2.1 Answer Set Programming

2.2 ASP with two quantifiers

A 2-ASP $^w$ (Q) program (Mazzotta et al. Reference Mazzotta, Ricca and Truszczynski2024) is an expression of the form $\Box _1 P_1 \Box _2 P_2: C : C^\omega$ , where $\Box _1,\Box _2$ are quantifiers in $\{\exists ^{st}, \forall ^{st}\}$ , $P_1,P_2$ are ASP programs possibly with weak constraints, $C$ is a stratified program (Ceri et al. Reference Ceri, Gottlob and Tanca1990) with hard constraints, and $C^\omega$ is a set of weak constraints such that $B_{C^{\omega }}\subseteq B_{P_1}$ . Weak constraints appearing $P_1$ and $P_2$ are said local; whereas those in $C^{\omega }$ are said global. A 2-ASP $^w$ (Q) program $\Pi$ is said to be existential if $\Box _1 = \exists ^{st}$ , otherwise it is universal. Moreover, $\Pi$ is said to be alternating if $\Box _1\neq \Box _2$ , and plain if it contains no weak constraints. We now define the semantics of 2-ASP $^w$ (Q) programs. Let $P$ be an ASP program and $M \subseteq B_P$ , then $\mathit{fix_{P}(M)}$ denotes the set of facts and hard constraints of the form $\{a\leftarrow \mid a \in M\} \cup \{\leftarrow a\mid a \in B_P\setminus M\}$ . Then, $M \in AS(P)$ satisfies a program $P'$ if $P'\cup \mathit{fix_{P}(M)}$ is coherent.

At this point the coherence of 2-ASP $^w$ (Q) can be defined as follows:

• • $\exists ^{st} P: C :C^{\omega }$ is coherent iff there exists $M \in OptAS(P)$ such that $M$ satisfies $C$ .

• • $\forall ^{st} P: C :C^{\omega }$ is coherent iff for each $M \in OptAS(P)$ , $M$ satisfies $C$ .

• • $\exists ^{st} P_1 \Box _2 P_2: C :C^{\omega }$ is coherent iff there exists $M_1 \in OptAS(P_1)$ such that $\Box _2 P_2 \cup \mathit{fix_{P_1}(M_1)}: C :C^{\omega }$ is coherent.

• • $\forall ^{st} P_1 \Box _2 P_2: C :C^{\omega }$ is coherent iff for each $M_1 \in OptAS(P_1)$ , $\Box _2 P_2 \cup \mathit{fix_{P_1}(M_1)}: C :C^{\omega }$ is coherent.

For an existential 2-ASP $^w$ (Q) program $\Pi$ , an optimal answer set $M_1 \in OptAS(P_1)$ is a quantified answer set of $\Pi$ if $\Box _2 P_2\cup \mathit{fix_{P_1}(M_1)}:C:C^{\omega }$ is coherent. We denote by $QAS(\Pi )$ the set of quantified answer sets of $\Pi$ . Let $\Pi$ be an existential 2-ASP $^w$ (Q) program, $l$ be an integer, and $M \in QAS(\Pi )$ , then the cost of $M$ at level $l$ is defined as $\mathcal{C}(M,\Pi ,l) = \mathcal{C}(M,P_1\cup C^{\omega },l)$ . Let $M_1,M_2 \in QAS(\Pi )$ , then $M_1$ is dominated by $M_2$ if there exists $l$ such that $\mathcal{C}(M_1,\Pi ,l) \gt \mathcal{C}(M_2,\Pi ,l)$ and for each $l'\gt l$ , $\mathcal{C}(M_1,\Pi ,l') = \mathcal{C}(M_2,\Pi ,l')$ . Thus, $M$ is an optimal quantified answer set if $M$ is not dominated by any $M'\in QAS(\Pi )$ . We denote by $OptQAS(\Pi )$ the set of optimal quantified answer sets.

Example 1. Let $\Pi$ be a 2-ASP $^w$ (Q) program of the form $\exists ^{st} P_1\forall ^{st}P_2:C$ , where $C=\{\leftarrow nb,\ nc\}$ and

\begin{equation*} P_1 = \left \{\begin{array}{lr} a \leftarrow {\scriptstyle \mathtt {\sim }} na\qquad\qquad & b \leftarrow {\scriptstyle \mathtt {\sim }} nb\\ na \leftarrow {\scriptstyle \mathtt {\sim }} a\qquad\qquad & nb \leftarrow {\scriptstyle \mathtt {\sim }} b\\ \end{array}\right \}\qquad P_2 = \left \{\begin{array}{ll} c \leftarrow {\scriptstyle \mathtt {\sim }} nc\qquad\qquad & \leftarrow ^{\omega } a,\ {\scriptstyle \mathtt {\sim }} c\ [1@1]\\ nc \leftarrow {\scriptstyle \mathtt {\sim }} c\qquad\qquad & \leftarrow ^{\omega } b,\ {\scriptstyle \mathtt {\sim }} nc\ [1@1]\\ \end{array}\right \} \end{equation*}

