We propose a QUBO encoding for the Subjective Acceptance Problem (SBA) in Value-Based Argumentation Frameworks (VAFs), where argument acceptance depends on audience-specific value preferences. SBA, an NP-complete problem, decides whether an argument is accepted by at least one audience in the preferred extension. By translating SBA into the QUBO formalism (which is solvable with quantum annealing), we enable new computational approaches to value-sensitive reasoning. This is the first QUBO encoding of a preference-based problem in Argumentation, and we validate it through initial empirical testing.
In many real-world debating and reasoning scenarios, simply identifying arguments is not suficient
because multiple arguments often conflict and support each other in complex ways, particularly in
moral and legal contexts [
For example, Abstract Argumentation Frameworks (AF s) [
In debates, arguments can conflict without necessarily defeating each other, as participants may
prioritize/prefer diferent underlying values. For instance, one person may argue for higher taxes to
promote equality, while another argues against it to reward enterprise; in this case, both people could
accept the merit of the other’s point but rank values diferently. For this reason, to model debates
efectively, it could be necessary to link arguments to the values they promote and to allow these values
to be ordered according to the audience’s preferences. Thus, the audience who attends the debate
becomes a crucial factor in evaluating the arguments presented. Value-based Argumentation Frameworks
(VAF s) [
Quadratic Unconstrained Binary Optimization (QUBO) [
The connection between QUBO problems and quantum annealing is fundamental. Quantum
annealers1 [8] are physical machines explicitly built to solve problems that can be expressed in QUBO (or its
equivalent, the Ising model [
This paper aims to propose and test a QUBO encoding of the Subjective Acceptance Problem (SBA) defined in [ 9] to be NP-complete. Formally, given a VAF and an argument , SBA is a decision problem that tests the existence of any audience such that is included in the unique preferred extension determined by that audience.2 Subjective acceptance captures the notion that an argument can be persuasive or justifiable, but only to specific groups of people or under certain value assumptions.
This paper continues the line of research initiated in [10, 11, 12, 13] by encoding and testing diferent
NP-Complete problems in Argumentation to a QUBO formulation. In this paper, we introduce and
encode problems related to preference-based AFs (i.e., VAFs) for the first time. As we have previously
discussed in this section, preferences (or values) are necessary to make reasoning in debates more
efective than using the original formulation [
This section provides the essential background required to understand the remainder of this work.
Section 2.1 introduces key concepts from Abstract Argumentation [
An Abstract Argumentation Framework (AF, for short) [
The collective acceptability of arguments depends on the definition of diferent semantics [
ℱ • ∈ gr(ℱ ) if ∈ co(ℱ ) and there is no ′ ∈ co(ℱ ) s.t. ′ ⊂ .
For a more detailed view of these semantics, please refer to [14]. Note that the grounded semantics
always uniquely identify a single extension [
Value-based Argumentation Frameworks [
Definition 2.1 (Value-based Argumentation Frameworks [
In Fig. 2 we show an example of VAF ℱ = ((A, →), , ) obtained from the AF in Fig. 1, where A = {, , , , }, →= {(, ), (, ), (, ), (, ), (, )}, = {1, 2, 3}, and () = 1, () = 2, () = 1, () = 3, () = 2.
The introduction of audiences afects the classical notion of attacks given for AFs in [
Therefore, an attack only “works” if the attacker’s value is not less preferred than the one being attacked. Otherwise, the argument is no longer considered part of the underlying ℱ , and the definition of the semantics presented before changes accordingly. For example, if we consider the VAF in Fig. 2 and the audience 1 ≥ 2 ≥ 3, the successful attacks are only {(, ), (, ), (, )}: {, } is now considered conflict-free and {} is no longer admissible or complete, compared to Fig. 1, for example.
The SBA problem [9] is a decision problem that answers the question of whether there is at least one perspective or value system under which a given argument can be accepted as justified. This has implications in real-world reasoning and decision-making, particularly in areas where people or institutions have diferent values or priorities, as, ethical debates (e.g., euthanasia, privacy vs. security), political discourse (e.g., freedom, equality, security), or in general systems where diferent agents have individual preferences or objectives: SBA helps determine whether a proposal is acceptable to at least one agentev en if it is not universally persuasive.
