Semi-Abstract Value-Based Argumentation

• SAVAF is a formal framework that integrates propositional logic and multi-valued preferences to evaluate argument acceptability in complex domains.
• It extends classical models by incorporating logical attack principles and a value-dependent defeat relation to reveal implicit conflicts.
• Applications include moral and legal reasoning, multi-agent negotiation, and decision support, demonstrating both subjective and objective analysis.

A Semi-Abstract Value-Based Argumentation Framework (SAVAF) is a formal framework for modeling structured argumentation with values, unifying and generalizing previous models such as Dung’s Abstract Argumentation Frameworks (AF), Bench-Capon’s Value-based Argumentation Frameworks (VAF), and Corsi–Fermüller’s Semi-Abstract Frameworks (SAF). SAVAFs allow for both the explicit structuring of argument content via propositional formulae and the assignment of ordered values, facilitating nuanced reasoning about the acceptability of arguments under both objective and subjective value orderings. The framework also introduces logical attack principles to systematically expose implicit attacks and defines a value-dependent defeat relation, making it suitable for formalizing complex domains such as moral and legal reasoning (Jeromela, 2023).

1. Formal Specification

A SAVAF is a tuple

$\mathcal{F} = (\Args,\,\Att,\,\Phi,\,V,\,\pi,\,\succ)$

where:

• $\Args$ is a finite set of argument identifiers.
• $\Att\subseteq \Args\times\Args$ denotes attacks; $a\to b$ iff $(a,b)\in\Att$.
• $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$ assigns each argument a propositional formula as its claim.
• $V$ is a finite set of values.
• $\succ$ is a strict total order on $V$.
• $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$ maps each formula to its set of associated values.

Special cases (recovering known frameworks) are:

• Dung’s AF: $\Args$0, $\Args$1 for all $\Args$2.
• VAF: $\Args$3 and $\Args$4 treated as abstract as in Bench-Capon (2002).
• SAF: $\Args$5 with nontrivial $\Args$6 and attack-closure using logical principles.

2. Logical Attack Principles and Defeat

SAVAFs extend attack relations using closure under logical attack principles to reveal implicit conflicts from propositional structure, employing selected subsets $\Args$7 of the following:

• (A.∧) If $\Args$8 and $\Args$9 then $\Att\subseteq \Args\times\Args$0.
• (A.∨) If $\Att\subseteq \Args\times\Args$1 then $\Att\subseteq \Args\times\Args$2 and $\Att\subseteq \Args\times\Args$3.
• (A.⊃) If $\Att\subseteq \Args\times\Args$4 then $\Att\subseteq \Args\times\Args$5.
• (B.⊃) If $\Att\subseteq \Args\times\Args$6 and $\Att\subseteq \Args\times\Args$7 then $\Att\subseteq \Args\times\Args$8.
• (B.¬) If $\Att\subseteq \Args\times\Args$9 then $a\to b$0.
• (C.∧) If $a\to b$1 then $a\to b$2 or $a\to b$3.
• (C.∨) If $a\to b$4 and $a\to b$5 then $a\to b$6.
• (C.⊃) If $a\to b$7 then $a\to b$8 or $a\to b$9.
• (C.¬) If $(a,b)\in\Att$0 then $(a,b)\in\Att$1.

Admissible choices for $(a,b)\in\Att$2 include the full set $(a,b)\in\Att$3 or the modally justified subset $(a,b)\in\Att$4.

Defeat: For order $(a,b)\in\Att$5 on $(a,b)\in\Att$6,

$(a,b)\in\Att$7

Consequence: $(a,b)\in\Att$8 holds if every argument $(a,b)\in\Att$9 whose claim is attacked and which defeats some argument with claim $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$0, itself defeats at least one of $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$1. No underlying truth valuation is required.

3. Semantics, Acceptability, and Value Ordering

Let $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$2:

• Conflict-free: No $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$3 such that $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$4 defeats $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$5.
• Acceptability: $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$6 is acceptable w.r.t.\ $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$7 if for every $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$8 defeating $\Phi:\Args\to\mathcal{L}_{\mathrm{prop}}$9 there exists $V$0 defeating $V$1.
• Admissible set: $V$2 is conflict-free and every $V$3 is acceptable w.r.t.\ $V$4.
• Complete extension: An admissible set containing each argument acceptable w.r.t.\ $V$5.
• Grounded extension: The least complete extension.
• Preferred extension: A maximal (by inclusion) admissible set.
• Stable extension: Conflict-free and defeats every argument outside it.

Subjectivity: Acceptability and consequence may be subjective (under some ordering $V$6) or objective (with respect to all total orderings on $V$7). An argument is subjectively acceptable if it belongs to a preferred extension for some ordering, and objectively acceptable if it appears in all preferred extensions for all orderings.

4. Properties and Computational Complexity

• Existence: Every SAVAF admits at least one complete extension; the grounded extension is unique.
• No-monochromatic-cycle theorem: If every directed cycle in the colored attack graph includes at least two distinct values, there is a unique nonempty preferred extension for any value ordering.
• Complexity:
  • Credulous acceptance (preferred semantics): NP-complete.
  • Skeptical acceptance (preferred semantics): coNP-complete.
  • Grounded acceptance (credulous/skeptical): P.

Problem	Semantics	Complexity
Credulous acceptance	Preferred	NP-complete
Skeptical acceptance	Preferred	coNP-complete
Cred./Skept. acceptance	Grounded	P

5. Illustrative Application: Autonomous Car Moral Dilemma

Given arguments $V$8 through $V$9 with claims $\succ$0 and values $\succ$1 (Deontology) or $\succ$2 (Consequentialism):

Argument	Claim	Value
$\succ$3	$\succ$4	$\succ$5
$\succ$6	$\succ$7	$\succ$8
$\succ$9	$V$0	$V$1
$V$2	$V$3	$V$4
$V$5	$V$6	$V$7
$V$8	$V$9	$\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$0
$\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$1	$\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$2	$\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$3

Initial attacks: $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$4, $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$5, $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$6, $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$7, $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$8, $\pi:\mathcal{L}_{\mathrm{prop}}\to 2^V$9, $\Args$00. Logical closure under $\Args$01 exposes further implicit attacks.

Key outcomes:

• $\Args$02 and $\Args$03 become indefensible,
• $\Args$04 and $\Args$05 are always objectively acceptable,
• For $\Args$06, $\Args$07 joins $\Args$08 in the unique preferred extension,
• For $\Args$09, $\Args$10 and $\Args$11 join $\Args$12 in the unique preferred extension.

The resolved acceptability of arguments is sensitive to the value ordering, demonstrating subjective rationality. However, arguments $\Args$13 and $\Args$14 are accepted objectively.

6. Expressivity, Applications, and Research Directions

SAVAFs strictly generalize prior frameworks by enabling propositional structuring and multi-valued preferences, supporting sophisticated analysis of the grounds for argument strength or weakness. Applications include moral and legal reasoning, multi-agent negotiation, preference-driven decision support, and argument mining in value-laden debates.

Advancing SAVAFs involves:

• Extending $\Args$15 to richer logical languages (modal, first-order, probabilistic, fuzzy),
• Developing efficient computational solvers leveraging propositional structure and value stratification,
• Investigating framework dynamics, including the impact of argument and value revisions,
• Analyzing parameterized complexity, e.g., in terms of attack graph treewidth (Jeromela, 2023).