Assumption-Based & Argumentation-Theoretic Foundation

• Assumption-based and argumentation-theoretic foundation is a formal framework that unifies defeasible reasoning, structured argumentation, and nonmonotonic inference through explicit assumptions and attack relations.
• It translates ABA into bipolar and constrained frameworks, preserving Dung-style semantics and linking to logic programming, preference reasoning, and adaptive logics.
• The framework supports applications in planning, paraconsistent reasoning, and learning, while addressing scalability and computational challenges in multi-agent systems.

Assumption-based and argumentation-theoretic foundation unifies several strands of defeasible reasoning, structured argumentation, and logic-based nonmonotonic inference by recognizing that the interplay of explicit, defeasible hypotheses (“assumptions”) and their systematic contestation/defense best captures many forms of automated and multi-agent reasoning under uncertainty. Central to this approach are the formal apparatus of Assumption-Based Argumentation (ABA), the generalization to bipolar and constrained settings, the translation into Dung-style frameworks, and the correspondences to logic programming, preference reasoning, argument-based planning, and adaptive logics.

1. Core Concepts of Assumption-Based Argumentation Frameworks

An assumption-based argumentation (ABA) framework is a tuple $\mathcal{F} = (L, R, A, \overline{\cdot})$, where $L$ is a (ground or propositional) language, $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$ is a set of (strict) rules, $A\subseteq L$ is a nonempty set of assumptions, and $\overline{\cdot}: A\to L$ is a contrariness mapping assigning each assumption a unique contrary (Proietti et al., 2023). In typical structured argumentation, derivations are represented by finite trees: an argument (for claim $s$) consists of a set of top-level supporting assumptions and a finite proof (using $R$) deriving $s$ whose leaves are assumptions or facts.

Attack relations in ABA principally take the form of undercutting: an argument $\alpha: \Delta \vdash s$ undercuts argument $\beta: \Delta' \vdash s'$ iff $L$0 for some $L$1. Rebutting attacks (where $L$2) can be encoded via auxiliary assumptions and contraries.

ABA frameworks support a range of Dung-style extension-based semantics. An admissible set of arguments is conflict-free (no internal attacks) and defends itself against all attackers (by counterattacking). Stable, grounded, and preferred extensions are defined as in abstract argumentation, yielding sets of mutually supporting and undefeated arguments/assumptions (Proietti et al., 2023, Ulbricht et al., 2023).

2. Structured Foundation: From ABA to Bipolar and Constrained Frameworks

While “flat” ABA restricts assumptions to non-derivable literals, general (or "non-flat") ABA permits inference rules having assumptions in their heads, yielding rich dependency networks among hypotheses. The formalization $L$3 supports tree-based deduction: $L$4 holds if there is a rooted proof tree with leaves in $L$5 or axioms and interior nodes justified by $L$6 (Ulbricht et al., 2023, Lehtonen et al., 2024).

Non-flat ABA is systematically abstracted to bipolar argumentation frameworks (BAFs), where arguments (nodes) are derived subtrees and edges encode collective attack and deductive support (the latter forming closure conditions on extensions). Formally, a BAF is $L$7 with $L$8 the set of arguments, $L$9 an attack relation (typically derived from ABA's contrariness), and $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$0 a support relation reflecting deductive closure or assumption inheritance (Ulbricht et al., 2023, Karamlou et al., 2019). This translation preserves complete, grounded, and stable semantics (and, with enhancements, admissible and preferred) via explicit theorems (Ulbricht et al., 2023, Lehtonen et al., 2024).

Recent generalizations include constrained ABA (CABA), which allows the underlying language, rules, and assumptions to range over variables and constraints, enabling infinite or parametric domains (Angelis et al., 13 Feb 2026). Attack relations are defined both at the ground and schematic levels via logical entailment over constraints.

3. Argumentation-Theoretic Semantics and Computation

Dung-style semantics apply directly to (bipolar, constrained) ABA and their BAF translations through notions of conflict-freeness, closure, and defense. Specifically, for bipolar ABA:

• Admissible: $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$1 is closed ($$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$2), conflict-free ($$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$3), and self-defending (for all closed $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$4 attacking $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$5, $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$6 attacks $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$7).
• Preferred: Maximal admissible sets.
• Stable: Closed, conflict-free, and attacks every $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$8 (Karamlou et al., 2019).

Correspondence theorems guarantee that for every semantics $$R\subseteq\{s_0\leftarrow s_1,\dots,s_m\mid m\geq 0, s_i\in L\}$$9 and ABA $A\subseteq L$0, $A\subseteq L$1 (where $A\subseteq L$2 is the BAF translation and $A\subseteq L$3 collects all assumptions supporting arguments in $A\subseteq L$4) (Ulbricht et al., 2023). Modularization properties (decomposition into extensions via reducts) and fixpoint characterizations apply to both standard and non-standard admissibility (weak, strong) in non-flat frameworks (Berthold et al., 15 Aug 2025).