Definition 2.3 (Subjective Acceptance (SBA) [9]). Given a VAF ℱ = ((A, →), , ) and an argument ∈ A in this framework, the SBA problem accepts the instance (ℱ, ) if there exists at least one audience ≤ such that ∈ ≤ , where ≤ is the preferred extension of ℱ (i.e., { } = pr(ℱ ).) relative to the audience ≤ .
In SBA, the preferred semantics is used because it identifies arguments that are defensible under specific value orderings, allowing for multiple acceptable viewpoints that reflect the diversity of audience preferences. Unlike stable semantics, it is not overly strict, and, unlike the grounded semantics, it is not excessively skeptical, making it a balanced and suitable approach for capturing the notion of subjective acceptability. Note that Def. 2.3 relies on the existence of a single preferred extension in (A, →) given an audience ≤ because of Prop. 2.1.
Proposition 2.1 (Unicity of preferred/stable [15]). Let ℱ = ((A, →), , ) be a VAF, and let ≤ be an audience. We assume that every directed cycle in the argument graph (A, →) involves at least two distinct values: i.e., there are no cycles consisting only of arguments associated with the same value by . There exists a unique non-empty preferred extension ⊆ A (i.e., { } = pr(ℱ )) considering ≤ , that is, there exists only one ≤ . Moreover, is also the only stable extension given ≤ : { } = st(ℱ ), whose existence is thus guaranteed in this case.
The SBA problem is shown to be NP-complete in [9], where the proof is obtained by means of a reduction from a 3-SAT problem: the reduction constructs a VAF such that an argument is subjectively acceptable if and only if the 3-CNF formula is satisfiable. 3
The no-cycle condition on the values in Prop. 2.1 is not stringent because single-value cycles typically represent paradoxes or irrational structures [15]. Such cycles rarely occur in practical reasoning, where conflicts commonly arise between difering values, rather than within the same value. Figure 3 shows a VAF that respects the no-cycle condition on the values, while Fig. 4 shows a VAF that does not respect this condition. 3A 3CNF formula is a Boolean formula in conjunctive normal form where each clause contains exactly three literals.
As an example, given the VAF ℱ in Fig. 2 and the argument , the SBA accepts the instance (ℱ, ) because there exists an audience, for example 1 ≥ 2 ≥ 3, for which the preferred/stable extension is ≥ = {, , }, that is, ∈ ≥ as requested by Def. 2.3. Note that with the VAF in Fig. 2, all the arguments are subjectively acceptable: is always acceptable with any audience, is acceptable if () ≥ () ∧ () ≥ () (e.g., 2 ≥ 1), is acceptable if () ≥ () (e.g., 1 ≥ 3), is acceptable if () ≥ () (e.g., 3 ≥ 1), and finally is acceptable if () ≥ () (e.g., 2 ≥ 3). Given the VAF ℱ in Fig. 2 and the argument instead, the SBA does not accept the instance (ℱ, ) as for any audience ≥ the attack (, ) is always successful since () = () = 1.
Quantum Annealing is an approach [16] that uses a quantum computer to tackle optimization problems by identifying the lowest energy state configuration. This technique relies on the Quantum Adiabatic Theorem and leverages quantum tunneling to deliver optimal or nearly optimal solutions for discrete optimization problems [16]. The process begins with an initial function , for which the solution that minimizes the energy is easily determined. It gradually transitions to the final function , which corresponds to the function to be optimized. If the transition is slow enough, the Quantum Adiabatic Theorem guarantees that the solution with the minimum energy adapts to the change of the objective function.
Quantum Annealing is implemented by contemporary architectures such as D-Wave, as highlighted in Sect. 1. This company develops a line of quantum annealing devices that execute quantum annealing algorithms. In particular, their most sophisticated machines can handle problems involving thousands of variables.
Quantum annealers can be “programmed” by encoding the problems as Quadratic Unconstrained
Binary Optimization [
In a QUBO problem, binary variables 1, . . . , and an × upper-triangular matrix are used to formulate the task, which involves minimizing (or sometimes maximizing) the second-order function () = ∑︁ , + ∑︁ , .
=1 <
The diagonal terms , are the linear coeficients, and the non-zero of-diagonal terms quadratic coeficients. This can be expressed more concisely as , are the min ∈{0,1} . where denotes the transpose of the vector . The square matrix of coeficients can be organized in a symmetric way, where for all , except = , , is replaced by (, + ,)/2, or, as stated before, in an upper-diagonal form where for all , s.t. > , , is replaced by , + , and then all , are replaced by 0 for < .
