Base-Extension Semantics in ADFs

• Base-extension semantics is a framework that generalizes Dung’s argumentation by incorporating arbitrary acceptance conditions, merging attack and support relations.
• It employs positive dependency evaluations to control cycles, distinguishing between cycle-permitting (cc-semantics) and cycle-free (aa-semantics) approaches.
• The approach lifts classical properties like conflict-freeness and defense to abstract dialectical frameworks, enabling consistent modular reasoning in argumentation systems.

Base-extension semantics (B-eS) is a family of proof-theoretic semantic frameworks for abstract argumentation in which extensions are evaluated with respect not only to the classical attack relation but to a richer space of arbitrary acceptance conditions. The development and classification of extension-based semantics for abstract dialectical frameworks (ADFs) incorporates managing cycles in support dependencies, generalizing Dung’s graph-based semantics, and formalizing properties such as conflict-freeness and defense over generalized acceptance conditions.

1. Abstract Dialectical Frameworks and Acceptance Conditions

An abstract dialectical framework (ADF) is defined as $D = (S, C)$, where $S$ is a finite set of arguments and each $s \in S$ is equipped with an acceptance condition $C_s$. Each $C_s$ specifies, by a propositional formula over the parents $par(s) \subseteq S$ (the arguments with a link to $s$), exactly which sets $X \subseteq par(s)$ are sufficient for $s$ to be accepted (“in”). The assignment $C_s(X)$ yields either “in” or “out,” thus fusing the roles of attack and support into a unified notion of acceptance. Extension-based semantics for ADFs must concurrently accommodate “coexistence” (compatibility of arguments) and success/failure of acceptance via these conditions.

This generalization subsumes Dung's abstract argumentation frameworks (AFs), where each $C_s$ is defined by a “being unattacked” condition. In ADFs, arbitrary dependencies (including positive support, conjunctions, disjunctions, and negations) may be expressed at the semantic level.

2. Controlling Support Cycles via Positive Dependency Evaluations

A fundamental methodological advance in this framework is the management of positive dependency (pd) cycles—situations where one or more arguments’ acceptance is, directly or indirectly, self-dependent through the acceptance conditions. The central construct is the pd-evaluation machinery:

• For $A \subseteq S$ and $a \in A$, a pd-function $pd$ maps $a$ to a minimal two-valued interpretation $v \in min\_dec(in,a)$, such that the true set $v^t \subseteq A$. If no such $v$ exists, $pd(a) = \mathcal{N}$, a placeholder for “undefined”.
• An acyclic positive dependency evaluation for $a$ in $A$ is a pair $((a_0, \ldots, a_n), B)$, with $(a_0, \ldots, a_n)$ a sequence of distinct arguments in $A$ culminating at $a = a_n$, and $B = \bigcup_{i=0}^n pd(a_i)^f$ (the union of all arguments mapped to false in the sequence).
• The sequence must satisfy: every $pd(a_i) \neq \mathcal{N}$; $pd(a_0)^t = \emptyset$; and for $i \geq 1$, $pd(a_i)^t \subseteq \{a_0, ..., a_{i-1}\}$.

A blocking interpretation $v$ occurs if any $b \in B$ has $v(b) = t$ or any $a_i$ in the sequence is assigned $f$. If so, the pd-evaluation is “blocked,” precluding self-support via cycles. This acyclicity is then enforced in acyclic conflict-free extensions and, transitively, in stricter variants of admissible, complete, or preferred semantics.

3. Permissive and Restrictive Approaches to Cycles

The paper introduces two axes for classifying extension-based semantics based on what is permitted internally to an extension and externally (in attackers):

• cc-semantics: Both the set itself and attackers can contain cycles (cyclic-cyclic). These are less restrictive and resemble the classical Dung semantics, supporting, for example, self-supporting cycles.
• aa-semantics: Both the internal set and the attackers must be cycle-free (acyclic-acyclic). In these semantics, cyclic justifications are invalid, enforced via the pd-evaluation apparatus.

This “inside-outside” classification fills crucial gaps left by previous treatments, where the treatment of cycles (especially as attackers in defense relations) was inconsistent or ambiguous.

4. Lifting Classical Dung Properties to ADFs

The translation of Dung’s fundamental properties to the ADF setting proceeds via careful generalization:

• Conflict-freeness: An extension $E$ is conflict-free iff for every $s \in E$, $C_s(E \cap par(s)) = in$.
• Defense and Decisiveness: The defense of $a$ is generalized by “decisiveness”—$a$ is decisively “in” with respect to a range interpretation $v_E$ induced by $E$ if, under all extensions/attackers admissible in the condition $C_a$, $a$’s acceptance is robust.
• Fundamental Lemma: For cc-admissible (resp. aa-admissible) semantics, if $E$ is admissible and $a$ and $b$ are both decisively in, then $E \cup \{a\}$ retains admissibility, and $b$ remains decisively in with respect to the new range interpretation.

These generalizations ensure that, for the relevant sub-families of semantics, the relationships and fixed-point structures familiar from Dung’s framework persist, now instantiated over richer acceptance conditions and variable cycle policies.

5. Systematic Taxonomy and Relation to Labeling Semantics

The introduced classification makes the family of extension-based ADF semantics explicit along the two (inside/outside) acyclicity axes. The notation $xy$-admissible/complete/preferred (with $x, y \in \{\text{a}, \text{c}\}$) denotes the requirement for acyclicity or allowance of cycles in accepted arguments and attackers, respectively.

This enables structured comparison:

Prefix	Internal Defense	Attackers Checked For Cyclicity	Example
aa–	acyclic	acyclic	aa–admissible, aa–complete
cc–	cyclic	cyclic	cc–admissible, cc–complete

Further, the extension-based and labeling-based (three-valued, operator-based) semantics can be related: for many properties, range interpretations induced by extensions yield compatible labelings, and the grounded/fixed-point semantics coincide. Differences appear for more refined properties, notably preferred semantics, especially in the presence or absence of cycles.

6. Implications and Further Directions

This family of extension-based semantics for ADFs offers robust, modular means to generalize classical argumentation semantics in the presence of arbitrary acceptance conditions and support cycles. By introducing explicit acyclicity controls via pd-evaluations, a principled distinction can be made between self-supporting and cycle-free semantics, clarifying a previously open methodological issue.

Lifting Dung’s properties, including the Fundamental Lemma, to this setting provides assurance that the formal relationships between semantics and extension properties are preserved under this generalization. The taxonomy built from the inside-outside acyclicity policy, together with known labeling-based approaches, enables context-sensitive adoption of semantics—e.g., choosing aa– variants in contexts disallowing positive dependency cycles.

These methods significantly advance the expressivity of argumentation systems, handling not only attacks but also arbitrary acceptance dependencies in a linear-time (in $|S|$) local-checking setting for each acceptance condition. Potential applications include AI domains where argument dependencies (both attack and support) are intricate, as well as the implementation of argumentation frameworks in systems managing conflicting, cyclic, or mutually supportive evidence.

Continued research aims to integrate these sub-semantics with richer labeling-based frameworks, investigate computational properties of particular semantics under constraints, and further refine the relationships between extension-based semantics and practical reasoning architectures (Polberg, 2014).