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From Contexts to Values: Context-Dependent Defeat in Abstract Argumentation
Albert Sadowski, Jarosław A. Chudziak
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 8/22/2026, 2:25:20 AM
Summary
This paper introduces Context-Dependent Argumentation Frameworks (CDAFs), a model where defeat relations vary by context rather than audience values. It addresses the 'representability problem': determining if a CDAF can be reduced to a Value-Based Argumentation Framework (VAF). The authors provide a polynomial-time decision procedure (O(n^3k + nmk)) based on a structural criterion involving value-consistent partitions. They also analyze the complexity of related problems, noting that single-context realisability is in NP (grounded) or Σ3p (preferred), and present a validated implementation showing that VAF representability is rare and decreases rapidly with the number of contexts.
Entities (10)
Relation Signals (7)
VAF-Representability → determinedby → Value-Consistent Partition
confidence 95% · Theorem 1 A CDAF is VAF-representable iff it admits a value-consistent partition.
CDAF → generalizes → Dung Framework
confidence 95% · Each context c induces an ordinary Dung framework AF_c over the same arguments, so a CDAF is a finite family of frameworks over one shared attack structure.
VAF-Representability → hascomplexity → Polynomial Time
confidence 95% · We present a polynomial-time decision procedure for this question... O(n^3k + nmk) time.
Algorithm 1 → solves → VAF-Representability
confidence 95% · Algorithm 1 Deciding VAF-representability of a CDAF... Theorem 2 VAF-representability is decidable in O(n^3k + nmk) time.
CDAF → canbereducedto → VAF
confidence 90% · This raises a reduction question: is context genuinely new, or can one value assignment with per-context orderings reproduce the defeat function, collapsing the CDAF into a VAF?
Single-context realisability → hascomplexity → NP
confidence 90% · For grounded semantics... this puts the problem in NP.
Single-context realisability → hascomplexity → Sigma3p
confidence 90% · For preferred semantics... the same scheme gives Σ3p.
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Abstract
Abstract:In value-based argumentation, an audience's ordering of values decides which attacks succeed as defeats. In many settings the deciding factor is not the audience but the circumstances: the same attack may succeed at one procedural stage, or under one regulation, and fail at another. Context-dependent argumentation frameworks (CDAFs), a model we recently introduced, capture this directly: one set of arguments, one attack relation, and a defeat function that switches each attack on or off per context, so every context induces an ordinary Dung framework. This raises a reduction question: is context genuinely new, or can one value assignment with per-context orderings reproduce the defeat function, collapsing the CDAF into a VAF? We present a polynomial-time decision procedure for this question and map the harder neighbouring problems, with upper bounds from NP to $\Sigma^p_3$. We also present a validated reference implementation and a measurement: representability is rare and falls fast with the number of contexts.
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- Source: https://arxiv.org/abs/2608.15536v1
- Canonical: https://arxiv.org/abs/2608.15536v1
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From Contexts to Values: Context-Dependent Defeat in Abstract Argumentation Albert Sadowski Affiliation: Faculty of Electronics and Information Technology Affiliation: Warsaw University of Technology Affiliation: Warsaw, Poland Email: albert.sadowski.stud@pw.edu.pl Jarosław A. Chudziak Affiliation: Faculty of Electronics and Information Technology Affiliation: Warsaw University of Technology Affiliation: Warsaw, Poland Email: jaroslaw.chudziak@pw.edu.pl Abstract In value-based argumentation, an audience’s ordering of values decides which attacks succeed as defeats. In many settings the deciding factor is not the audience but the circumstances: the same attack may succeed at one procedural stage, or under one regulation, and fail at another. Context-dependent argumentation frameworks (CDAFs), a model we recently introduced, capture this directly: one set of arguments, one attack relation, and a defeat function that switches each attack on or off per context, so every context induces an ordinary Dung framework. This raises a reduction question: is context genuinely new, or can one value assignment with