Non-deterministic approximation operators: ultimate operators, semi-equilibrium semantics and aggregates (full version)

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, i.e.\ operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola, et al., and (3) we generalize the characterisations of disjunctive logic programs to disjunctive logic programs with aggregates.


Introduction
Knowledge representation and reasoning (KRR), by its very nature, is concerned with the study of a wide variety of languages and formalisms.In view of this, unifying frameworks that allow for the language-independent study of aspects of KRR is essential.One framework with strong unifying potential is approximation fixpoint theory (AFT) (Denecker et al. 2000), a purely algebraic theory which was shown to unify the semantics of, among others, logic programming default logic and autoepistemic logic.The central objects of study of AFT are (approximating) operators and their fixpoints.For logic programming for instance, it was shown that Fitting's three-valued immediate consequence operator is an approximating operator of Van Emden and Kowalski's two-valued immediate consequence operator and that all major semantics of (normal) logic programming can be derived directly from this approximating operator.Moreover, this observation does not only hold for logic programming: also for a wide variety of other domains, it is straightforward how to derive an approximating operator, and the major semantics can be recovered n i=1 p i ← ψ, where the head n i=1 p i is a non-empty disjunction of atoms, and the body ψ is a formula not containing ←.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, e.g. a program is normal if all its rules are normal.The set of atoms occurring in P is denoted A P .
The semantics of dlps are given in terms of four-valued interpretations.A four-valued interpretation of a program P is a pair (x, y), where x ⊆ A P is the set of the atoms that are assigned a value in {T, C} and y ⊆ A P is the set of atoms assigned a value in {T, U}.We define −T = F, −F = T and X = −X for X = C, U. Truth assignments to complex formulas are as follows: T if p ∈ x and p ∈ y, U if p ∈ x and p ∈ y, F if p ∈ x and p ∈ y, C if p ∈ x and p ∈ y.
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 ⊆ y.Interpretations are compared by two order relations which form a pointwise extension of the structure F OUR consisting of T, F, C and U with U < i F, T < i C and F < t C, U < t T. The pointwise extension of these orders corresponds to the information order, which is equivalently defined as (x, y) ≤ i (w, z) iff x ⊆ w and z ⊆ y, and the truth order, where (x, y) ≤ t (w, z) iff x ⊆ w and y ⊆ z.
Thus, IC 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 P whose body is x-satisfied.Denoting by ℘(S) the powerset of S, IC P is an operator on the lattice ℘(A P ), ⊆ . 3iven a dlp P a consistent interpretation (x, y) is a (three-valued) model of P, if for every φ ← ψ ∈ P, (x, y)(φ) ≥ t (x, y)(ψ).The GL-transformation P (x,y) of a disjunctively normal dlp P with respect to a consistent (x, y), is the positive program obtained by replacing in every rule in P of the form p

Approximation Fixpoint Theory
We now recall basic notions from approximation fixpoint theory (AFT), as described by Denecker, Marek and Truszczyński (2000).We restrict ourselves here to the necessary formal details, and refer to more detailed introductions by Denecker, Marek and Truszczyński (2000) and Bogaerts (2015) for more informal details.AFT introduces constructive techniques for approximating the fixpoints of an operator O over a lattice L = L, ≤ .5 Approximations are pairs of elements (x, y).Thus, given a lattice L = L, ≤ , the induced bilattice is the structure An approximating operator O : L 2 → L 2 of an operator O : L → 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 Approximating operators induce a family of fixpoint semantics.Given a complete lattice L = L, ≤ and an approximating operator O :