Algorithmically, enumeration of extensions in general (bipolar) ABA is realized using backtracking and labelling with propagation and pruning based on closure and attack status. Though the general case is exponential, practical heuristics and redundant-argument elimination yield tractable reasoning for many applications (Karamlou et al., 2019, Lehtonen et al., 2024).

4. Logical Correspondences, Learning, and Preference Handling

ABA frameworks admit logic programming and nonmonotonic reasoning as special cases. Flat ABA can be mapped to normal logic programs (with NAF), while non-flat ABA and its attack/support structure are crucial for modelling more general disjunctive, extended, and abductive logic programs (Proietti et al., 2023, Heyninck et al., 2023, Schulz et al., 2014). Under precise translations, stable extensions of the ABA framework yield answer sets or minimal models, and attack trees constructed in ABA correspond to justifications for answer set membership (Heyninck et al., 2023, Schulz et al., 2014).

Paraconsistent and conflict-minimal reasoning is also captured by ABA-based frameworks, where assumptions are derived from tableau non-closure and attack relations encode minimality of conflict sets; stable extensions correspond to conflict-minimal interpretations (Qiao et al., 2014).

ABA can also be learned from data: positive and negative examples are incorporated by sequence of transformations—rote learning, equality elimination, folding, subsumption, and assumption introduction—resulting in frameworks that explain observed examples while minimally extending their defeasible hypotheses (Proietti et al., 2023).

Preferences, as modelled in ABA$A\subseteq L$5, are integrated directly into the attack relation: attacks by less-preferred assumptions are reversed (“reverse attack”), and the system supports weak forms of contraposition to retain desirable semantic properties (Čyras et al., 2016). The resulting extensions generalize standard (preference-free) semantics.

5. Admissibility Notions and Rationality Properties

Classical admissibility for ABA interprets a set of assumptions as admissible if it is closed, conflict-free, and defends every member. Non-flat (general) settings motivate alternatives:

• Weak admissibility relaxes defense: an assumption set is weakly admissible if there is no subset that, after removing elements under a reduct, can attack defended members (Berthold et al., 15 Aug 2025).
• Strong admissibility requires that every member is defended by a strongly admissible subset excluding itself.

While standard admissibility supports modularization and realizes the “fundamental lemma” (extending with defended members preserves admissibility), neither strong nor weak admissibility preserve all desirable rationality properties in the non-flat case—uniqueness of maximal extensions, relative containment, or closure under defense often fail and only partial remedies are known. Γ-closure (defense-based closure) partially restores these properties for strong but not weak semantics (Berthold et al., 15 Aug 2025).

6. Applications: Planning, Paraconsistency, and Gradual Semantics

Assumption-based and argumentation-theoretic principles underpin assumption-based planning, where agent plans are constructed by explicit conjecture/refutation cycles framed as a Dung-style dialogue. Here, unmet preconditions are treated as explicit assumptions to be discharged (via delegation or subplanning) or refuted (by counter-planning), and a plan is accepted only when all assumptions have been defended or resolved. This schema formalizes collaborative, multi-agent plan synthesis under incomplete knowledge (Pellier et al., 2018).

For paraconsistent reasoning (e.g., in ALC), ABA argumentation constructs are employed to handle three-valued (conflict) logics: assumptions correspond to presumed absence of conflict, arguments to tableau-derived closure, and stable extensions to conflict-minimal interpretations (Qiao et al., 2014).

Recent work advances gradual semantics for ABA, equipping assumptions with dialetical strengths computed via fixed-point iterations over set-attack/set-support hypergraphs, generalizing modular QBAF semantics, and supporting continuous degrees of acceptability. Experimental evidence demonstrates robustness and convergence properties for both direct and argument-based approaches (Rapberger et al., 14 Jul 2025).

7. Unification with Other Nonmonotonic Formalisms and Future Directions

ABA, ASPIC$A\subseteq L$6, adaptive logics, and default assumption consequence relations are intertranslatable under suitable conditions, via mappings of strict rules, assumptions, and contraries; consequence relations correspond directly to minimal-abnormality, reliability, and normal-selection strategies of adaptive logics (Heyninck et al., 2016). This theoretical unity enables the transfer of properties such as cumulativity, computational results, and dialogical proof procedures across formalisms.

Ongoing challenges include managing complexity in general (non-flat) frameworks, refining admissibility and closure notions for non-flat and supported settings, and developing modular, scalable algorithms for real-world applications involving large, richly-structured knowledge bases, or integrating symbolic and learning-based components. Open research continues on extension-based, labelling, and gradual semantics for ever more expressive frameworks.