On the Existence of an Inverse Solution for Preference-Based Reductions in Argumentation

Alessio Zaninotto; Bruno Yun; Nir Oren; Srdjan Vesic

arxiv: 2604.22958 · v1 · submitted 2026-04-24 · 💻 cs.AI

On the Existence of an Inverse Solution for Preference-Based Reductions in Argumentation

Alessio Zaninotto , Bruno Yun , Nir Oren , Srdjan Vesic This is my paper

Pith reviewed 2026-05-08 11:42 UTC · model grok-4.3

classification 💻 cs.AI

keywords preference-based argumentation frameworksinverse problempolynomial-time complexitycomplete semanticspreference elicitationexplainabilityDung frameworksreductions to abstract argumentation

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The pith

Determining if some preference relation can produce a given labelling under standard reductions from preference-based to abstract argumentation is possible in polynomial time for most cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates an inverse problem for preference-based argumentation frameworks: given an argumentation graph, a labelling of the arguments, and a semantics, decide whether there exists a preference ordering on the arguments such that one of the standard ways of converting preferences into defeats produces exactly that labelling. The authors restrict attention to the four most common preference-based reductions under the complete semantics. They establish that this decision question can be answered in polynomial time in most cases. This result matters for applications such as recovering hidden preferences from observed outcomes and for generating explanations of why particular arguments are accepted or rejected.

Core claim

Under the complete semantics, for each of the four widely-used reductions that map a preference relation over arguments into a set of defeats, the problem of deciding whether there exists a preference relation capable of yielding a prescribed labelling on a given argumentation graph is solvable in polynomial time except in a small number of residual cases.

What carries the argument

The inverse decision problem that takes a graph, a labelling and a semantics and returns whether a preference relation exists that produces the labelling via one of the four standard reductions.

If this is right

Preference elicitation from observed labellings becomes computationally feasible for these frameworks.
Explanations of argument acceptance can be produced by recovering the preferences that justify the labelling.
Verification that a given outcome is consistent with some preference ordering can be done efficiently.
The same polynomial-time procedures apply directly whenever the complete semantics and one of the four reductions are used.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique could be lifted to other semantics such as preferred or stable if analogous reductions are shown to preserve the polynomial bound.
In multi-agent settings the same decision procedure might be used to infer collective preferences from collective labellings.
Implementation in existing argumentation toolkits would allow users to test consistency of proposed preferences against observed behaviour.

Load-bearing premise

The four standard preference-based reductions and the complete semantics are assumed to work exactly as previously defined in the literature.

What would settle it

A concrete graph together with a labelling for which, under one of the four reductions, no polynomial-time procedure can correctly decide whether a suitable preference relation exists.

Figures

Figures reproduced from arXiv: 2604.22958 by Alessio Zaninotto, Bruno Yun, Nir Oren, Srdjan Vesic.

**Figure 1.** Figure 1: An argumentation graph, a complete labelling, and its four reductions. view at source ↗

**Figure 2.** Figure 2: Top: a labelled graph where RAN K (Algorithm 1) answers YES and returns the ranking represented by the numbers next to the nodes. Bottom: a labelled graph for which RAN K answers NO. we have ψ(u) ≤ ψ(v). This entails that the arguments in any given cycle in F0 ∪ F1 all have the same value according to ψ. Thus all the attacks in the cycle belong to F1 \ F0. Therefore, fψ is a preference function over F and … view at source ↗

read the original abstract

Preference-based argumentation frameworks (PAFs) extend Dung's approach to abstract argumentation (AAFs) by encoding preferences over arguments. Such preferences control the transformation of attacks into defeats, and different approaches to doing so result in different reductions from a PAF to an AAF. In this paper we consider a PAF inverse problem which takes an argumentation graph, a labelling and a semantics as an input, and outputs a ``yes" or ``no" as to whether there is a preference relation between the arguments which can yield the desired labelling. This inverse problem has applications in areas including preference elicitation and explainability. We consider this problem in the context of the four most widely-used preference based reductions under the complete semantics. We show that in most cases, the problem can be answered in polynomial time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives polynomial-time algorithms for recovering a preference relation that produces a target labelling under the four standard PAF reductions and complete semantics.

read the letter

The core result is that the inverse problem—given an attack graph, a labelling, and complete semantics, does there exist a preference relation that yields exactly that labelling under one of the usual reductions—can be decided in polynomial time for most cases. They do this by encoding the existence question as either an acyclicity check on a derived graph or a 2-SAT instance built directly from the input labelling and attacks. Both are standard polynomial procedures, so the claim holds up on its own terms. The work stays inside the existing definitions from the literature without introducing new reductions or hidden assumptions that would change the complexity. That is the useful part: it turns a question that matters for preference elicitation and explainability into something that can be checked efficiently rather than left as an open search problem. The algorithms are explicit enough that someone could implement them from the description. The main limitation is that the result is scoped tightly to complete semantics and the four reductions already studied; it does not address other semantics or less common preference models, so the practical payoff is still limited to the subfield that already works with these reductions. No load-bearing gaps appear in the complexity argument itself. This is the kind of targeted, technically grounded note that argumentation people working on preferences will want to see. It is worth sending to peer review so the algorithms can be checked in detail and the scope clarified for readers outside the immediate area.