Non-deterministic approximation fixpoint theory
AFT was generalized to non-deterministic operators, i.e. operators which map elements of a lattice to a set of elements of that lattice (like the operator IC P for DLPs) by Heyninck et al. (2022).We recall the necessary details, referring to the original paper for more details and explanations.
A non-deterministic operator on L is a function O : L → ℘(L) \ {∅}.For example, the operator IC P from Definition 1 is a non-deterministic operator on the lattice ℘(A P ), ⊆ .
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 = L, ≤ be a lattice, and let X, Y ∈ ℘(L).Then: X S L Y if for every y ∈ Y there is an x ∈ X such that x ≤ y; and )), and is exact (i.e., for every x ∈ L, O(x, x) = O l (x, x) × O l (x, x)).We restrict ourselves to ndaos ranging over consistent pairs (x, y).
We finally define the stable operator (given an ndao O) as follows.The complete lower stable operator is defined by (for any y Other semantics, e.g. the well-founded state and the Kripke-Kleene fixpoints and state are defined by Heyninck et al (2022) and can be immediately obtained once an ndao is formulated.

Example 1
An example of an ndao approximating IC P (Definition 1) is defined as follows (given a dlp P and an interpretation (x, y)): , and IC P (x, y) = (IC l P (x, y), IC u P (x, y)).Consider the following dlp: P = {p ∨ q ← ¬q}.The operator IC l P behaves as follows: • For any interpretation (x, y) for which q ∈ x, HD l P (x, y) = ∅ and thus IC l P (x, y) = {∅}.• For any interpretation (x, y) for which q ∈ x, HD l P (x, y) = {{p, q}} and thus IC l P (x, y) = {{p}, {q}, {p, q}}.
In general, (total) stable fixpoints of IC P correspond to (total) stable models of P, and weakly supported models of IC P correspond to fixpoints of IC P .(Heyninck et al. 2022).

Ultimate Operators
Approximation fixpoint theory assumes an approximation operator, but does not specify how to construct it.In the literature, one finds various ways to construct a deterministic approximation operator O that approximates a deterministic operator O.Of particular interest is the ultimate operator (Denecker et al. 2002), which is the most precise approximation operator.In this section, we show that non-deterministic approximation fixpoint theory admits an ultimate operator, which is, however, different from the ultimate operator for deterministic AFT.
We first recall that for a deterministic operator O : L → L, the ultimate approximation O u is defined by Denecker et al. (2002) as follows: Where O[x, y] := {O(z) | x ≤ z ≤ y}.This operator is shown to be the most precise operator approximating an operator O (Denecker et al. 2002).In more detail, for any (deterministic) approximation operator O approximating O, and any consistent (x, y), O(x, y) < i O DMT d (x, y).
The ultimate approximator for IC P for non-disjunctive logic programs P looks as follows: Definition 2 Given a normal logic program P, we let: In this section, we define the ultimate semantics for the non-deterministic operators.In more detail, we constructively define an approximation operator that is most precise and has non-empty upper and lower bounds.Its construction is based on the following idea: we are looking for an operator O U s.t. for any ndao O that approximates O, O l (x, y) S L O U l (x, y) (and similarly for the upper bound).As we know that O l (x, y) S L O(z) for any x ≤ z ≤ y, we can obtain O U l by simply gathering all applications of O to elements of the interval [x, y] i.e. we define: The upper bound can be defined in the same way as the lower bound.Altogether, we obtain: The following example illustrates this definition for normal logic programs: Example 2 Let P = {q ← ¬p; p ← p}.Then IC P (∅) = IC P ({q}) = {q} and IC P ({p}) = IC P ({p, q}) = {p}.Therefore, IC U P (∅, {p, q}) = {{p}, {q}} × {{p}, {q}} whereas The ultimate approximation is the most precise ndao approximating the operator O: Proposition 1 Let a non-deterministic operator O over a lattice L, ≤ be given.Then O U is an ndao that approximates O. Furthermore, for any ndao O that approximates O and for every x, y ∈ L s.t.
x ≤ y, it holds that O(x, y) A i O U (x, y).In conclusion, non-deterministic AFT admits, just like deterministic AFT, an ultimate approximation.However, as we will see in the rest of this section, the ultimate non-deterministic approximation operator O U does not generalize the deterministic ultimate approximation operator defined by Denecker et al (2002).In more detail, we compare the non-deterministic ultimate operator IC U P with the deterministic ultimate IC DMT P from Definition 2. Somewhat surprisingly, even when looking at normal logic programs, the operator IC DMT d P does not coincide with the ultimate ndao IC U P (and thus IC DMT d P is not the most precise ndao, even for non-disjunctive programs).The intuitive reason is that the additional expressivity of non-deterministic operators, which are not restricted to single lower and upper bounds in their outputs, allows to more precisely capture what is derivable in the "input interval" (x, y).

