\models_{\textit{FPL}} \mathit{SUM}(\{1:p, -1:q\}) \geq 0$ . Similarily to $\models_{\textit{MR}}$ , the $\textit{FPL}$ -semantics leads to $\{p, q, s\}$ as an answer set. It is obvious that if $(I, J)\models_{\textit{FPL}} \psi$ , then $(I, J)\models_{\textit{MR}} \psi$ , since $I \subseteq I$ . Accordingly $\models_{\textit{FPL}}$ is less precise then $\models_{\textit{MR}}$ . Analogously, if $(I, J)\models_{\textit{ult}} \psi$ , then $(I, J)\models_{\textit{FPL}} \psi$ since the interpretations I and J are both elements of [I, J].

Proposition 12 For convex aggregate atoms, the TSR $\models_{\textit{FPL}}$ behaves lower-regular and equivalent to $\models_{\textit{ult}}$ .

Reference Ferraris Ferraris (2011) . This semantics is closely related to the $\textit{FPL}$ -semantics. Actually, they coincide for negation-free programs. Ferraris (Reference Ferraris2011) also cover more extensive instances of answer set programs, such as arbitrary propositional theories. It obtains the reduct $P^J$ for a program P and interpretation J by replacing all maximal subformulas in P that are unsatisfied in J by $\bot$ . This corresponds to the TSR $\models_{\textit{F}}$ :

Definition 15 ( $\models_{\textit{F}}$ ) $\models_{\textit{F}}$ extends $\models_{\textit{GL}}$ with: Let $a^{\textit{Aggr}} = \textit{Agg}(\{a_1:cond_1. \ldots, a_n:cond_n\})*w$ , $(I, J) \models_{\textit{F}} a^{\textit{Aggr}}$ iff $J \models_2 a^{\textit{Aggr}}$ and $I \models_2 \textit{Agg}(\{a_i:cond_i \in \{a_1:cond_1, \ldots, a_n:cond_n\}|J \models_2 cond_i\})*w$ .

Since $\models_{\textit{F}}$ and $\models_{\textit{FPL}}$ coincide for negation-free programs, an analogous discussion of Example 1 leads to the conclusion that the satisfaction relation $\models_{\textit{F}}$ is not lower-regular due to the lack of monotonicity.

Proposition 13 $\models_{\textit{F}}$ extends $\models_2$ , that is, $(I,I) \models_{\textit{F}} \psi $ iff $I \models_2 \psi$ .

Proposition 14 For convex aggregate atoms, $\models_{\textit{F}}$ behaves lower-monotone, that is, if $I \subseteq I'$ and $(I, J) \models_{\textit{F}} \psi$ , then $(I', J)\models_{\textit{F}} \psi$ .

Proposition 15 For anti-monotone aggregate atoms, $\models_{\textit{F}}$ behaves lower-regular and equivalent with $\models_{\textit{ult}}$ .

Table 1 gives an overview. Row 1 indicates for which aggregate atoms the ternary satisfaction relation behaves lower-regular. Row 2 shows which semantics from Pelov et al. (Reference Pelov, Denecker and Bruynooghe2007) coincides with this semantics for the subclass of programs such that the semantics behaves lower-regular. Row 3 indicates for which aggregate atoms the semantics is monotone in the first component such that the answer sets can be constructed as the $\textit{lfp}$ of ${\Phi_{P}}$ . Row 4 gives the set $S_a$ of semantics discussed in this paper that are strictly more precise than the considered instance a. Consequently every a-answer will always be an answer set for every semantics in $S_a$ . This does not generally hold the other way around. Interestingly, all non-regular semantics coincide with the ult-semantics of Pelov (Reference Pelov2004) for the subclass of programs in which they behave lower-regular. ^{Footnote 10} For programs outside this subclass, the non-regular semantics differ and may derive more answer sets than the ult-semantics.

Table 1. A summary of the different ternary satisfaction relations

Supplementary material

To view supplementary material for this article, please visit http://10.1017/S1471068422000126.