A Model-Theoretic Counterpart of Loop Formulas

Abstract

In an important recent paper, Lin and Zhao introduced the concept of a loop formula, and showed that the answer sets for a logic program are exactly the models of Clark’s completion of the program that satisfy the loop formulas. Just as supported sets are a model-theoretic account of completion, “externally supported” sets, defined in this paper, are a model-theoretic counterpart of loop formulas. This reformulation of loop formulas shows that they are related to assumption sets (Sacc ́a and Zaniolo) and to unfounded sets (Van Gelder, Ross and Schlipf; Leone, Rullo and Scarcello), invented many years earlier. Other contributions of this paper includes a simplification of the definition of a loop, extending it to programs with classical negation and infinite programs, and a generalization of the definition of a loop formula.