Example 3 (Example 2 continued)
Consider again P = {q ← ¬p; p ← p}.Applying the DMT d -operator gives: IC DMT d P (∅, {p, q}) = (∅, {p, q}).Intuitively, the ultimate semantics IC U P (∅, {p, q}) = {{p}, {q}} × {{p}, {q}} gives us the extra information that we will always either derive p or q, which is information a deterministic approximator can simply not capture.Such a "choice" is not expressible within a single interval, hence the deterministic ultimate approximation is (∅, {p, q}).This example also illustrates the fact that, when applying the ultimate ndao-construction to (nonconstant) deterministic operators O, O U might be a non-deterministic approximation operator.The upper and lower immediate consequences operator are then straightforwardly defined, that is: by taking all interpretations that only contain atoms in HD DMT, † P (x, y) and contain at least one member of every head ∆ ∈ HD DMT, † P (x, y) (for † ∈ {u, l}): Finally, the DMT-ndao is defined as: A slightly extended program P = {q ← ¬q; p∨q ← q} shows some particular but unavoidable behavior of this operator.IC DMT,l P (∅, {q}) = {∅} as HD P (∅) = {{q}} and HD P ({q}) = {{p, q}}.Note that the lower bound for is not the stronger {p}.This would result in a loss of A i -monotonicity, as the lower bound {{q}} for the less informative (∅, {q}) would be S Lincomparable to the lower bound {{p}, {q}, {p, q}} of the more informative ({q}, {q}).
We have shown in this section that non-deterministic AFT admits an ultimate operator, thus providing a way to construct an ndao based on a non-deterministic operator.We have also shown that the ultimate ndao diverges from the ultimate operator for deterministic AFT, but that this deterministic ultimate operator can be generalized to disjunctive logic programs.Both operators will be used in Section 5 to define semantics for DLP's with aggregates.