To formulate a discrete constrained optimization problem into a QUBO, one must: i) determine a binary representation of the potential solutions, and ii) define a penalization function to deter nonfeasible solutions, specifically those that violate a constraint.
Except for the 0/1 limitations on the decision variables, QUBO is an unconstrained model, where the matrix contains all problem data. Rather than applying constraints in the conventional sense, classical constrained models can be successfully reformulated as QUBO models by inserting quadratic penalties into the objective function. In constrained optimization problems, selecting a penalty that is too small may lead to infeasible solutions. In contrast, employing a significant penalty to ensure constraints are met can make it challenging for the optimization algorithm to find feasible solutions. This strategy may also introduce challenges, such as sensitivity to penalty values, increased computational demands during optimization, and instability in the iterative process of the solver. For this reason, a substantial amount of literature has been produced to find good coeficients [ 20], or techniques to model classical constraints into a function, e.g., a logical and constraint between 1 and 2 as a multiplication 1 · 2.
The literature on precise approaches to QUBO on conventional computers includes various algorithms [18], all of which converge to a globally optimal solution given suficient time and memory. Most of these techniques use a standard branch-and-bound tree search, although alternative methods are also available. For example, in [21], a QUBO solution is based on the inherent geometric properties of the minimum circumscribed sphere that contains the ellipsoidal contour of the objective function, while in [22], the authors adopt Lagrangean decompositions. The NP-hardness of QUBO and its various potential applications have led to a significant number of research papers being published in recent years. These documents outline a variety of heuristic methods designed to rapidly identify high-quality solutions to medium- to large-scale problem cases. Although some of these techniques may be simple enough to be regarded as heuristics, the most efective ones are metaheuristic processes. These involve more complex compound strategies that outperform basic heuristics. For example, in addition to simulated annealing [23], the work in [24] presents a guided tabu-search algorithm alternating between a basic tabu-search procedure and a variable fix/free phase. In contrast, [ 25] presents a hybrid metaheuristic approach that combines crossover and update operators with tabu search to evaluate the ofspring solutions.
The encoding of Argumentation problems in QUBO uses a set of binary variables 1, . . . , associated with arguments {1, . . . , } in A. The variables 1, . . . , represent a subset of A: ∈ A if and only if = 1 [10]. Each semantics will be associated with a quadratic penalty function such that assumes its minimum value in the values (1, . . . , ) if and only if the corresponding set = { ∈ A : = 1} is an extension valid for .
For the SBA problem, we also use 2 − binary variables , with , = 1, . . . , , with ̸= , with the meaning that = 1 if and only if is preferred to in the audience ≤ . To enforce that the variables encode a total order, three quadratic terms are employed.
The first term encodes that for each , = 1, . . . , , with ̸= , is preferred to or viceversa. This term is:
The second term is used to encode the transitivity constraint: − = ∑︁( + − 1)2.
̸= =
∑︁ expresses the conjunction = ( and ) of binary variables , as a quadratic function.
We denote by = − + + the sum of these three quadratic terms. To encode the constraint of a stable extension, we first need to express that the set is complete. Hence, we define a penalty function that enforces this property, which is the sum of 4 terms. The first term forces the set to be conflict-free : where the binary variables , for , , = 1, . . . , are used to represent the product (or conjunction) · . Here and in other equations, we use sets of additional binary variables (and corresponding penalty terms to avoid cubic terms, which are products of three variables).
The third term has the purpose of encoding the constraint on :
(, , ) = + + + − 2( + ) is used to express the constraint that the binary variable is the disjunction = ( or ) of the binary variables , by means of a quadratic function, as shown in [26].
Each product of the form (), () must be replaced by a single binary variable , for = 1, . . . , such that → . The constraints = (), () for = . . . , are enforced by the penalization term = ∑︁ (, , (), ()).
→
Each penalty function requires max{ℎ − 2, 0} auxiliary variables , where ℎ is the number of possible attackers of , to represent the ⋁︀ operator as a composition of functions. If ℎ ≤ 2, no additional variable is required, as explained in [10]. On the other hand, if ℎ > 2 and the possible attackers of are 1 , . . . , ℎ , then
The value of corresponds to the number of internal attacks in that respect the audience ≤ , and its value is 0 if and only if is conflict-free.