Let us consider $M_1 = \{na,nb\} \in AS(P_1)$ . In this case, $\{na,nb,nc\} \in OptAS(P_2\cup \mathit{fix_{P_1}(M_1)})$ violates the constraint $\leftarrow nb,nc \in C$ . Thus, $M_1\notin QAS(\Pi )$ . Conversely, $M_1' = \{a,nb\}\in AS(P_1)$ is such that $P_2\cup \mathit{fix_{P_1}(M_1')}$ admits only one optimal answer set $M_2 = \{a,nb,c\}$ which satisfies the constraint in $C$ and so $M_1\in QAS(\Pi )$ . If we remove weak constraints from $P_2$ , then $AS(P_2\cup \mathit{fix_{P_1}(M_1)}) = \{M_2, M_2'\}$ where $M_2' = \{a,nb,nc\}$ . Here, $M_2'$ violates the constraint in $C$ and so $M_1$ is not a quantified answer set anymore.

4.1 The 2-ASP $^w$ (Q) game

Let $\Box \in \{\exists ^{st},\forall ^{st}\}$ be a quantifier, then the opponent of $\Box$ is $\overline {\Box } = \forall ^{st}$ if $\Box = \exists ^{st}$ ; otherwise $\overline {\Box } = \exists ^{st}$ . Thus, for any alternating 2-ASP $^w$ (Q) program $\Box _2$ is the opponent of $\Box _1$ , that is, $\Box _2 = \overline {\Box _1}$ . In what follows, w.l.o.g., we consider 2-ASP $^w$ (Q) programs of the form:

(1) \begin{equation} \Box P_1\overline {\Box } P_2:C:C^{\omega } \end{equation}

Let $\Pi$ be a 2-ASP $^w$ (Q) program of the form (1), then a move for $\Box$ is an answer set $M_1 \in OptAS(P_1)$ . Given a move $M_1$ for $\Box$ , for each $M_2 \in OptAS(P_2 \cup \mathit{fix_{P_1}(M_1)})$ , $M_2|_{\mathcal{H}(P_2)}$ is a candidate countermove to $M_1$ for $\overline {\Box }$ . Let $M_2 \in OptAS(P_2\cup \mathit{fix_{P_1}(M_1)})$ be a candidate countermove then $M_2|_{\mathcal{H}(P_2)}$ is effectively a countermove to $M_1$ for $\overline {\Box }$ if: ( $i$ ) $\Box =\exists ^{st}$ and $M_2$ does not satisfies $C$ ; or ( $ii$ ) $\Box =\forall ^{st}$ and $M_2$ satisfies $C$ . A winning move for $\Box$ is a move $M_1$ such that no countermove to $M_1$ for $\overline {\Box }$ exists. Thus, $\Box = \exists ^{st}$ wins if there exists $M_1 \in OptAS(P_1)$ such that no countermove to $M_1$ for $\overline {\Box }$ exists.

Intuitively, the notion of winning moves closely corresponds to the definition of coherence of 2-ASP $^w$ (Q) programs. Let $\Pi$ be a 2-ASP $^w$ (Q) program of the form (1), and let $M_1$ be a winning move for $\Box$ . If $\Box = \exists ^{st}$ , then there is no $M_2 \in OptAS(P_2 \cup \mathit{fix_{P_1}(M_1)})$ such that $M_2$ violates $C$ . This implies that $\forall ^{st} P_2 \cup \mathit{fix_{P_1}(M_1)} : C$ is coherent, and hence $\Pi$ is coherent. Conversely, if $\Box = \forall ^{st}$ , then there is no $M_2 \in OptAS(P_2 \cup \mathit{fix_{P_1}(M_1)})$ such that $M_2$ satisfies $C$ . Therefore, $\exists ^{st} P_2 \cup \mathit{fix_{P_1}(M_1)} : C$ is incoherent, and thus $\Pi$ is incoherent.

4.2 Counterexample search in 2-ASP $^w$ (Q)

Given a 2-ASP $^w$ (Q) program $\Pi$ of the form (1), from Proposition1, the coherence of $\Pi$ can be decided by searching for a winning move for $\Box$ . Thus, the program $P_1$ , whose optimal answer sets coincide with the possible moves of $\Box$ , is a natural abstraction for $\Pi$ . Hence, given $M_1\in OptAS(P_1)$ , the counterexample search aims at contradicting $M_1$ , which means finding a countermove to $M_1$ for $\overline {\Box }$ . To this end, we define the countermove program whose optimal answer sets correspond to countermoves to $M_1$ for $\overline {\Box }$ .