Semi-Equilibrium Semantics
To further extend the reach of non-deterministic AFT, we generalize yet another semantics for dlp's, namely the semi-equilibrium semantics (Amendola et al. 2016).The semi-equilibrium semantics is a semantics for disjunctive logic programs that has been studied for disjunctively normal logic programs.This semantics is a three-valued semantics that fulfills the following properties deemed desirable by Amendola et al. (2016): (1) Every (total) answer set of P corresponds to a semi-equilibrium model; (2) If P has a (total) answer set, then all of its semiequilibrium models are (total) answer sets; (3) If P has a classical model, then P has a semiequilibrium model.We notice that these conditions can be seen as a view on approximation of the total stable interpretations alternative to the well-founded semantics.We do not aim to have the last word on which semantics is the most intuitive or desirable.Instead, we will show here that semi-equilibrium models can be represented algebraically, and thus can be captured within approximation fixpoint theory.This leaves the choice of exact semantics to the user once an ndao has been defined, and allows the use of the semi-equilibrium semantics for formalisms other than nlps, such as disjunctive logic programs with aggregates (see below) or conditional ADFs.
Semi-equilibrium models are based on the logic of here-and-there (Pearce 2006).An HTinterpretation is a pair (x, y) where x ⊆ y (i.e. a consistent pair in AFT-terminology).Satisfaction of a formula φ, denoted |= HT , is defined recursively as follows: Semi-equilibrium models are a special class of HT-models.They are obtained by performing two minimization steps on the set of HT-models of a program.The first step is obtained by minimizing w.r.t.≤ t .11The second step is obtained by selecting the maximal canonical models.For this, the gap of an interpretation is defined as gap(x, y) = y \ x,12 and, for any set of interpretations X, the maximally canonical interpretations are mc(X) = {(x, y) ∈ X | ∃(w, z) ∈ X : gap(x, y) ⊃ gap(w, z)}.The semi-equilibrium models of P are then defined as: SEQ(P) = mc (min ≤t (HT(P)).
Before we capture the ideas behind this semantics algebraically, we look a bit deeper into the relationship between HT(P)-models and the classical notion of three-valued models of a program (see Section 2.1).We first observe that HT-models of a program are a proper superset of the three-valued models of a program: Proposition 3 Let a disjunctively normal logic program P and a consistent intepretation (x, y) be given.Then if (x, y) is a model of P, it is an HT-model of P.However, not every HT-model is a model of P.
We now define the concept of a HT-pair algebraically, inspired by Truszczyński ( 2006 x.This simple definition faithfully transposes the ideas behind HT-models to an algebraic context.Indeed, applying it to IC P gives use exactly the HT-models of P: Proposition 4 Let some normal disjunctive logic program P be given.Then: HT(P) = HT(IC P ).
We now show that exact ≤ t -minimal HT-models of O are stable interpretations of O in our algebraic setting.The opposite direction holds as well: total stable fixpoints are ≤ t -minimal HTpairs of O.In fact, every total fixpoint of O is a HT-pair of O.We assume that O is upwards coherent, i.e. for every x, y ∈ L, O l (x, y) S L O u (x, y).In the appendix, we provide more details on upwards coherent operators.Notice that all ndaos in this paper are upwards coherent.

Proposition 5
Given an upwards coherent ndao O, (1) if (x, x) ∈ O(x, x) then (x, x) ∈ HT(O); and (2) The second concept that we have to generalize to an algebraic setting is that of maximal canonical models.Recall that gap(x, y) consists of the atoms which are neither true nor false, i.e. it can be used as a measure of the informativeness or precision of a pair.For the algebraic generalization of this idea, it is useful to assume that the lattice under consideration admits a difference for every pair of elements. 13In more detail, z ∈ L is the difference of y w.r.t.x if z ⊓ x = ⊥ and x ⊔ y = x ⊔ z.If the difference is unique we denote it by x ⊘ y.As an example, note that any Boolean lattice admits a unique difference for every pair of elements.We can then define mc(X) = argmin (x,y)∈X {y ⊘ x}.This allows us to algebraically formulate the semi-equilibrium models of an ndao O as The properties mentioned at the start of this section are preserved, and this definition generalizes the semi-equilibrium models for disjunctive logic programs by Amendola et al. (2016): Proposition 6 Let an upwards coherent ndao O over a finite lattice be given s.t.every pair of elements admits a unique difference.Then SEQ(O) = ∅.Furthermore, if there is some Corollary 1 Let a disjunctively normal logic program P be given.Then SEQ(IC P ) = SEQ(P).
In this section, we have shown that semi-equilibrium models can be characterized algebraically.This means semi-equilibrium models can now be obtained for other ndao's (e.g.those from Section 5, as illustrated in Appendix C), thus greatly enlarging the reach of these semantics.
We end this section by making a short, informal comparison between the semi-equilibrium models and the well-founded state for ndaos (Heyninck et al. 2022).Both constructions have a similar goal: namely, approximate the (potentially non-existent) total stable interpretations.In the case of the semi-equilibrium models, the set of semi-equilibrium models coincides with the total stable interpretations if they exist, whereas the well-founded state approximates any stable interpretation (and thus in particular the total stable interpretations), but might not coincide with them.When it comes to existence, we have shown here that the semi-equilibrium models exist for any ndao, just like the well-founded state.Thus, the well-founded state and semi-equilibrium models seem to formalize two different notions of approximation.Which notion is most suitable is hard to decide in abstracto but will depend on the exact application context.

