Refining Gelfond Rationality Principle: Towards More Comprehensive Foundational Principles for Answer Set Semantics

TL;DR

This paper refines the Gelfond answer set principles to develop more comprehensive foundational criteria for answer set semantics, addressing limitations of existing conditions and proposing new semantics based on well-supportedness and minimality.

Contribution

It introduces refined GAS principles that extend well-supportedness and minimality, providing a new foundation for answer set semantics and evaluating existing semantics against these principles.

Findings

Refined GAS principles ensure answer sets are free of circular justification.

New semantics based on these principles better capture expected answer sets.

Analysis of computational complexity of the proposed semantics.

Abstract

Non-monotonic logic programming is the basis for a declarative problem solving paradigm known as answer set programming (ASP). Departing from the seminal definition by Gelfond and Lifschitz in 1988 for simple normal logic programs, various answer set semantics have been proposed for extensions. We consider two important questions: (1) Should the minimal model property, constraint monotonicity and foundedness as defined in the literature be mandatory conditions for an answer set semantics in general? (2) If not, what other properties could be considered as general principles for answer set semantics? We address the two questions. First, it seems that the three aforementioned conditions may sometimes be too strong, and we illustrate with examples that enforcing them may exclude expected answer sets. Second, we evolve the Gelfond answer set (GAS) principles for answer set construction by refining the Gelfond's rationality principle to well-supportedness, minimality w.r.t. negation by default and minimality w.r.t. epistemic negation. The principle of well-supportedness guarantees that every answer set is constructible from if-then rules obeying a level mapping and is thus free of circular justification, while the two minimality principles ensure that the formalism minimizes knowledge both at the level of answer sets and of world views. Third, to embody the refined GAS principles, we extend the notion of well-supportedness substantially to answer sets and world views, respectively. Fourth, we define new answer set semantics in terms of the refined GAS principles. Fifth, we use the refined GAS principles as an alternative baseline to intuitively assess the existing answer set semantics. Finally, we analyze the computational complexity.