This paper develops a logic programming language, GI-log, that extends answer set programming language with a new graded modality Kω where ω is an interval satisfying ω ⊆ [0, 1]. The modality is used to precede a literal in rules bodies, and thus allows for the representation of graded introspections in the presence of multiple belief sets: KωF intuitively means: it is known that the proportion of the belief sets where F is true is in the interval ω. We define the semantics of GI-log, study the relation to the languages of strong introspections, give an algorithm for computing solutions of GI-log programs, and investigate the use of GI-log for formalizing contextual reasoning, conformant planning with threshold, and modeling a graph problem.
By incorporating the methods of Answer Set Programming (ASP) and Markov Logic Networks (MLN), LP MLN becomes a powerful tool for non-monotonic, inconsistent and uncertain knowledge representation and reasoning. To facilitate the applications and extend the understandings of LP MLN , we investigate the strong equivalences between LP MLN programs in this paper, which is regarded as an important property in the field of logic programming. In the field of ASP, two programs P and Q are strongly equivalent, iff for any ASP program R, the programs P ∪ R and Q ∪ R have the same stable models. In other words, an ASP program can be replaced by one of its strong equivalent without considering its context, which helps us to simplify logic programs, enhance inference engines, construct human-friendly knowledge bases etc. Since LP MLN is a combination of ASP and MLN, the notions of strong equivalences in LP MLN is quite different from that in ASP. Firstly, we present the notions of p-strong and w-strong equivalences between LP MLN programs. Secondly, we present a characterization of the notions by generalizing the SE-model approach in ASP. Finally, we show the use of strong equivalences in simplifying LP MLN programs, and present a sufficient and necessary syntactic condition that guarantees the strong equivalence between a single LP MLN rule and the empty program.
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