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 study hedonic games with dichotomous preferences. Hedonic games are cooperative games in which players desire to form coalitions, but only care about the makeup of the coalitions of which they are members; they are indifferent about the makeup of other coalitions. The assumption of dichotomous preferences means that, additionally, each player's preference relation partitions the set of coalitions of which that player is a member into just two equivalence classes: satisfactory and unsatisfactory. A player is indifferent between satisfactory coalitions, and is indifferent between unsatisfactory coalitions, but strictly prefers any satisfactory coalition over any unsatisfactory coalition. We develop a succinct representation for such games, in which each player's preference relation is represented by a propositional formula. We show how solution concepts for hedonic games with dichotomous preferences are characterised by propositional formulas.

Preference models lie at the core of the formalization for several related notions, such as non-monotonic reasoning,obligations, goals, beliefs, etc. Recently, the interest in integrating dynamic operators in the logics of belief, preference and obligation has gained momentum.This integration sheds light on similarities among several change operations traditionally studied independently of each other. While a prolific approach, important operations, such as the well-known contraction of beliefs or derogation of norms studied in the AGM tradition,have not received proper attention in this framework.In this work, we study codifications of contraction operations, stemming from the work on iterate dbelief change, in the logic of preferences, by means of both semantically defined operations and their counterpart in syntactical priority structures.

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.

Knowledge compilation has been successfully used to solve beyond NP problems, including some PP-complete and NP PP -complete problems for Bayesian networks. In this work we show how knowledge compilation can be used to solve problems in the more intractable complexity class PP^PP.  This class contains NP PP and includes interesting AI problems, such as non-myopic value of information. We show how to solve the prototypical PP PP -complete problem MajMajsat in linear-time once the problem instance is compiled into a special class of Sentential Decision Diagrams. To show the practical value of our approach, we adapt it to answer the Same-Decision Probability (SDP) query, which was recently introduced for Bayesian networks. The SDP problem is also PP PP P-complete. It is a value-of-information query that quantifies the robustness of threshold-based decisions and comes with a corresponding algorithm that was also recently proposed. We present favorable experimental results, comparing our new algorithm based on knowledge compilation with the state-of-the-art algorithm for computing the SDP.

In this paper, we propose a general framework, both parameterized and parameter-free, for defining a family of fine-grained inconsistency measures for propositional knowledge bases. The parameterized approach allows to encompass several existing inconsistency mea- sures as specific cases, by properly setting its parameter. And the parameter-free approach is defined to avoid the difficulty in choosing a suitable parameter in practice but still keeps a desired ranking for knowledge bases by their inconsistency degrees. The fine granularity of our framework is based on the notion of MIS partition that considers the inner structure of all the minimal inconsistent subsets of a knowledge base. Moreover, MinCostSAT-based encodings are provided, which enable the use of efficient SAT solvers for the computation of the proposed measures. We implement these algo- rithms and test them on some real-world datasets. The preliminary experimental results for a variety of inputs show that the proposed framework gives a wide range of possibilities for evaluating large knowledge bases.