We recall that countermoves correspond to optimal answer sets of $P_2\cup \mathit{fix_{P_1}(M_1)}$ that either satisfy or not the program $C$ according to $\overline {\Box }$ . Since $C$ is stratified with hard constraints, it admits one answer set (Dantsin et al. Reference Dantsin, Eiter, Gottlob and Voronkov2001) iff hard constraints are satisfied. Thus, the following transformation can be used to capture the incoherence of $C$ .

At this point, if $\Box = \exists ^{st}$ (resp. $\Box = \forall ^{st}$ ) one might be tempted to define the counterexample program as $P_2 \cup \neg C \cup \mathit{fix_{P_1}(M_1)}$ (resp. $P_2 \cup C \cup \mathit{fix_{P_1}(M_1)}$ ). However, an optimal answer set of such a counterexample program may correspond to a non-optimal answer set of $P_2\cup \mathit{fix_{P_1}(M_1)}$ that violates (resp. satisfies) $C$ , which does not correspond to a countermove for $\overline {\Box }$ to $M_1$ . To address this issue, we need to relax hard constraints from $C$ into weak constraints with a lower priority level w.r.t. weak constraints in $P_2$ .

Definition 2 (Relaxed program). Let $P$ be a stratified program with hard constraints, $l$ be an integer, and $unsat$ be a fresh atom not appearing anywhere else. Then relaxed $(P,l)$ is defined as:

\begin{equation*} relaxed(P,l) = \left \{\begin{array}{lllr} H_r & \leftarrow & B_r. & \forall r \in P s.t. H_r \ne \emptyset \\ unsat & \leftarrow & B_r. & \forall r \in P s.t. H_r = \emptyset \\ & \leftarrow ^w & unsat. [1@l] & \\ \end{array}\right \} \end{equation*}

Intuitively, $relaxed(\cdot , \cdot )$ is similar to $\neg C$ but the fresh atom introduced as head of the hard constraints of $P$ is used to assign a penalty if $P$ is incoherent. As a result, $relaxed(\cdot , \cdot )$ has the following property, which is fundamental in the computation of countermoves.

Thanks to the above property, the program $relaxed(\neg C,l)$ (resp. $relaxed(C,l)$ ) can be combined with $P_2$ in such a way that rules in $\neg C$ (resp. $C$ ) do not filter out any optimal answer set of $P_2\cup \mathit{fix_{P_1}(M_1)}$ . Thus, we are now ready to define the countermove program.

Note that, the above definition aligns with the countermove definition from Section 4.1. Specifically, when $\Box = \exists ^{st}$ , a countermove must violate $C$ , so the countermove program incorporates rules from $relaxed(\neg C, l_{\min}-1)$ . Conversely, when $\Box = \forall ^{st}$ , a countermove must satisfy $C$ , so the countermove program incorporates rules from $relaxed(C,l_{\min}-1)$ . Consequently, countermoves to a move for $\Box$ are given by the optimal answer sets of the counterexample program.

4.3 Refining abstractions in 2-ASP $^w$ (Q)

The next step in designing our CEGAR-based approach for 2-ASP $^w$ (Q) is to define a transformation for refining the abstraction according to known countermoves. Specifically, let $M_1$ be a move for $\Box$ and $M_2$ be a countermove for $\overline {\Box }$ , then the goal of the refinement, namely $Ref(\Pi , M_2)$ , is to obtain a set of rules such that optimal answer sets of $P_1 \cup Ref(\Pi , M_2)$ correspond to moves $M_1'$ for $\Box$ such that $M_2$ is not a countermove to $M_1'$ for $\overline {\Box }$ . Note that, $M_2$ is not a countermove to $M_1'$ for $\overline {\Box }$ if one of the these condition holds:

1. 1. there is no $M_2' \in AS(P_2\cup \mathit{fix_{P_1}(M_1')})$ such that $M_2'|_{\mathcal{H}(P_2)} = M_2$ ;

2. 2. there exists $M_2' \in AS(P_2\cup \mathit{fix_{P_1}(M_1')})$ such that $M_2'|_{\mathcal{H}(P_2)} = M_2$ but $M_2'\notin OptAS(P_2\cup \mathit{fix_{P_1}(M_1')})$ (i.e., $M_2'$ is dominated by some answer set of $P_2\cup \mathit{fix_{P_1}(M_1')}$ );

3. 3. there exists $M_2' \in OptAS(P_2\cup \mathit{fix_{P_1}(M_1')})$ such that $M_2'|_{\mathcal{H}(P_2)} = M_2$ but $M_2'$ satisfies $C$ and $\Box = \exists ^{st}$ (resp. violates $C$ and $\Box =\forall ^{st}$ ).

