Warranted Derivations of Preferred Answer

We are aiming at a semantics of logic programs with preferences defined on rules, which always selects a preferred answer set, if there is a non-empty set of (standard) answer sets of the given program. It is shown in a seminal paper by Brewka and Eiter that the goal mentioned above is incompatible with their second principle and it is not satisfied in their semantics of prioritized logic programs. Similarly, also according to other established semantics, based on a prescriptive approach, there are programs with standard answer sets, but without preferred answer sets. According to the standard prescriptive approach no rule can be fired before a more preferred rule, unless the more preferred rule is blocked. This is a rather imperative approach, in its spirit. In our approach, rules can be blocked by more preferred rules, but the rules which are not blocked are handled in a more declarative style, their execution does not depend on the given preference relation on the rules. An argumentation framework (different from the Dung’s framework) is proposed in this paper. Argu- mentation structures are derived from the rules of a given program. An attack relation on argumentation structures is defined, which is derived from attacks of more preferred rules against the less preferred rules. Preferred answer sets correspond to complete argumentation structures, which are not blocked by other complete argumentation structures.

💡 Research Summary

The paper addresses a fundamental limitation of existing semantics for prioritized logic programs: many approaches, notably the Brewka‑Eiter framework and subsequent “prescriptive” semantics, can fail to produce any preferred answer set even when the underlying program has non‑empty standard answer sets. This incompatibility stems from the strict procedural requirement that a more preferred rule must fire before any less preferred rule unless it is blocked, which can eliminate all candidate preferred models.

To overcome this, the authors propose a novel, more “descriptive” approach based on an argumentation framework that is distinct from Dung’s abstract argumentation. They introduce argumentation structures, each derived from a rule of the program. A basic argumentation structure captures the rule’s head together with its default‑negated premises (the “assumptions”) and its positive premises. Three derivation rules (R1, R2, R3) allow these basic structures to be combined, extended, or unfolded, yielding derived argumentation structures that correspond to larger fragments of the program, including those representing entire answer sets.

The preference relation on rules is transferred to an attack relation between argumentation structures: a more preferred rule attacks a less preferred one when its conclusion contradicts the latter’s premises or conclusion. Attacks can be blocked if a yet more preferred structure already neutralizes them. The key semantic objects are complete argumentation structures (those that contain every literal that can be derived) and, among them, those that are not blocked by any other complete structure. Such unblocked, complete structures are called warranted.

A preferred answer set is defined as the set of objective literals associated with a warranted complete argumentation structure. The authors prove that whenever a program has at least one standard answer set, it necessarily has at least one warranted structure, and thus at least one preferred answer set. This satisfies a modified version of the second principle originally proposed by Brewka and Eiter, while abandoning the restrictive procedural ordering of the prescriptive approach.

The paper details the formal definitions of literals, rules, consistency, and the construction of the closure operator Cn≪P(W). It then rigorously defines basic and derived argumentation structures, presents the three derivation rules, and shows how attacks are propagated through derivations. An illustrative example with three rules (r₁, r₂, r₃) and a single preference (r₂ ≺ r₁) demonstrates how basic structures are combined, how attacks are generated, and how the warranted structure corresponds to the preferred answer set {a, b}.

In addition to the theoretical contribution, the authors discuss how their framework modifies the two original principles for preferred answer set specification, ensuring that a non‑empty set of standard answer sets always yields a non‑empty set of preferred answer sets. They also note that the current paper extends a preliminary version, adding more detailed proofs and a richer discussion of the relationship to existing argumentation approaches.

Overall, the work offers a compelling alternative to prescriptive semantics, providing a declarative mechanism that respects preferences without sacrificing the existence of preferred models. It opens avenues for further research on algorithmic implementation, complexity analysis, and integration with broader argumentation theories.