Application to DLPs with Aggregates
We apply non-deterministic AFT to disjunctive logic programs with aggregates by studying three ndaos: the ultimate, DMT and the trivial operators.We show the latter two generalize the ultimate semantics (Pelov et al. 2007) respectively the semantics by Gelfond and Zhang (2019).

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 [t : Conj ] with t a list of constants and 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.#Sum, #Count or #Max).An aggregate atom is an expression of the form f (S) * w where f (S) is an aggregate function, * ∈ {<, ≤, ≥, >, =} and w is a numerical constant.We denote by At(f (S) * w) the atoms occuring in S.
A disjunctively normal aggregate program consists of rules of the form (where ∆ is a set of propositional atoms, and α 1 , . . ., α n , β 1 , . . ., β m are aggregate or propositional atoms): An aggregate symbol is evaluated w.r.t. a set of atoms as follows.First, let x(S) denote the multiset [t 1 | t 1 , . . ., t n : Conj ∈ S and Conj 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)) = where is a fixed symbol not occuring in P.An aggregate atom f (S) * w is true w.r.t.x (in symbols, x(f * w) = T) if: (1) x(f (S)) = and (2) x(f (S)) * w holds; otherwise, f (S) * w is false (in symbols, x(f * w) = F).¬f (S) * w is true if: (1) x(f (S)) = and (2) x(f (S)) * w does not hold; otherwise, ¬f (S) * 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 HD P to take into account aggregate formulas as described above: HD P (x) = {∆ | ∆ ← φ ∈ P, x(φ) = T}.IC P from Definition 1 is then generalized straightforwardly by simply using the generalized HD P .Thus, the only difference with the immediate consequence operator for dlp's is that the set of activated heads HD P now takes into account the truth of aggregates as well.
The first semantics we consider is the one formulated by Gelfond and Zhang (2019) (defined there only for logic programs with aggregates occurring positively in the body of a rule): Definition 4 Let a disjunctively normal aggregate logic program P s.t. for every ∆ ← n i=1 α i ∧ m j=1 ¬β j ∈ P, β j is a normal (i.e.non-aggregate) atom.Then the GZ-reduct of P w.r.t.x is defined by doing, for every r = ∆ ← n i=1 α i ∧ m j=1 ¬β j ∈ P, the following: (1) if an aggregate atom α i is false or undefined for some i = 1, . . ., n, delete r; (2) otherwise, replace every aggregate atom α i = f (S) * w by {Conj occurs in S | x(Conj) = T}.We denote the GZ-reduct of P by P x GZ .Notice that this is a disjunctively normal logic program.A set of atoms x ⊆ A P is a GZ-answer set of P if (x, x) is an answer set of P x GZ .
We now move to the semantics by Denecker et al. (2002).They are defined only for nondisjunctive aggregate programs.They are defined on the basis of the ultimate (deterministic) approximator IC DMT P (Definition 2).In more detail, an interpretation (x, y) is DMT d -stable if and only if (x, y) ∈ S(IC We first explain why {p} is not a GZ-stable model.First, we construct P {p} GZ = {p ← p}.Since {p} is not a stable model of P {p} GZ , we see that {p} is not a GZ-stable model.Likewise, since P ∅ GZ = {p ← ∅}, we see that ∅ is not a stable model of P ∅ GZ and therefore not GZ-stable.To see {p} is a DMT d -stable model, observe that IC

