On the Complexity of Entailment for Cumulative Propositional Dependence Logics

Recent work in defeasible reasoning has seen notions of preferential semantics and entailment in the style of Kraus et al. applied to modal logics. However, work in this field has focussed primarily on satisfiability checking, and monotonic notions of entailment, which may be inferentially weak. One particular modal logic where this has been introduced is propositional standpoint logics, where modalities can express the views of different viewpoints. This has resulted in the formalisation of propositional defeasible standpoint logic (PDSL). In this paper, we propose a means of lifting the class of (non-monotonic) rational entailment relations from traditional KLM-style reasoning to a fragment of PDSL. In order to do so, we extend the expressivity of PDSL via situated standpoint conditionals, allowing us to talk about a defeasible conditional holding in the context of a given standpoint. This allows us to re-characterise the syntax of PDSL in terms of situated conditionals, and shows that a large fragment of PDSL is expressible as a set of situated conditionals. We then focus on characterising non-monotonic entailment in this fragment, defining a method to transport any ranking-based entailment relation from the propositional case into the PDSL case. This is first described in the general case and then considered in the specific cases of rational and lexicographic closures, providing a faithful translation of each inference into PDSL. We also show that entailment-checking in this fragment of PDSL can be done largely using algorithms from the propositional case, while preserving complexity bounds.

In this paper, we integrate the defeasible logic of Kraus, Lehmann and Magidor (KLM) with the standpoint logic framework of Gómez Álvarez and Rudolph. This is done with the goal of formally expressing knowledge taking into account multiple (possibly contradicting) viewpoints, which in turn may hold defeasible beliefs. In doing so, we utilise Defeasible Restricted Standpoint Logics (DRSL), introduced by Leisegang et al. Our work expands on previous work by providing a foundational representation result for DRSL semantics and systematically lifting several well-known entailment relations from the propositional case to the standpoint-enhanced setting. In particular, we characterise the semantics for DRSL through a set of KLM-style postulates adapted for the standpoints case. We furthermore provide a means to lift preferential entailment, and the class of entailment relations based on single ranking functions from the purely propositional to the standpoint-enhanced context, including rational and lexicographic closure. We show this can be done equivalently through semantic and algorithmic means. Furthermore, we show that, for each considered form of entailment, the complexity class of entailment checking does not change when moving from propositional KLM to DRSL.

Description logic programs are a powerful formalism for combining rules with ontologies. The well-supported semantics for description logic programs ensures that no answer sets rely on cyclic dependencies. Most popular semantics for logic programming have this property of well-supportedness. We recognize two limitations of the current well-supported semantics for DL programs: its increased computational complexity for the consistency problem and its lack of a reduct transformation characterization. In this work, we present a new semantics which evaluates ontological atoms more strictly than the current semantics. This keeps the complexity of its consistency problem NP-complete, rather than increasing it to the second level of the polynomial hierarchy. Additionally, we identify a syntactic class of description logic programs for which our new semantics is equivalent to the current semantics. We characterize our semantics using a fixpoint operator and a reduct-based transformation. Our new semantics is a strict subset of the current well-supported semantics, so it maintains the prior notion of well-supportedness while inducing its own stricter notion. We prefer our new notion of well-supportedness due to its similarities with logic programming.