Thus, we define some transformations encoding these conditions to refine our abstraction.

First of all, we need a predicate substitution function that maps each predicate $p$ to a fresh one of the form $p^{\alpha }_\beta$ , where $\alpha$ and $\beta$ are strings, to obtain rules over a fresh signature for each discovered countermove. More in detail, let $L$ be a set of literals and $\epsilon$ be an ASP expression (i.e., rule, program, etc.), then $\sigma ^{\alpha }_\beta (L,\epsilon )$ denotes the expression obtained from $\epsilon$ by mapping each positive (resp. negative) literal $p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \in L$ (resp. ${\scriptstyle \mathtt {\sim }} p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \in L$ ) with $p^{\alpha }_\beta (\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)$ (resp. ${\scriptstyle \mathtt {\sim }} p^{\alpha }_\beta (\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)$ ). Now, we are ready to define the rules which verify Condition 1.

Definition 4 (Check Answer Set). Let $P$ be an ASP program and $M$ be an interpretation, then $checkAS(P,M)$ is:

\begin{equation*} checkAS(P, M) = \left \{\begin{array}{lr} \sigma _M^-(\mathcal{H}(P),\{a \leftarrow \vert a \in M\})&\\ \sigma _M^+(\mathcal{H}(P), \sigma _M^-(\overline {\mathcal{H}(P)},r)) & \forall r \in P, H(r)\neq \emptyset \\ fail_M \leftarrow \sigma _M^+(\mathcal{H}(P), \sigma _M^-(\overline {\mathcal{H}(P)},B(r))) & \forall r \in P, H(r)=\emptyset \\ fail_M \leftarrow p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)_M^+,{\scriptstyle \mathtt {\sim }} p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)_M^- & \forall \ p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \in \mathcal{H}(P) \\ fail_M \leftarrow p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)_M^-,{\scriptstyle \mathtt {\sim }} p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)_M^+ & \forall \ p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \in \mathcal{H}(P)\\ as_M \leftarrow {\scriptstyle \mathtt {\sim }} fail_M\ & \ \end{array}\right . \end{equation*}

Basically, $checkAS(P, M)$ encodes the GL-reduct of a program $P$ w.r.t. an interpretation $M$ and checks whether $M$ is a $\subset$ -minimal model of the reduct. Specifically, $M$ is encoded as facts over a fresh negative signature (i.e., each atom $p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \in M$ is represented by a fact $p^{-}_M(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt) \leftarrow$ ). Atoms over the negative signature are used to rewrite negative body literals, yielding the GL-reduct of $P$ w.r.t. $M$ ; whereas head atoms and positive body literals are mapped to a fresh positive signature (i.e., $p(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)$ mapped to $p^{+}_M(\kern-0.6pt\vec {\kern0.2pt t}\kern1.5pt)$ ). Finally, if the true atoms over positive and negative signatures coincide then $M$ is an answer set of $P$ .

We can now focus on Condition 2 which requires checking answer set optimality. To this end, we define a transformation to compute the cost of answer sets of an ASP program.

Definition 5 (Cost Program). Let $P$ be an ASP program, $L$ be the set of priority levels in $\mathcal{W}(P)$ , and $v_P$ and $cl_P$ are fresh predicates not appearing in $P$ . Then, $cost(P)$ is defined as:

\begin{equation*} cost(P) = \left \{\begin{array}{lclr} v_P(w,l,\kern-1.5pt\vec {\kern1pt t}\kern1.3pt) & \leftarrow & l_1, \ldots , l_n & \forall \leftarrow ^{\omega } l_1, \ldots , l_n [w@l,\kern-1.5pt\vec {\kern1pt t}\kern1.9pt] \in P\\ cl_P({T},l) &\leftarrow & \#sum\{C,\vec {t}: v_P(C,l,\kern-1.5pt\vec {\kern1pt t}\kern1.3pt)\}={T} & \forall l \in L\\ \end{array}\right . \end{equation*}

Intuitively, weak constraints of the form $\leftarrow ^{\omega } l_1, \ldots , l_n [w@l,\kern-1.5pt\vec {\kern1pt t}\kern1.9pt]$ can be rewritten into rules whose head is a fresh atom encoding violation tuples (i.e., $(w,l,\kern-1.5pt\vec {\kern1pt t}\kern1.3pt)$ ) and introduces a rule for each level $l$ that sums up the the weights of the violation tuples at level $l$ . To verify the optimality of an answer set $M \in AS(P)$ , it is necessary to compare its cost with that of every $M' \in AS(P)$ . To this end, we use $clone(P)=\sigma _{clone}(\mathcal{H}(P)\cup \overline {\mathcal{H}(P)}, P)$ , which clones the program $P$ , enabling the comparison between the cost of $M$ and the cost of all answer sets of the cloned program.