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. 2007), which maps two-valued interpretations to their immediate consequences whereas three-valued interpretations are mapped to the least precise pair (∅, A P ) (or, in the non-deterministic case, {∅} × {A P }).We also study the ndao IC DMT P based on the deterministic ultimate approximation, and the ultimate ndao IC U P .
We now show that these operators are approximation operators with increasing orders of precision: IC GZ P is the least precise, IC DMT P holds a middle ground and IC U is the most precise: Proposition 7 Let some ξ ∈ {DMT, GZ, U} and a disjunctively normal aggregate logic program P be given.Then IC ξ P (x, y) is an ndao approximating IC P .For any (x, y), IC GZ P (x, y) A i IC DMT P (x, y) A i IC U P (x, y).
The following properties follow from the general properties shown by Heyninck et al. (2022): Proposition 8 Let some ξ ∈ {DMT, GZ, U} and a disjunctively normal aggregate logic program P be given.Then: (1) S(IC ǫ P )(x, y) exists for any x, y ⊆ A P , and (2) every stable fixpoint of IC ǫ P is a ≤ t -minimal fixpoint of IC ǫ P .
The ndao IC GZ P only admits two-valued stable fixpoints, and these two-valued stable fixpoints generalize the GZ-semantics (Gelfond and  We finally show that stable semantics based on IC DMT P generalize those for non-disjunctive logic programs with aggregates by Denecker et al. (2002).
Proposition 10 Let a non-disjunctive logic program P be given.Then (x, y) is a stable model according to Denecker et al. (2002) iff (x, y) ∈ S(IC DMT P )(x, y).
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. 2023) 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. 2022) and semi-equilibrium semantics (Section 4, as illustrated in Appendix C) 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 dlp's with aggregates.Other ndao's, 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. (2021) to characterise the semantics of Marek and Remmel (2004) or the bounded ultimate operator introduced by Pelov and Truszczynski (2004)).

Conclusion, in view of related work
In this paper, we have made three contributions to the theory of non-deterministic AFT: (1) definition of the ultimate operator, (2) an algebraic generalization of the semi-equilibrium semantics and (3) an application of non-deterministic AFT to DLPs with aggregates in the body.To the best of our knowledge, there are only a few other semantics that allow for disjunctive rules with aggregates.Among the best-studied is the semantics by Faber et al. (2004) (so-called FLP-semantics).
As the semantics we propose generalize the operator-based semantics for aggregate programs without disjunction, the differences between the FLP-semantics and the semantics proposed here essentially generalize from the non-disjunctive case (see e.g.(Alviano et al. 2023)).
Among the avenues for future work, an in-depth analysis of the computational complexity of the semantics proposed in this paper seems to be among the most pressing of questions.Other avenues of future work include the generalisation of the constructions in Section 5 to other semantics (Vanbesien et al. 2021;Alviano et al. 2023) and defining ndaos for rules with choice constructs in the head (Marek et al. 2008), which can be seen as aggregates in the head.

Proof of Proposition 1
It is immediate to see that O U is exact and approximates O (as x≤z≤x O(z) = O(x)).We now show it is A i -monotonic.For this, consider some (x 1 , y 1 ) ≤ Proof of Proposition 3 suppose (x, y) is a model of P and consider some Θ 1 ∪ Θ 2 → ∆ ∈ P (where Θ 1 consists of atoms and Θ 2 consists of negated atoms).We first show that (y, y) To see this, suppose that (y, y)( To see that the inclusion can be proper, consider P = {b ← ¬c}.Then (∅, {c}) ∈ HT(P) as {c} does not classically entail ¬c.However, (∅, {c})(¬c) = U whereas (∅, {c})(b) = F and thus (∅, {c}) is not a model of P.

Proof of Proposition 4
Proposition 4 follows from the following Lemma: y) is consistent) and thus ∆ ∈ HD l P (x, y), which means, since IC l P (x, y) S L (x, y), there is some (w, z) ∈ IC l P (x, y) s.t.∆ ∩ w = ∅ and w ⊆ x, i.e. (x, y) |= HT ∆.The ⇒-direction is analogous.