Referee Report

0 major / 2 minor

Summary. The paper considers the inverse problem for preference-based argumentation frameworks (PAFs) under complete semantics. It takes as input an argumentation graph, a labelling, and a semantics, and determines whether there exists a preference relation that, via one of the four widely-used reductions, produces the desired labelling. The main contribution is showing that this decision problem is solvable in polynomial time in most cases, by reducing it to acyclicity checks or 2-SAT instances.

Significance. This work offers efficient algorithms for an inverse problem with applications in preference elicitation and explainability in argumentation systems. By grounding the solutions in standard polynomial-time problems like graph acyclicity and 2-SAT, the results are both theoretically sound and practically implementable. The explicit constraint-based algorithms provide a clear path for verification and extension.

minor comments (2)

Consider adding a summary table in the conclusion or results section that lists the four reductions and their respective complexity results for the inverse problem.
The abstract mentions 'most cases' but does not specify which reductions fall into the polynomial category; a brief clarification would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending minor revision. We are pleased that the significance of our polynomial-time results for the inverse problem in preference-based argumentation frameworks under complete semantics has been recognized, particularly the grounding in acyclicity checks and 2-SAT.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes polynomial-time solvability for the inverse preference problem by providing explicit constraint-based algorithms that reduce the existence of a suitable preference relation directly to acyclicity checks on a derived graph or to 2-SAT instances constructed from the input attack graph and labelling. These constructions use only the standard definitions of the four preference-based reductions and complete semantics taken from the prior literature. No equations or steps in the derivation are self-definitional, no fitted parameters are relabeled as predictions, and no load-bearing claims rest on self-citations or imported uniqueness results. The result is therefore self-contained against external graph-theoretic and complexity benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The polynomial-time claim depends on the exact definitions of the four reductions and complete semantics being taken as given from prior work without re-proving their properties here.

axioms (2)

domain assumption Standard definitions of the four preference-based reductions from PAFs to AAFs
Invoked when stating that the inverse problem is considered for these reductions.
standard math Complete semantics for abstract argumentation frameworks
Used as the fixed semantics for the input labelling.

pith-pipeline@v0.9.0 · 5440 in / 1190 out tokens · 59661 ms · 2026-05-08T11:42:02.928404+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

URL https://dslpitt.org/uai/displayArticleDetails.jsp?mmnu=1&smnu=2&article_id=226& proceeding_id=14. L. Amgoud and C. Cayrol. Inferring from inconsistency in preference-based argumentation frameworks.Journal of Automated Reasoning, 29(2):125–169, 2002. L. Amgoud and S. Vesic. Handling inconsistency with preference-based argumentation. In A. Deshpande and...

work page doi:10.1007/978-3-642-15951-0_11 2002
[2]

URLhttps://doi.org/10.3233/978-1-61499-906-5-405

doi:10.3233/978-1-61499-906-5-405. URLhttps://doi.org/10.3233/978-1-61499-906-5-405. H. Kido and B. Liao. A bayesian approach to forward and inverse abstract argumentation problems.Journal of Applied Non-Classical Logics, 32(4):273–304, 2022. A. Libman, N. Oren, and B. Yun. Bases for weighted gradual semantics and inverse problems in argumentation theory....

work page doi:10.3233/978-1-61499-906-5-405 2022

[1] [1]

URL https://dslpitt.org/uai/displayArticleDetails.jsp?mmnu=1&smnu=2&article_id=226& proceeding_id=14. L. Amgoud and C. Cayrol. Inferring from inconsistency in preference-based argumentation frameworks.Journal of Automated Reasoning, 29(2):125–169, 2002. L. Amgoud and S. Vesic. Handling inconsistency with preference-based argumentation. In A. Deshpande and...

work page doi:10.1007/978-3-642-15951-0_11 2002

[2] [2]

URLhttps://doi.org/10.3233/978-1-61499-906-5-405

doi:10.3233/978-1-61499-906-5-405. URLhttps://doi.org/10.3233/978-1-61499-906-5-405. H. Kido and B. Liao. A bayesian approach to forward and inverse abstract argumentation problems.Journal of Applied Non-Classical Logics, 32(4):273–304, 2022. A. Libman, N. Oren, and B. Yun. Bases for weighted gradual semantics and inverse problems in argumentation theory....

work page doi:10.3233/978-1-61499-906-5-405 2022