Consequence-based calculi are a family of reasoning algorithms for description logics (DLs), and they combine hypertableau and resolution in a way that often achieves excellent performance in practice. Up to now, however, they were proposed for either Horn DLs (which do not support disjunction), or for DLs without counting quantifiers. In this paper we present a novel consequence-based calculus for SRIQ — a rich DL that supports both features. This extension is non-trivial since the intermediate consequences that need to be derived during reasoning cannot be captured using DLs themselves. The results of our preliminary performance evaluation suggest the feasibility of our approach in practice.

We review several different languages for collective decision making problems, in which agents express their judgments, opinions, or beliefs over elements of a logically structured domain. Several such languages have been proposed in the literature to compactly represent the questions on which the agents are asked to give their views. In particular, the framework of judgment aggregation allows agents to vote directly on complex, logically related formulas, whereas the setting of binary aggregation asks agents to vote on propositional variables, over which dependencies are expressed by means of an integrity constraint. We compare these two languages and some of their variants according to their relative succinctness and according to the computational complexity of aggregating several individual views expressed in such languages into a collective judgment. Our main finding is that the formula-based language of judgment aggregation is more succinct than the constraint-based language of binary aggregation. In many (but not all) practically relevant situations, this increase in succinctness does not entail an increase in complexity of the corresponding problem of computing the outcome of an aggregation rule.

We introduce the concept of weighted rules under the stable model semantics following the log-linear models of Markov Logic. This provides versatile methods to overcome the deterministic nature of the stable model semantics, such as resolving inconsistencies in answer set programs, ranking stable models, associating probability to stable models, and applying statistical inference to computing weighted stable models. We also present formal comparisons with related formalisms, such as answer set programs, Markov Logic, ProbLog, and P-log.

Many approaches for forgetting in Answer Set Programming (ASP) have been proposed in recent years, in the form of specific operators, or classes of operators, following different principles and obeying different properties. Whereas each approach was developed to somehow address some particular view on forgetting, thus aimed at obeying a specific set of properties deemed adequate for such view, we are lacking a comprehensive and uniform overview of existing operators and properties. We aim at overcoming this by thoroughly examining existing properties and (classes of) operators for forgetting in ASP, drawing a complete picture, which includes many novel (even surprising) results on relations between properties and operators. Our goal is to provide a guide to help users in choosing the most adequate operator for their application requirements.

This paper is set to study the applicability of AGM-like operations to probabilistic bases. We focus on the problem of consistency restoration, also called consolidation or contraction by falsity. We aim to identify the reasons why the set of AGM postulates based on discrete operations of deletions and accretions is too coarse to treat finely adjustable probabilistic formulas. We propose new principles that allow one to deal with the consolidation of inconsistent probabilistic bases, presenting a finer method called liftable contraction. Furthermore, we show that existing methods for probabilistic consolidation via distance minimization are particular cases of the methods proposed.

We survey a selection of inconsistency measures from the literature and investigate their computational complexity wrt. decision problems related to bounds on the inconsistency value and the functional problem of determining the actual value. Our findings show that those inconsistency measures can be partitioned into three classes related to their complexity. The first class contains measures whose complexity are located on the first level of the polynomial hierarchy, the second class contains measures on the second level of the polynomial hierarchy, and the third class is located beyond the second level of the polynomial hierarchy. We provide membership results for all the investigated problems and completeness results for most of them.

Lifting a recent proposal by Moreno-Centeno and Karp, we propose a general framework for so-called implicit hitting set algorithms for reasoning beyond NP. The framework is motivated by empirically successful specific instantiations of the approach---based on interactions between a Boolean satisfiability (SAT) solver and an integer programming (IP) solver---in the context of maximum satisfiability (MaxSAT). The framework opens up opportunities for developing implicit hitting set algorithms for various important reasoning problems in KR by implementing domain-specific reasoning modules with SAT and IP solvers. We detail instantiations of the framework for the minimum satisfiability problem---as a natural dual of MaxSAT---and, as a central KR problem, for propositional abduction, covering the second level of the polynomial hierarchy. We show empirically that an implementation of the instantiation for propositional abduction surpasses the efficiency of an approach based on encoding and solving propositional abduction instances as disjunctive logic programming under answer set semantics.We also study key properties of the general framework.

The formalisms for knowledge representation and reasoning(KR&R) typically have a variety of semantics, each one having its particular application scenarios. However, the KR&Rcommunity cannot readily benefit from such a variety due toa lack of efficient solver technology. This is partly caused bythe fact that solver development is laborious and even accomplishing a working prototype can form a major effort.In this paper, we introduce a new framework that enables us todeclaratively specify a given semantics in second-order logicand to automatically generate a solver from that specification. Hence, KR&R researchers can rapidly develop a solverprototype for their new/existing semantics with a minimal effort. Technically, our framework builds on a recent approachfor nesting SAT solvers based on lazy clause generation.We evaluate our framework in the context of Dung’s argumentation frameworks, logic programming, and propositionallogic subject to standard and non-standard semantics. Weshow for each of those formalisms that one can easily specify its semantics using a few second-order sentences and thatone can effectively obtain a solver for that semantics usingour automated solver generation procedure.For instance, in the case of argumentation frameworks, weobtain 16 different solvers, each solving one of four inference tasks for one of four major argumentation semantics andshow that our solvers (slightly) outperform the best solverfrom the last system competition despite not being tuned forargumentation instances.

A central question in knowledge representation is the following: given some knowledge representation formalism, is it possible, and if so how, to simplify parts of a knowledge base without affecting its meaning, even in the light of additional information? The term strong equivalence was coined in the literature, i.e. strongly equivalent knowledge bases can be locally replaced by each other in a bigger theory without changing the semantics of the latter. In contrast to classical (monotone) logics where standard and strong equivalence coincide, it is possible to find ordinary but not strongly equivalent objects for any nonmonotonic formalism available in the literature. This paper addresses these questions in the context of abstract argumentation theory. Much effort has been spent to characterize several argumentation tailored equivalence notions w.r.t. extension-based semantics. In recent times labelling-based semantics have received increasing attention, for example in connection with algorithms computing extensions, proof procedures, dialogue games, dynamics in argumentation as well as belief revision in general. Of course, equivalence notions allowing for replacements are of high interest for the mentioned topics. In this paper we provide kernel-based characterization theorems for semantics based on complete labellings as well as admissible labellings w.r.t. eight different equivalence notions including the aforementioned most prominent one, namely strong equivalence.

The ability to infer intentions, emotions, and other unobservable psychological states from people's behavior is a hallmark of human social cognition, and an essential capability for future Artificial Intelligence systems. The commonsense theories of psychology and sociology necessary for such inferences have been a focus of logic-based knowledge representation research, but have been difficult to employ in robust automated reasoning architectures. In this paper we model behavior interpretation as a process of logical abduction, where the reasoning task is to identify the most probable set of assumptions that logically entail the observable behavior of others, given commonsense theories of psychology and sociology. We evaluate our approach using Triangle-COPA, a benchmark suite of 100 challenge problems based on an early social psychology experiment by Fritz Heider and Marianne Simmel. Commonsense knowledge of actions, social relationships, intentions, and emotions are encoded as defeasible axioms in first-order logic. We identify sets of assumptions that logically entail observed behaviors by backchaining with these axioms to a given depth, and order these sets by their joint probability assuming conditional independence. Our approach solves almost all (91) of the 100 questions in Triangle-COPA, and demonstrates a promising approach to robust behavior interpretation that integrates both logical and probabilistic reasoning.