per-context orderings reproduce the defeat function, collapsing the CDAF into a VAF? We present a polynomial-time decision procedure for this question and map the harder neighbouring problems, with upper bounds from NP to Σ3p ^p_3. We also present a validated reference implementation and a measurement: representability is rare and falls fast with the number of contexts. Keywords abstract argumentation ⋅· value-based argumentation ⋅· computational argumentation 1 Introduction In abstract argumentation, a Dung framework fixes a set of arguments and an attack relation, and acceptance is computed from that fixed structure (7). Value-based argumentation frameworks (VAFs) added an influential source of variation: arguments promote values, an audience orders the values, and the ordering decides which attacks succeed as defeats (5). One structure, many audiences, many outcomes. In many settings the variation does not come from who evaluates, but from the circumstances under which the evaluation happens. A safety argument may defeat a cost argument during an incident and fail to defeat it during routine planning. The arguments and the attack stay the same; what changes is whether the attack succeeds. The circumstance can be a procedural stage, a regulatory regime, or the evidence available at the time; it is external to the arguments, and existing formalisms do not take it as a parameter. In recent work we introduced context-dependent argumentation frameworks (CDAFs) to model it directly (14). A CDAF is a tuple ⟨,R,,δ⟩ ,R,C,δ : arguments, attacks, a finite set of contexts, and a defeat function δ that marks each attack as operative or suppressed in each context. Each context c induces an ordinary Dung framework AFcAF_c over the same arguments, so a CDAF is a finite family of frameworks over one shared attack structure. The family view raises three questions. First, is contextual variation genuinely new? VAFs already vary defeat through audiences, so a CDAF might be a VAF in disguise: one value assignment and one ordering per context could reproduce δ exactly. When does this reduction hold, and how fast can it be decided? Second, when δ itself cannot be reproduced, can a VAF at least reproduce the extensions of every induced framework, and at what cost? Third, what does the model offer systems: can the decision procedure be implemented and validated, and how common is reducibility in the first place? Context and variation appear in argumentation in several roles. In argumentation context systems, contexts are modules that constrain one another (6); in context-based argumentation frameworks, contexts carry the agents’ backgrounds toward consensus (17). Closer to our question, other extensions vary which attacks succeed, but internally or epistemically: extended frameworks argue about preferences inside the framework (13), and incomplete frameworks induce a family of completions read as possible worlds (3). A CDAF’s variation is external: contexts are a parameter, and the family is evaluated as a whole. The paper answers the three questions in order. The reduction question has a complete answer: a structural criterion and a decision procedure that runs in O(n3k+nmk)O(n^3k+nmk) time for n arguments, m attacks, and k contexts. Section 3 presents the procedure as pseudocode and traces it on a small instance built to exercise every phase. The second question is harder: the known upper bounds are NP for grounded and Σ3p ^p_3 for preferred semantics, with no matching lower bounds, and Section 4 maps the landscape in a table. For the third, Section 5 presents a reference implementation, cross-validated against definitional brute force, and a measurement: representability is rare and collapses further with every added context. Section 6 lists open problems. 2 Preliminaries An argumentation framework (AF) is a pair F=⟨,R⟩F= ,R with A a finite set of arguments and R⊆×R ×A an attack relation (7). A semantics σ assigns to F a set σ(F)σ(F) of extensions, subsets of A. We use the grounded, complete, preferred, and stable semantics with their standard definitions, and refer to 2. Throughout, n=||n=|A| and m=|R|m=|R|. A value-based argumentation framework (VAF) extends an AF with a finite set of values V, an assignment :→Vval → V, and a set of audiences (5). Following 5, we take an audience p to be a strict total order ≻p _p on V. For audience p, an attack (a,b)∈R(a,b)∈ R succeeds as a defeat iff (b)⊁p(a)val(b) _pval(a). So an attack fails only if the attacked argument carries a strictly preferred value. Audience p induces the