Argumentation Semantics for Prioritised Default Logic

We endow prioritised default logic (PDL) with argumentation semantics using the ASPIC+ framework for structured argumentation, and prove that the conclusions of the justified arguments are exactly the prioritised default extensions. Argumentation semantics for PDL will allow for the application of argument game proof theories to the process of inference in PDL, making the reasons for accepting a conclusion transparent and the inference process more intuitive. This also opens up the possibility for argumentation-based distributed reasoning and communication amongst agents with PDL representations of mental attitudes.

💡 Research Summary

The paper “Argumentation Semantics for Prioritised Default Logic” establishes a bridge between Brewka’s Prioritised Default Logic (PDL) and the structured argumentation framework ASPIC+. The authors first recall Dung’s abstract argumentation theory, which provides a graph‑based model of attacks and various extensions (complete, preferred, grounded, stable). They then review ASPIC+, which enriches Dung’s framework by making the internal logical structure of arguments explicit, distinguishing strict and defeasible inference rules, and introducing a mechanism for lifting preferences over premises and defeasible rules to preferences over whole arguments.

A central technical contribution is the identification of a flaw in the original ASPIC+ preference lifting: the “elitist” set‑comparison relation used to compare the sets of defeasible rules does not satisfy the “reasonable inducing” property required for rationality postulates. To remedy this, the authors propose two refined set‑comparison relations: the “strict elitist” order, which demands that every element of one set be strictly less preferred than every element of the other, and the “disjoint elitist” order, which only compares sets that are disjoint, thereby avoiding the problematic situation where overlapping rule sets could lead to non‑reasonable outcomes.

With these refined preference relations in hand, the paper proceeds to instantiate ASPIC+ for PDL. The knowledge base consists of the factual premises and the set of defaults from a PDL theory. Each default is treated as a defeasible rule in ASPIC+, while strict logical consequences are modeled by strict rules. The attack relations (undermining, rebutting, undercutting) are kept as defined in ASPIC+, but the defeat relation is now governed by the disjoint elitist order, ensuring that a defeat occurs exactly when the attacking argument is not strictly less preferred than the sub‑argument it attacks.

The authors prove a representation theorem: the set of conclusions of all sceptically justified arguments (i.e., those appearing in every stable extension of the resulting ASPIC+ framework) coincides precisely with the set of formulas belonging to the prioritized default extensions of the original PDL theory. The proof proceeds by showing (1) that stable extensions are unique for the instantiated framework, (2) that the notion of “blocked” versus “non‑blocked” defaults in PDL aligns with the presence or absence of attacks in the argumentation graph, and (3) that the disjoint elitist order respects the original priority ordering over defaults. Consequently, the argumentation semantics faithfully reproduces PDL inference.

Beyond the representation result, the paper analyses the normative rationality of the instantiation. It verifies that the ASPIC+ framework with the disjoint elitist order satisfies the rationality postulates of closure, consistency, and sub‑argument closure, as defined by Caminada and Amgoud. The authors discuss well‑definedness (the preference relation is a total preorder on defeasible rules) and reasonableness (the order is “reasonable inducing”), showing that the new order meets both criteria. They also distinguish between “strict extensions” (extensions built only from strict arguments) and “non‑strict extensions,” and argue that the former correspond to the classical extensions of PDL when no conflicts arise.

The discussion highlights the practical implications: by endowing PDL with argumentation semantics, one can apply argument game proof procedures to PDL reasoning, making the justification of conclusions explicit and transparent. This opens the way for distributed reasoning among agents that hold PDL‑based mental attitudes (beliefs, obligations, intentions, desires – BOID). Agents can exchange arguments and counter‑arguments, negotiate conflicts, and reach consensus using well‑studied dialogue protocols from argumentation theory.

In conclusion, the paper makes three key contributions: (1) it identifies and corrects a deficiency in the original ASPIC+ preference lifting; (2) it proposes the disjoint elitist order that aligns ASPIC+ preferences with PDL priorities; and (3) it proves a representation theorem showing exact correspondence between justified arguments and PDL extensions, while confirming that the resulting instantiation satisfies the standard rationality postulates. This work thus provides the first argumentation‑based semantics for Prioritised Default Logic and paves the way for more intuitive, explainable, and distributed non‑monotonic reasoning in multi‑agent systems.