The 2 binary variables , for , = 1, . . . , are used to represent the product (or conjunction) = 1. Hence, an additional penalty function must be included: cf = ∑︁ (), (). → = ∑︁ ( , , ).
,=1
The constraints for modeling the notion of defense are more complex and require additional variables. The first set contains the variables 1, . . . , , which denote the arguments that are successfully attacked by : = 1 if and only if some argument of attacks with respect to ≤ .
For each argument , the penalty function enforces the logical constraint =
⋁︁ (), ().
→ To express this constraint in QUBO, three “ingredients” are needed. The function (3) (4) (5) (6) {} {} {} {} {} {} { } { } { }
Size 3 2 − ∑∑∑︀︀︀===111 ℎℎℎ ∑∑︀︀==11 mmaaxx((ℎℎ −− 2, 0) 2, 0) = (, [1, ], 1 ) + ( 1 , [2, ], 2 ) + . . . + ( ℎ− 3, [ℎ− 2, ],
ℎ− 2)+ ( ℎ− 2, [ℎ− 1, ], [ℎ , ]), (7)
The binary variables 1, . . . , denote the arguments that are defended by : = 1 if and only if is defended (from all successful attacks) by some arguments of .
The penalty function forces to be 1 if and only if is defended by respecting ≤ , i.e., = ⋀︀ →
( ∨ ¬ (), ()). by the binary variable , for = 1, . . . , such that → .
In fact, is defended if for each possible attacker , either is attacked by or the attack is not successful, i.e., (), () = 0. In addition, in this case, each term ( ∨ ¬ (), ()) must be replaced The constraints = ( ∨ ¬ (), ()), for = 1, . . . , , are enforced by the penalization term = ∑︁ ( , , 1 − (), ()) → express the ⋀︀ operator as a composition of the function.
Each penalty function requires max{ℎ − 2, 0} auxiliary variables , for = 1, . . . , ℎ − 2 to When ℎ > 2 the penalty function is = (, [1, ], 1) + ( 1, [2, ], 2) + . . . + ( ℎ− 3, [ℎ− 2, ],
ℎ− 2)+ (
ℎ− 2, [ℎ− 1, ], [ℎ , ]) (8) =1 =1 → To encode that is a complete extension, the constraint = must hold for = 1, . . . , . Hence, the penalty function for complete sets is = + + + ∑︁( + ) + ∑︁ ∑︁ ( + ) + ∑︁((1 − ) + (1 − )) =1 where ℎ = max ℎ.
The sets of binary variables appearing in are listed in Tab. 3. The number of these variables is = 2+2 − +3 +3 ∑︀ =1 ℎ +2 ∑︀
=1 max(ℎ − 2, 0), which can be simplified as = (ℎ+3),
Finally, the encoding of the stable semantics is obtained by adding an additional term to which forces all arguments not belonging to to be attacked by : =1 = + ∑︁(1 − )(1 − )
It is straightforward to prove that the minimum value of is 0 if and only if the corresponding is a stable extension. Hence, this encoding can be used to determine whether a stable extension exists.
Solving the SBA problem for a given argument is suficient to minimize after setting = 1. If the minimum is 0, then is accepted; the values of variables indicate the stable/preferred extension, and the values of reveal the order relation ≤ . (9) 20 25 30 35
As a proof of concept, we have conducted a preliminary series of experiments using the Simulated Annealing (SA) [27] algorithm present in the Ocean SDK4 package. As noted in Sect. 2.2, the Hamiltonian presented in Sect. 3 can be minimized using methods other than SA; however, in the future, we would like to test the implementation on actual quantum annealers (Sect. 5).
To achieve this aim, we have generated four sets of 50 random directed graphs, each with = 20, 25, 30, 35 nodes/arguments, for a total of 200 graphs. For every graph, representing a diferent VAF, we used the Erdős-Rényi random generator [28], and we used an edge/attack probability of = 0.09. In each VAF, we have randomly assigned a value () to all arguments among the ⌊/2⌋ possible values 1, . . . , ⌊/2⌋. In this way, the average number of arguments that share the same value is two. Then, we have tested whether, in all the cycles of the graph, the arguments are assigned to at least two diferent values, as required by Prop. 2.1. In the negative case, we generate another argument-value assignment until the VAF satisfies the condition.