defeat relation Rp=(a,b)∈R:(b)⊁p(a)R_p=\(a,b)∈ R:val(b) _pval(a)\ and the framework ⟨,Rp⟩ ,R_p . Definition 1 (14) A context-dependent argumentation framework (CDAF) is a tuple D=⟨,R,,δ⟩D= ,R,C,δ , where A is a finite set of arguments, R⊆×R ×A is an attack relation, C is a non-empty finite set of contexts, and δ:×R→0,1δ × R→\0,1\ is the defeat function. Each context c∈c induces the defeat relation Rc=(a,b)∈R:δ(c,(a,b))=1R_c=\(a,b)∈ R:δ(c,(a,b))=1\ and the Dung framework AFc=⟨,Rc⟩AF_c= ,R_c . Attacks in R are latent conflicts; attacks in RcR_c are the operative defeats in c. We write k=||k=|C|. Cross-context acceptance classifies arguments across the family. Under a semantics σ, an argument is universally accepted if it belongs to every σ-extension of every AFcAF_c, indefensible if it belongs to none, and context-contingent otherwise. 3 Representability and Its Algorithm The representability problem asks whether audiences suffice to explain a given CDAF’s variation. VAF-Representability takes as input a CDAF D=⟨,R,,δ⟩D= ,R,C,δ and asks: are there a finite value set V, an assignment :→Vval → V, and a map φ from C to strict total orders on V such that for every c∈c and (a,b)∈R(a,b)∈ R: δ(c,(a,b))=1δ(c,(a,b))=1 iff (b)⊁φ(c)(a)val(b) _ (c)val(a)? The VAF must reproduce every entry of δ exactly; the weaker question, matching only extensions, is the realisability problem of Section 4. 3.1 The criterion and the algorithm The criterion is a partition condition. For a partition P of A, write [a][a] for the class of argument a. Definition 2 (value-consistent partition) A partition P of A is value-consistent for D if the following hold. (I) Intra-class unanimity: for all (a,b)∈R(a,b)∈ R with [a]=[b][a]=[b] and all c∈c : δ(c,(a,b))=1δ(c,(a,b))=1. (U) Cross-class uniformity: for all (a1,b1),(a2,b2)∈R(a_1,b_1),(a_2,b_2)∈ R with a1,a2∈[α]a_1,a_2∈[α], b1,b2∈[β]b_1,b_2∈[β], and [α]≠[β][α]≠[β]: δ(c,(a1,b1))=δ(c,(a2,b2))δ(c,(a_1,b_1))=δ(c,(a_2,b_2)) for all c∈c . (C) Cross-class complementarity: for all (a,b),(b′,a′)∈R(a,b),(b ,a )∈ R with a,a′∈[α]a,a ∈[α], b,b′∈[β]b,b ∈[β], and [α]≠[β][α]≠[β]: δ(c,(a,b))+δ(c,(b′,a′))=1δ(c,(a,b))+δ(c,(b ,a ))=1 for all c∈c . (A) Cross-class acyclicity: for each c∈c the relation ⊳c _c on P is acyclic, where for distinct classes [α]⊳c[β][α] _c[β] iff some attack from [α][α] to [β][β] succeeds in c, or some attack from [β][β] to [α][α] fails in c. Theorem 1 A CDAF is VAF-representable iff it admits a value-consistent partition. The conditions unfold the arithmetic of strict total orders. Equal values never block a defeat, so intra-class attacks must always succeed, which is (I). An audience compares values rather than individual attacks, so parallel cross-class attacks share one outcome, which is (U), and opposite cross-class attacks have exactly one winner, which is (C). A successful attack from [α][α] to [β][β] and a failed attack from [β][β] to [α][α] both witness that the value of [α][α] is strictly preferred in c; the relation ⊳c _c collects these forced preferences, and by Szpilrajn’s extension theorem (15) they are compatible with some strict total order iff they contain no cycle, which is (A). Conversely, any total order extending ⊳c _c reproduces δ exactly, checked case by case against the defeat rule. Theorem 1 reduces representability to the search for a partition, and the search is polynomial because all merges are forced. Algorithm 1 starts from singleton classes and only ever merges. Algorithm 1 Deciding VAF-representability of a CDAF ⟨,R,,δ⟩ ,R,C,δ 1: ←a:a∈P←\\a\:a \ ⊳ union-find over A 2: Phase 1: co-equality merges 3: for all mutual pairs (a,b),(b,a)∈R(a,b),(b,a)∈ R do 4: if δ(c,(a,b))=1δ(c,(a,b))=1 and δ(c,(b,a))=1δ(c,(b,a))=1 for every c∈c then 5: merge the classes of a and b 6: end if 7: end for 8: repeat 9: Phase 2: intra-class check (the only failure point) 10: for all (a,b)∈R(a,b)∈ R with [a]=[b][a]=[b] do 11: if δ(c,(a,b))=0δ(c,(a,b))=0 for some c∈c then 12: return not representable 13: end if 14: end for 15: Phase 3: per-context cycle merges 16: ←falsemerged 17: for all c∈c do 18: build ⊳c _c on the classes of P; record its non-trivial SCCs 19: end for 20: for all recorded non-trivial SCCs S do 21: merge the classes in S; ←truemerged 22: end for 23: until =falsemerged=false 24: Phase 4: verification 25: check (U), (C), (A) for P 26: return P ⊳ the finest value-consistent partition Theorem 2 VAF-representability is decidable in O(n3k+nmk)O(n^3k+nmk) time. The correctness argument rests on two invariants; we sketch it here. At every stage the current partition refines