Definition 6 (Dominated program). Let $P_1$ and $P_2$ be two ASP programs, then $checkDom(P_1,P_2)$ is the following program:

\begin{equation*} \begin{array}{l@{\quad}c@{\quad}lr} diff_{P_1,P_2}(L) & \leftarrow & cl_{P_1}(C1,L), cl_{P_2}(C2,L), C1 \neq C2. & \\ hasHigher_{P_1,P_2}(L) & \leftarrow & diff_{P_1,P_2}(L), diff_{P_1,P_2}(L1), L\lt L1.& \\ highest_{P_1,P_2}(L) & \leftarrow & diff_{P_1,P_2}(L), {\scriptstyle \mathtt {\sim }} hasHigher_{P_1,P_2}(L).& \\ dom_{P_1,P_2} & \leftarrow & highest_{P_1,P_2}(L), cl_{P_1}(C1,L), cl_{P_2}(C2,L), C2 \lt C1 & \\ \end{array} \end{equation*}

where $cl_{P_1}$ and $cl_{P_2}$ are, respectively, the predicates introduced by $cost(P_1)$ and $cost(P_2)$ ; whereas $diff_{P_1,P_2}$ , $hasHigher_{P_1,P_2}$ , $highest_{P_1,P_2}$ , and $dom_{P_1,P_2}$ are fresh predicates.

Intuitively, let $M \in AS(P)$ and $M_{c}\in AS(clone(P))$ , then $checkDom(P, clone(P))$ can be used to compare the cost of $M$ and $M_{c}$ . More precisely, $cost(P)$ and $cost(clone(P))$ compute, respectively, the cost of $M$ and $M_c$ , for each level $l$ , as atoms of the form $cl_P(C,l)$ and $cl_{clone(P)}(C,l)$ . Thus, rules from $dom(P,clone(P))$ derive $dom_{P,clone(P)}$ if and only if $M$ is dominated by $M_{c}$ , which means $M$ is not an optimal answer set of $P$ .

Finally, we can focus on Condition 3. Recall that Condition 3 requires to verify the coherence (resp. incoherence) of the program $C$ . To this end, it is possible to leverage $relaxed(C,l)$ (resp. $relaxed(\neg C,l)$ ), as we have seen for counterexample program.

Now that, we have transformations for each condition (i.e., Conditions 1, 2, and 3), we need to control the activation of the different transformations in the refinement process.

Intuitively, $or(P,l)$ controls the activation of $P$ according to the literal $l$ . If $l$ is true, it can be removed from rules in $or(P,l)$ , obtaining back the program $P$ ; otherwise rules in $or(P,l)$ are satisfied as $l$ falsifies rules’ body. We can now define the refinement program.

Definition 8 (Refinement Program). Let $\Pi$ be a 2-ASP $^w$ (Q) program of the form (1), $M$ be a move for $\Box$ and let $CE$ be a countermove to $M$ for $\overline {\Box }$ . The refinement program is defined as follows:

\begin{equation*} ref(\Pi , CE) = \left \{\begin{array}{lr} checkAS(P_2,CE) & \\ \leftarrow ^w as_{CE}\ [1@l_{\min}-1] & \\ or(cost(P_2^{CE}),as_{CE}) & \\ or(clone(\mathcal{R}(P_2)),as_{CE}) & \\ or(cost(clone(P_2)),as_{CE}) & \\ or(checkDom(P_2^{CE}, clone(P_2)),as_{CE}) & \\ \leftarrow ^w as_{CE}, {\scriptstyle \mathtt {\sim }} dom_{P_2^{CE},clone(P_2)}\ [1@l_{\min}-2] & \\ or(\sigma _{CE}^-(lits(P_2)\cup lits(C'), C'),as_{CE}) & \\ \end{array}\right \} \end{equation*}

where $P_2^{CE} = \sigma _{CE}^-(lits(P_2),P_2)$ , $lits(P_2) = \mathcal{H}(P_2)\cup \overline {\mathcal{H}(P_2)}$ , $as_{CE}$ is the predicate introduced by $checkAS(P_2, CE)$ , $C' = relaxed(C,l_{\min}-3)$ if $\Box =\exists ^{st}$ ; otherwise $C' = relaxed(\neg C,l_{\min}-3)$ , $lits(C') = \mathcal{H}(C')\cup \overline {\mathcal{H}(C')}$ , and $l_{\min}$ is the smallest priority level among weak constraints of $P_1$ .