For each VAF, we have tested the acceptance of each argument by running the SA on 1000 reads of the corresponding QUBO model. We denote by the minimum value of the energy (i.e., the objective function) obtained in all reads for a given VAF and a given argument .
The aggregate results are shown in Tab. 1, where is the number of arguments, is the possible number of not accepted arguments, #AFs denotes the number of AFs of arguments in which the number of not accepted arguments is , min and max are respectively the minimum value of and the maximum value of computed for all not accepted arguments.
The results can be summarized as follows. For = 20, in 19 VAFs all the arguments are accepted, in 4Ocean SDK: https://docs.dwavequantum.com/en/latest/ocean/index.html. 16 VAFs only one argument is not accepted, in 14 VAFs two arguments are not accepted, and in one VAF three arguments are not accepted. It is interesting to see that the = 1 in the AFs when the argument is not accepted.
For = 25, again in 19 VAFs all arguments are accepted. In the remaining 24 VAFs, the number of arguments not accepted ranges from 1 to 4. Also, in these cases, = 1 when the argument is not accepted. The same value for the minimum energy was found in an AF with = 8 and in an AF with = 7 (the latter data does not appear in Tab. 1). The situation is diferent for the remaining 5 VAFs with = 25. In fact, we have found that in these VAFs, is 2 for at least one argument. Currently, we do not know whether 2 is the correct value of the objective function when the argument is not accepted. Hence, we cannot determine whether the value 2 indicates that either the argument is unacceptable or the SA was unable to find a lower-energy solution.
For = 30, apparently, in no are all arguments accepted. We have found 2 VAFs with one argument not accepted and 3 VAFs with 3 arguments not accepted. In these cases, the minimum energy is 1. In all the other AFs, the number of arguments not accepted is unclear. In particular, in 36 VAFs, no argument was accepted, which is highly unlikely. In these combinations, the minimum energy ranges from 5 to 212. Moreover, in 20 VAFs, we have > 90 for all arguments, which is a very large value and a strong indication that the SA has not been able to optimize the objective function. Finally, all experiments with = 35 have a unique outcome: in each AF, none of the arguments are accepted. This outcome is likely due to the non-optimal solutions found by the SA.
A partial justification can be given by looking at Tab. 2, in which the number of variables appearing in the corresponding QUBO models is reported for each value of . It is evident that the average value of nearly doubles when increases from 25 to 30. The SA is likely trapped in a local minimum when exceeds 2000.
In fact, a significant correlation between and the average value of can be observed in Fig. 6.
This paper elaborates on the research initiated in [10, 11, 12, 13] and introduces the first QUBO encoding for a preference-based problem in Argumentation, explicitly addressing the Subjective Acceptance problem in Value-based Argumentation Frameworks. By translating the NP-complete SBA problem into the QUBO formalism, we enable novel computational approaches to energy-minimization reasoning, particularly allowing the use of quantum annealers to solve the problem.
The encoding involves defining binary variables for arguments and value preferences and constructing quadratic penalty functions to enforce various constraints, such as conflict-freeness and defense.
Future research will focus on several directions. First, we plan to explore the scalability of this QUBO encoding for larger and more complex VAFs, investigating its performance on current and future quantum annealing hardware. The experimental results presented in this article are preliminary, and further investigation is necessary in the following steps of this work. For example, we will check the correctness of the solutions we find against the outputs provided by ASPARTIX [29],5 which consists of a set of AF/VAF encodings to be solved by Answer-set Programming (ASP) solvers.
M. Baioletti has been partially supported by: European Union - NextGenerationEU, Mission 4, Component 2, under the Italian Ministry of University and Research (MUR) National Innovation Ecosystem grant ECS00000041 - VITALITY - CUP J97G22000170005.
F. Santini has been partially supported by: European Union - Next Generation EU PNRR MUR PRIN Project J53D23007220006 EPICA: “Empowering Public Interest Communication with Argumentation”, MUR PNRR project SERICS (PE00000014 AQuSDIT: CUP_H73C22000880001), funded by the European Union – Next Generation EU; University of Perugia - Fondo Ricerca di Ateneo (2022) – Project RATIONALISTS, WP4.1 on “AI Data Management and Data Science”.
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