every value-consistent partition, because Phase 1 merges are forced by (C) and Phase 3 merges by (A): a ⊳c _c cycle over distinct classes can only be repaired by making them coincide. And any partition that survives to Phase 4 is value-consistent, so Phase 4 never fails. Hence the algorithm rejects exactly the non-representable instances, and on acceptance it returns the unique finest value-consistent partition. Phase 3 tests each ⊳c _c separately, never their union: condition (A) is imposed per context, and edges from different contexts may form a union cycle while every single ⊳c _c stays acyclic. For the running time: each round costs O(mk)O(mk) to run Phase 2 and build the relations ⊳c _c, plus O(n2k)O(n^2k) for their strongly connected components, and there are at most n−1n-1 rounds because every round loses at least one class. 3.2 A worked trace We trace Algorithm 1 on a fresh instance. The instance is constructed so that a single run exercises every phase: a Phase 1 merge, a Phase 3 cycle merge, and a Phase 2 rejection. Let =w,x,y,zA=\w,x,y,z\ and =c1,c2C=\c_1,c_2\, with R=(w,x),(x,w),(x,y),(y,z),(z,x).R=\(w,x),\ (x,w),\ (x,y),\ (y,z),\ (z,x)\. The attack graph is a mutual pair between w and x plus a directed cycle x→y→z→x→ y→ z→ x. Context c1c_1 activates every attack. Context c2c_2 suppresses exactly one attack, (x,y)(x,y). Figure 1 shows the graph and the full defeat function. wwxxyyzz Attack c1c_1 c2c_2 (w,x)(w,x) 1 1 (x,w)(x,w) 1 1 (x,y)(x,y) 1 0 (y,z)(y,z) 1 1 (z,x)(z,x) 1 1 Figure 1: The worked instance. Left: the attack relation R; the dashed attack (x,y)(x,y) is the only attack suppressed in some context. Right: the defeat function δ. The run proceeds as follows. 1. Phase 1. The only mutual pair is w,x\w,x\. Both (w,x)(w,x) and (x,w)(x,w) succeed in both contexts, so w and x merge. Now =w,x,y,zP=\\w,x\,\y\,\z\\. 2. Phase 2. The intra-class attacks are (w,x)(w,x) and (x,w)(x,w), and both succeed in every context. Pass. 3. Phase 3, building the relations. On the classes [wx][wx], [y][y], [z][z]. In c1c_1 every attack succeeds, so the first clause of ⊳c1 _c_1 yields [wx]⊳c1[y][wx] _c_1[y] from (x,y)(x,y), then [y]⊳c1[z][y] _c_1[z] from (y,z)(y,z), and [z]⊳c1[wx][z] _c_1[wx] from (z,x)(z,x). The three edges form a cycle, one non-trivial strongly connected component. In c2c_2 the attack (x,y)(x,y) fails, so the second clause yields [y]⊳c2[wx][y] _c_2[wx]; together with [y]⊳c2[z][y] _c_2[z] and [z]⊳c2[wx][z] _c_2[wx] from the first clause, ⊳c2 _c_2 is acyclic. The cycle lives in ⊳c1 _c_1 alone; c2c_2 forces nothing. 4. Phase 3, merging. The c1c_1 component merges all three classes: =w,x,y,zP=\\w,x,y,z\\. Back to Phase 2. 5. Phase 2, second pass. Every attack is now intra-class. The check reaches (x,y)(x,y) and finds δ(c2,(x,y))=0δ(c_2,(x,y))=0. The algorithm returns not representable. The run ends at the algorithm’s only failure point. The rejection tracks a semantic impossibility that can be stated without the algorithm. Suppose a value-consistent partition existed. In c1c_1 all three attacks of the cycle x→y→z→x→ y→ z→ x succeed. If x, y, z did not all share one class, the distinct classes among them would carry a ⊳c1 _c_1 cycle, violating (A). So [x]=[y]=[z][x]=[y]=[z]. But then (x,y)(x,y) is an intra-class attack, and δ(c2,(x,y))=0δ(c_2,(x,y))=0 violates (I). In words: c1c_1 forces the three values into a loop of strict preferences that only equality can resolve, equal values make defeat unconditional, and c2c_2 needs (x,y)(x,y) to fail. The two contexts place contradictory demands on one shared value assignment. Neither context is pathological on its own. For c1c_1 alone, give every argument the same value; then every attack succeeds, which matches Rc1=R_c_1=R. For c2c_2 alone, set (w)=(x)=vval(w)=val(x)=v, give y and z fresh values, and order (y)≻(z)≻vval(y) (z) v; checking the five attacks against the defeat rule reproduces Rc2R_c_2 exactly. So each context is representable in isolation, and non-representability is a coupling phenomenon: it appears only when one value assignment must serve both contexts. 