Let $\Pi$ be a 2-ASP $^w$ (Q) of the form (1), $M$ be a move for $\Box$ and $CE$ be a countermove to $M$ for $\overline {\Box }$ , then we aim at computing a new move $M'$ for $\Box$ such that $CE$ is not a countermove to $M'$ for $\overline {\Box }$ . To this end, we use $ref(\Pi ,CE)$ to check Conditions 1–3. More precisely, rules in $checkAS(P_2,CE)$ encode the reduct of $P_2$ w.r.t. $CE$ and derive the atom $as_{CE}$ iff there exists $M_2 \in AS(P_2\cup \mathit{fix_{P_1}(M')})$ such that $M_2|_{\mathcal{H}(P_2)} = CE$ (i.e., Condition 1 is not satisfied). Thus, the weak constraint $\leftarrow ^{\omega } as_{CE}\ [1@l_{\min}-1]$ expresses the preference to satisfy Condition 1. If Condition 1 is not satisfied, then there exists $M_2 \in AS(P_2\cup \mathit{fix_{P_1}(M')})$ such that $M_2|_{\mathcal{H}(P_2)} = CE$ and so $as_{CE}$ is derived as true. At this point, $as_{CE}$ activates the following blocks which check Conditions 2 and 3. To check Condition 2, $or(cost(P_2^{CE}),as_{CE})$ computes the cost of $M_2$ w.r.t. weak constraints in $P_2$ , $or(clone(\mathcal{R}(P_2)),as_{CE})$ clones the program $P_2$ to compare pair of answer set of $P_2\cup \mathit{fix_{P_1}(M')}$ , $or(cost(clone(P_2)),as_{CE})$ computes the cost of an answer set of the cloned program, and finally $or(checkDom(P_2^{CE}, clone(P_2)),as_{CE})$ derive the atom $dom_{P_2,clone(P_2)}$ iff $M_2$ is not optimal as $M_2$ is dominated by some answer set of the cloned program. As a result, the weak constraint $\leftarrow ^w as_{CE}, {\scriptstyle \mathtt {\sim }} dom_{P_2^{CE},clone(P_2)}\ [1@l_{\min}-2]$ prefers answer sets of the cloned program that dominate $M_2$ (i.e., $M_2$ is not optimal). If $M_2$ cannot be dominated by any answer set of the cloned program then $M_2$ is optimal and so $or(\sigma _{CE}^-(lits(P_2)\cup lits(C'), C'),as_{CE})$ checks Condition 3. In particular, if $\Box = \exists ^{st}$ (resp. $\Box = \forall ^{st}$ ), then the atom $unsat$ is derived iff $M_2$ does not satisfy $C$ (resp. $\neg C$ ), which means Condition 3 is not satisfied. Thus, the weak constraint $\leftarrow ^{\omega } unsat, as_{CE}\ [1@l_{\min}-3]$ in $C'$ models the preference of satisfying $C$ (resp. $\neg C$ ) (i.e., Condition 3 is satisfied). Thus, if there exists an optimal answer set of $P_1\cup ref(\Pi ,CE)$ that satisfies at least one weak constraint in $ref(\Pi ,CE)$ then this corresponds to a move $M'$ that does not admit $CE$ as countermove. A detailed example is provided in the online appendix.

5.1. Quantified answer set computation

The computation of quantified answer sets in presence of weak constraints can be obtained by plugging the transformations defined in previous section in the CEGAR for 2-ASP(Q) algorithm by Cuteri et al. (Reference Cuteri, Mazzotta and Ricca2026). In particular, we update both counterexample search and refinement procedure with those introduced in Section 4.2 (i.e., Definitions3 and 8) and update the conditions of existence of a winning move accordingly. Indeed, in the original algorithm it was sufficient to check for existence of an answer set of the refined or counterexample programs, whereas here we have to look at the cost of their optimal answer sets. More in detail, at the least three levels the cost 1 if no winning exists; whereas for the counterexample program at the last level the cost is 1 if the current move is winning (full algorithm in the online appendix).

5.2. Upper-bound improving

Optimal quantified answer sets can be computed starting from a quantified answer set and iteratively searching for better ones until the incoherence is met (Alviano et al. Reference Alviano, Dodaro, Marques-Silva and Ricca2020). A quantified answer set $M$ of the input program $\Pi$ is computed applying CEGAR for 2-ASP $^w$ (Q), and the cost of $M$ is our initial upper bound. Note that if $M$ does not exist, no optimal quantified answer set exists and the search stops immediately. Next, to improve on this bound a constraint is added to $\Pi$ to enforce a preference for answer sets of $P_1$ (i.e., candidate quantified answer sets) with a lower cost, and the solver is called again. The process repeats until the upper bound cannot be improved anymore, and the last quantified answer set is optimal.

