An empirical study of the 4‐valued Kripke–Kleene and 4‐valued well‐founded semantics in random propositional logic programs

Abstract

We empirically investigated the difficulty of finding stable models for logic programs using backtracking, by trying to identify what makes random instances easy or hard. Additionally, we empirically investigated the effectiveness of the 4‐valued Kripke–Kleene semantics (4KK) and the 4‐valued well‐founded semantics (4WF) in the Niemelä and Simons’ backtracking algorithm, smodels, for finding stable models. We studied the behavior of 4KK and 4WF in a parameterized distribution of random propositional logic programs of fixed rule‐length k. In all of our experiments, 4KK and 4WF (both modified to extend an input partial truth assignment) were computed with respect to a fixed percentage of proposition letters (randomly chosen) initially assigned TRUE and a fixed percentage (randomly chosen) initially assigned FALSE. There exists a region, R, in the parameter space of our distribution where smodels required a large number of recursive calls to determine if programs generated in this region have any stable models. Hence, the “hardest” programs for smodels to determine if a stable model exists lie in R. Additionally, there exists a subregion of R where smodels made significantly fewer recursive calls when using 4WF as a pruning technique than when using 4KK. To gain a deeper insight into the causes for the “hardness” of programs in R and the differences between 4WF and 4KK as pruning techniques in smodels, we examined more closely the behavior of 4KK and 4WF. There exists a region in which a very small percentage of inconsistent models were produced by both 4KK and 4WF, thereby providing very little information useful for smodels to immediately backtrack. This region roughly corresponded to the above region where smodels required a large number of recursive calls. Also, there exists a region in which both 4KK and 4WF produced a high percentage of inconsistent models, thereby providing information useful for smodels to immediately backtrack.