4 The Complexity Landscape Table 1 collects the decision problems that a CDAF generates and what is known about them. Table 1: Decision problems for a CDAF with n arguments, m attacks, and k contexts. Cross-context bounds follow from single-framework classifications (9) with a factor k. Problem Semantics Bound Status Cross-context acceptance (all classes) grounded in P closed Universal acceptance preferred Π2p ^p_2-complete closed Indefensibility preferred coNP-complete closed Audience inducibility of R′⊆R R (defeat relation) O(n+m)O(n+m) closed VAF-representability (defeat function) O(n3k+nmk)O(n^3k+nmk) closed Single-context realisability grounded in NP lower bound open Single-context realisability preferred in Σ3p ^p_3 lower bound open Family realisability any decidable criterion open Cross-context acceptance. These queries inherit their complexity from single-framework reasoning, with a factor k for iterating over contexts: all three classes are polynomial under grounded semantics, while under preferred semantics universal acceptance is Π2p ^p_2-complete and indefensibility is coNP-complete (9); the corresponding audience-dependent questions for VAFs are classified by 4. The family structure adds bookkeeping, not hardness. Single-audience inducibility. The basic building block is a linear-time test. Given ⟨,R⟩ ,R and a subset R′⊆R R, is R′=RpR =R_p for some audience p of some VAF over ⟨,R⟩ ,R ? Build a mixed graph H on the arguments: a weak edge b→ab→ a for each (a,b)∈R′(a,b)∈ R , and a strict edge a→ba→ b for each (a,b)∈R∖R′(a,b)∈ R R . Read an edge u→vu→ v as “the rank of u is at most the rank of v”, strictly so for strict edges. The constraint system is satisfiable iff H has no directed cycle through a strict edge, which is checkable in O(n+m)O(n+m) time: contract the strongly connected components and reject iff some strict edge lies inside one. Single-context realisability. Realisability weakens representability: the VAF only has to reproduce the extensions. For a single context c and semantics σ, the question is whether some audience p over ⟨,R⟩ ,R satisfies σ(⟨,Rp⟩)=σ(AFc)σ( ,R_p )=σ(AF_c). Guess and check gives the upper bounds. For grounded semantics, guess an audience, compute both grounded extensions, and compare; this puts the problem in NP. For preferred semantics, comparing extension sets is a Π2p ^p_2 predicate, so the same scheme gives Σ3p ^p_3. Neither bound is known to be tight. The natural criterion for the grounded case fails: acyclicity of the latent attacks inside the target extension is necessary but not sufficient, and the smallest counterexamples have three arguments and four attacks. The counterexamples were found by exhaustive enumeration up to four arguments. Grounded and preferred realisability also come apart: there are instances realisable under grounded but not under preferred semantics. Family realisability. The family-level version asks for one VAF and one audience per context that reproduce σ(AFc)σ(AF_c) for every context simultaneously. It is decidable by brute force, since n values suffice and, up to renaming, there are finitely many assignments and orderings, but no better upper bound and no criterion in the style of Theorem 1 are known; the coupling constraint is the shared value assignment, and this is the first open problem of Section 6. Finally, the strategic variant is the subject of 14: an agent chooses which perspectives are active in an evaluation, thereby selecting which attacks succeed, and asks whether some choice makes a target argument credulously accepted; that problem is NP-complete under preferred and stable semantics. 5 A Reference Implementation and a Measurement This section takes the systems view: the tool, its validation against definitional brute force, its behaviour beyond toy sizes, and a measurement made with it.11 1 We release the implementation, the validation suite, and the enumeration data at https://github.com/albsadowski/cdaf, archived at https://doi.org/10.5281/zenodo.21320967. Small complete instance spaces are useful for conjecture testing beyond CDAFs, as in the realisability and signature programme for abstract argumentation (8; 12); the grounded counterexample of Section 4 was found by exhaustive enumeration of exactly this kind. We implemented Algorithm 1 and the mixed-graph inducibility test of Section 4 as a Python script. The tool reads a small extension of the apx format used in ICCMA (16; 10): a context block declares the contexts and lists, per context, the suppressed attacks. The worked instance of Section 3.2 reads: ⬇ arg(w). arg(x). arg(y). arg(z). att(w,x). att(x,w). att(x,y). att(y,z). att(z,x). context(c1). context(c2). off(c2,x,y). Every attack is operative unless switched off, so a plain apx file is a CDAF with one context, and projecting to one context yields a plain apx instance on which any ICCMA solver runs unchanged; cross-context queries are thin orchestration over k solver calls. Correctness is machine-checked against the definitions. One brute-force oracle enumerates all partitions