5.3. Lower-bound improving

The core idea of these strategies is to fix an initial lower-bound cost and, if no answer set within this bound exists, iteratively relax the program to raise the bound until an optimal solution is obtained. Traditional ASP solvers start by treating all weak constraints as hard, aiming for a zero-cost lower bound. If no answer set exists, unsatisfiable cores guide the progressive relaxation of the program, incrementally raising the lower bound (roughly admitting some weak constraint must be violated) until an optimal solution is found (Alviano et al. Reference Alviano, Dodaro, Marques-Silva and Ricca2020). Unluckily, this strategy cannot be ported as it is in our setting, since a notion of unsatisfiable core has never been defined for ASP $^w$ (Q). Thus, we propose alternative ways for targeting a lower bound and also for relaxing the program so that the lower bound improves iteratively until the optimum is found. Intuitively, we aim to compute an answer set of $P_1$ that is optimal with respect to the global weak constraints. This is achieved by adding the global weak constraints $C^{\omega }$ to $P_1$ with the lowest priority and searching for the optimum answer sets of the resulting program $P_1^{lower}$ . Note that, an optimum answer set of $P_1^{lower}$ is a reasonable lower-bound candidate, since it is either an optimum for $\Pi$ or it does not satisfy the subsequent quantifiers. In the latter case, following the CEGAR approach, an oracle call is used to compute a countermove, and the corresponding refinement is added to $P_1^{lower}$ . This enables the next iteration to search for a new candidate optimal answer set. The procedure repeats until a winning move is found, incrementally improving the lower bound by discarding candidates that are not quantified answer sets of $\Pi$ . To ensure the correctness of the approach the global weak constraints $W$ are moved at the lowest priority levels w.r.t. both local weak constraints in $P_1$ and weak constraints that would be added to $P_1^{lower}$ by the refinement procedure. More precisely, the initial abstraction is defined as $P_1 \cup W$ , where $W = \{\leftarrow ^\omega l_1,\ldots ,l_k\ [w@\lambda +l,\kern-1.5pt\vec {\kern1pt t}\kern1.9pt] \mid \leftarrow ^\omega l_1,\ldots ,l_k\ [w@l,\kern-1.5pt\vec {\kern1pt t}\kern1.9pt] \in C^{\omega }\}$ , with $\lambda$ being an integer such that levels from $C^{\omega }$ are remapped to strictly lower priority levels w.r.t. $l_{\min}\!-\!3$ (i.e., the smallest priority level added by the refinement), and $l_{\min}$ being the lowest priority level in $P_1$ . In this way, the first winning move will be a quantified answer set which is also optimal. Note that the proposed ordering of priority levels in the (refined) abstraction is essential: it first favors moves that admit no known countermove and only then prefers moves that are optimal with respect to $C^{\omega }$ . Under this ordering, the first winning move obtained is an optimal quantified answer set.

6.1. Implementation

The proposed approach was implemented on top of the CASPER (Cuteri et al. Reference Cuteri, Mazzotta and Ricca2026). More in detail, CASPER is written in Python and uses CLINGO (Gebser et al. Reference Gebser, Kaminski, Kaufmann, Ostrowski, Schaub and Wanko2016) as an oracle for computing moves and countermoves. For non-alternating 2-ASP $^w$ (Q) programs, our implementation applies the transformations proposed by Mazzotta et al. (Reference Mazzotta, Ricca and Truszczynski2024) for removing weak constraints, as the 2-ASP $^w$ (Q) game is defined for alternating programs. Specifically, non-alternating 2-ASP $^w$ (Q) programs are translated into 2-ASP(Q) program with two alternating quantifiers and no local weak constraints.

6.2. Benchmarks

We considered several benchmarks from diverse ASP(Q) applications (Faber et al. Reference Faber, Mazzotta and Ricca2023; Azzolini et al. Reference Azzolini, Mazzotta, Ricca and Riguzzi2025, Reference Azzolini, Leone, Mazzotta, Ricca, Amin and Arias2026): Propositional Abduction Problem (PAP) (Eiter and Gottlob Reference Eiter and Gottlob1995), Minmax Clique (MMC) (Cao et al. Reference Cao, Du, Gao, Wan and Pardalos1995), Max Term Deletion (MTD) (Schaefer and Umans Reference Schaefer and Umans2002), Most Probable Explanation in Probabilistic ASP (MPE) (Azzolini et al. Reference Azzolini, Mazzotta, Ricca and Riguzzi2025), and Clique Coloring (CC) (Schaefer and Umans Reference Schaefer and Umans2002). For PAP, we consider three reasoning tasks: PAP-OPT, PAP-REL, and PAP-NEC, corresponding to computing cardinality-minimal solutions, and relevant and necessary hypotheses, respectively. For MMC we considered both the decision and optimization variant of the problem, namely MMC-BOUND and MMC-OPT. For MTD, we considered the optimization version. Finally, for MPE we considered both COLORING and SMOKERS domains (Azzolini et al. Reference Azzolini, Mazzotta, Ricca and Riguzzi2025). For the CC benchmark, we considered graphs of varying size (from 10 to 120 nodes) and edge density (25%, 50%, and 75%), generated according to the Erdős–Rényi model provided by NetworkX library (https://pypi.org/project/networkx). For each combination of size and density, we generated 10 instances. For the other benchmarks, instances were drawn from previous experiments (Azzolini et al. Reference Azzolini, Mazzotta, Ricca and Riguzzi2025, Reference Azzolini, Leone, Mazzotta, Ricca, Amin and Arias2025; Cuteri et al. Reference Cuteri, Mazzotta and Ricca2026).