Proof of Proposition 5
Ad (1).We will make use of the following Lemma: Lemma 12 (Heyninck et al. (2022)) We now proceed to the proof of the proposition.We first show that min ≤t (HT(O)) ⊆ S(O)(x, x).Suppose (x, x) ∈ min ≤t (HT(O)).
We first show that x ∈ C(O l )(x).As O l (x, x) S t x, i.e. x is a pre-fixpoint of O l (., x).Suppose towards a contradiction that there is some z < x s.t.O l (z, x) S t z.Then (z, x) ∈ HT(O), contradicting (x, x) ∈ min ≤t (HT(O)).Thus, x is a ≤-minimal prefixpoint, and thus, with Lemma 12, x is a ≤-minimal fixpoint of O l (., x), which implies x ∈ C(O l )(x).
We now show that x ∈ C(O u )(x).As O l (x, x) = O u (x, x) (since an ndao O is exact), x ∈ O u (x, x).Suppose now towards a contradiction there is some y < x s.t.y ∈ O u (x, y).As O is upwards coherent, O l (x, y) S L O u (x, y) which implies that there is some y ′ ∈ O u (x, y), i.e.O u (x, y) S L {y ′ } S L {y}, i.e. y, which contradicts (x, x) ∈ min ≤t (HT(O)).We now show that min ≤t (HT(O)) ⊇ S(O)(x, x).Suppose (x, x) ∈ S(O)(x, x) and suppose towards a contradiction there is some (w, z) ∈ HT(O) s.t.w ≤ x and z ≤ x, and either z = x or w = x.Suppose first the latter.Then O l (w, z) S L w and thus w is a pre-fixpoint of O l (w, z).Since z ≤ x, O(w, x) S L O(w, z) and thus O l (w, x) S L w, i.e. w is a pre-fixpoint of O l (., x).But this contradicts x being a minimal fixpoint of O l (., x) (with Lemma 12).Suppose now z ≤ x.In view of the previous case, we can assume w = x.But then (w, z) ∈ HT(O) contradicts z ≤ x = w, contradiction.

Proof of Proposition 6
We first recall some background from Heyninck et al. (2022).
Finally, for a complete lattice L, let O : L 2 → L 2 be an approximating operator.We denote: O l (•, y) = λx.O l (x, y) and similarly for O u .The stable operator for O is then defined as S(O)(x, y) = (lfp(O l (., y)), lfp(O u (x, .)),where lfp(O) denotes the least fixpoint of an operator O.
However, one can still generalize the operator IC DMT d P to disjunctive logic programs.We first generalize the idea behind IC DMT d ,l P to an operator gathering the heads of rules that are true in every interpretation z in the interval [x, y].Similarly, IC DMT d ,u P is generalized by gathering the heads of rules with bodies that are true in at least one interpretation in [x, y]: HD DMT,l P (x, y) = x⊆z⊆y HD P (z) and HD DMT,u P (x, y) = x⊆z⊆y HD P (z)}.
Zhang 2019): If (x, y) ∈ min ≤t (IC GZ P (x, y)) then x = y.Let a disjunctively normal aggregate aggregate logic program P s.t. for every ∆ ← ∈ P, β i is a normal atom be given.(x, x) ∈ S(IC GZ P )(x, x) iff x is a GZ-answer set of P.
y 2 ).The case for the upper bound is entirely identical.We now show that any ndao O that approximates O be given and for every x, y ∈ L s.t.x ≤ y, O(x, y) A i O U (x, y).Indeed, consider an operator O that approximates O and somex, y ∈ L s.t.x ≤ y.Consider some w ∈ O U l (x, y), i.e. w ∈ O(z) for some z ∈ [x, y].Since x ≤ z ≤ y, (x, y) ≤ i (z, z) and thus O(x, y) A i O(z, z) (with A i -monotonicity) which means O l (x, y) S L O l (z, z) = O(z) (the latter equality since O approximates O) and thus, in particular, O l (x, y) S L {w}.Likewise, to show that O U l (x, y) H L O u (x, y), it suffices to observe that, since O(x, y)

Table D 2
: List of the preorders used in this paper.Table D 3: List of the operators used in this paper.