Normal forms for Answer Sets Programming

TL;DR

This paper introduces two normal forms for logic programs under answer set semantics, called kernel and 3-kernel, which simplify the analysis of program consistency and answer set properties.

Contribution

It presents novel normal forms for logic programs that facilitate the study of their properties, especially consistency, using graph-based representations.

Findings

Kernel form is more compact and useful for studying existence of answer sets.

3-kernel form aids static analysis of program consistency.

Consistency can be checked via colorings of the Extended Dependency Graph.

Abstract

Normal forms for logic programs under stable/answer set semantics are introduced. We argue that these forms can simplify the study of program properties, mainly consistency. The first normal form, called the {\em kernel} of the program, is useful for studying existence and number of answer sets. A kernel program is composed of the atoms which are undefined in the Well-founded semantics, which are those that directly affect the existence of answer sets. The body of rules is composed of negative literals only. Thus, the kernel form tends to be significantly more compact than other formulations. Also, it is possible to check consistency of kernel programs in terms of colorings of the Extended Dependency Graph program representation which we previously developed. The second normal form is called {\em 3-kernel.} A 3-kernel program is composed of the atoms which are undefined in the Well-founded semantics. Rules in 3-kernel programs have at most two conditions, and each rule either belongs to a cycle, or defines a connection between cycles. 3-kernel programs may have positive conditions. The 3-kernel normal form is very useful for the static analysis of program consistency, i.e., the syntactic characterization of existence of answer sets. This result can be obtained thanks to a novel graph-like representation of programs, called Cycle Graph which presented in the companion article \cite{Cos04b}.