6.3. Compared methods

In our evaluation we compared the approach by Azzolini et al. (Reference Azzolini, Leone, Mazzotta, Ricca, Amin and Arias2026) implemented in PYQASP. Note that, PYQASP implements the upper-bound improving algorithm on top of a rewriting in QBF and it uses QUABS (https://github.com/ltentrup/quabs) as backend solver. Then, we consider also the proposed techniques denoted as CASPER-U (upper-bound improving) and CASPER-L (lower-bound improving).

6.4. Results

Obtained results are summarized in Table 1 which reports, for each benchmark, the number of instances (#inst), the type of task (TT) ( $opt$ for optimal answer set and $coh$ for coherence), and, for each system, the number of solved instances (Sol.), total execution time (Sum t,(s)), and number of optimization steps (#Opt.). This latter is the sum of the number of quantified answer sets found to reach the solution; it is omitted for lower-bound improving and is not meaningful for local-weak-constraints-only problems.

Table 1.

Overall results

Boldface is used to highlight the best performing approach.

Fig 1.

Overall execution time – $opt$ .

Fig 2.

Solving time for CC.

We first focus on the benchmarks containing only local weak constraints, namely: PAP-NEC, PAP-REL, and MMC-BOUND. Observe that, in these cases, PYQASP could not be run (it does not support local weak constraints), and CASPER-L and CASPER-U clearly coincide, since the first winning move is the solution. For these benchmarks, nearly all instances were solved by CASPER within 2 min (only 12 of 813 timed out), confirming effectiveness of the systems herein introduced. To further assess the scalability of CASPER, we considered the CC benchmark. In particular, we compare CASPER with an enumeration-based solver, denoted as NESTED-ASPQ, which evaluates programs by enumerating answer sets of each subprogram according to the 2-ASP $^w$ (Q) semantics. Figure 2 reports the execution times of the systems for graphs with fixed edge density. Instances are sorted by increasing graph size, and each point $(x, y)$ represents the average runtime $y$ (over 10 generated instances) required by a system on graphs with $x$ nodes and a given edge density.

The generated instances are hard to solve and NESTED-ASPQ is unable to scale beyond graphs with 30 nodes, regardless of the edge density. In contrast, CASPER exhibits significantly better scalability, solving instances with up to 120 nodes. Furthermore, the edge density has a noticeable impact on CASPER runtime. For dense graphs, the runtime is considerably small, as dense graphs admits few maximal cliques and so the number of possible counterexample is reduced. Conversely, for sparse graphs, the runtime increases, due to the larger number of maximal cliques, which in turn leads to a higher number of counterexamples to be explored.

Let’s now shift the attention to optimum quantified answer set search. For MTD and COLORING, the upper-bound improving strategy, implemented in PYQASP and CASPER-U, is preferred over the lower-bound improving strategy adopted by CASPER-L. The ASP-based CASPER-U is preferable to the QBF-based PYQASP in MTD, whereas the opposite holds for COLORING. Diving in the details, PYQASP could compute for COLORING quantified answer sets that are closer to the optimum so requiring few optimization steps; on the other hand, CASPER-U shows a faster computation of quantified answer sets leading to better performance. On the other hand, CASPER-L likely computes many locally optimal moves that are not quantified answer sets, thus resulting slower than upper-bound improving alternatives. Conversely, for PAP-OPT, MMC-OPT, and SMOKERS, CASPER-L solves substantially more instances than the others (146 more than PYQASP and 172 more than CASPER-U), since locally optimal moves frequently correspond to optimal solutions.

Overall, CASPER-L solves 134 more instances than PYQASP, and 151 more instances than CASPER-U with a lower average runtime. Figure 1 reports a traditional cactus plot on all the instances, confirming CASPER-L is the most effective system overall.