of A and checks the four conditions of Definition 2 literally; a second decides inducibility by trying every ranking. The implementation agrees with the oracles on all 547,795 instances with three arguments and up to three contexts, on 180,000 uniformly sampled instances with four and five arguments, and on all 532,180 inducibility pairs with up to four arguments. A third oracle decides representability directly from the defeat rule, trying every value assignment with at most n values and every per-context strict total order. It agrees with Algorithm 1 on all 547,795 three-argument instances; since this oracle does not use the criterion, the agreement also checks Theorem 1 itself, not only the algorithm. On every representable instance the returned partition is verified to be value-consistent and finest, and the worked trace of Section 3.2 is asserted event by event. On VAF-generated instances, which are representable by construction and must be accepted, the reference code handles 10310^3 arguments, 10510^5 attacks, and 20 contexts in five seconds, and a mask-based variant included in the release handles 10410^4 arguments, 2×1062× 10^6 attacks, and the same 20 contexts in under three minutes. 5.1 How common is representability? The same machinery yields a measurement: the fraction of small CDAFs that is VAF-representable. Table 2 gives it for labeled instances without self-attacks. Table 2: Percentage of VAF-representable CDAFs among labeled instances without self-attacks, by arguments n and contexts k. The cells with n=3n=3, k≤3k≤ 3 are exact (exhaustive enumeration); the others are estimates from 2×1052× 10^5 uniform samples per cell, all 95% confidence intervals below 0.20.2 percentage points. Cells with no representable sample are bounded above by 0.0020.002 percent. k=1k=1 k=2k=2 k=3k=3 k=4k=4 n=3n=3 54.954.9 10.510.5 1.511.51 0.180.18 n=4n=4 20.320.3 0.520.52 0.0090.009 <0.002<0.002 n=5n=5 4.14.1 0.0030.003 <0.002<0.002 <0.002<0.002 Representability collapses quickly in both directions: at three arguments it falls from 55 percent for one context to 1.5 percent for three, and at five arguments and two contexts fewer than one instance in ten thousand is representable. We stop at n=5n=5: beyond it representability falls under the detection limit of uniform sampling, and larger instances would add empty bounds, not measurements. Two readings. First, audience structure is a strong constraint: generic context-dependence is almost never reducible to value orderings, at least under uniform sampling; structured instances arising from applications, such as attack graphs extracted from text by argument-mining pipelines (1), may behave differently. The polynomial test of Section 3 thus decides membership in a thin, structured class. Second, instances near the representability boundary, satisfying three of the four conditions of Definition 2, are too rare to sample and must be constructed; the enumeration data locates them. Realisability labels, by contrast, are not free at generation time: they need extension-set comparisons, at the second level of the polynomial hierarchy for preferred semantics, a problem type that current competition tracks do not exercise. 6 Open Problems The main structural question is family-level realisability: a criterion in the style of Theorem 1, or any upper bound better than brute force, where one value assignment couples all contexts. For grounded single-context realisability, acyclicity of latent attacks inside the target extension is necessary but not sufficient, and a characterisation would settle whether the NP upper bound can be lowered. More broadly, the NP and Σ3p ^p_3 upper bounds for realisability have no matching lower bounds. The perspective-labeled fragment of 14 derives δ from perspective activation and priorities; whether that structure admits faster algorithms is unexplored. Finally, cross-context queries currently materialise k frameworks and run a solver per context; declarative encodings that work on δ directly, in the spirit of the ASPIC+ encodings of 11, could avoid the materialisation and are a natural next step for this community. 7 Conclusion A CDAF is one attack structure under many contexts. We presented its decision problems from the algorithmic side: VAF-representability is solved by a polynomial-time partition-refinement algorithm, given as pseudocode and traced on a fresh instance, while the surrounding realisability problems have upper bounds from NP to Σ3p ^p_3 and no matching lower bounds. 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