In the recent years, we have linked a large corpus of formal mathematics with automated theorem proving (ATP) tools, and started to develop combined AI/ATP systems working in this setting. In this paper we first relate this project to the earlier large-scale automated developments done by Quaife with McCune's Otter system, and to the discussions of the QED project about formalizing a significant part of mathematics. Then we summarize our adventure so far, argue that the QED dreams were right in anticipating the creation of a very interesting semantic AI field, and discuss its further research directions.

Answer Set Programming (ASP) is a declarative problem solving approach, initially tailored to modeling problems in the area of Knowledge Representation and Reasoning (KRR). More recently, its attractive combination of a rich yet simple modeling language with high-performance solving capacities has sparked interest in many other areas even beyond KRR. This book presents a practical introduction to ASP, aiming at using ASP languages and systems for solving application problems. Starting from the essential formal foundations, it introduces ASP's solving technology, modeling language and methodology, while illustrating the overall solving process by practical examples.

We propose a Bayesian methodology for one-mode projecting a bipartite network that is being observed across a series of discrete time steps. The resulting one mode network captures the uncertainty over the presence/absence of each link and provides a probability distribution over its possible weight values. Additionally, the incorporation of prior knowledge over previous states makes the resulting network less sensitive to noise and missing observations that usually take place during the data collection process. The methodology consists of computationally inexpensive update rules and is scalable to large problems, via an appropriate distributed implementation.

We present a growing dimension asymptotic formalism. The perspective in this paper is classification theory and we show that it can accommodate probabilistic networks classifiers, including naive Bayes model and its augmented version. When represented as a Bayesian network these classifiers have an important advantage: The corresponding discriminant function turns out to be a specialized case of a generalized additive model, which makes it possible to get closed form expressions for the asymptotic misclassification probabilities used here as a measure of classification accuracy. Moreover, in this paper we propose a new quantity for assessing the discriminative power of a set of features which is then used to elaborate the augmented naive Bayes classifier. The result is a weighted form of the augmented naive Bayes that distributes weights among the sets of features according to their discriminative power. We derive the asymptotic distribution of the sample based discriminative power and show that it is seriously overestimated in a high dimensional case. We then apply this result to find the optimal, in a sense of minimum misclassification probability, type of weighting.

We propose an efficient method for Bayesian network inference in models with functional dependence. We generalize the multiplicative factorization method originally designed by Takikawa and D Ambrosio(1999) FOR models WITH independence OF causal influence.Using a hidden variable, we transform a probability potential INTO a product OF two - dimensional potentials.The multiplicative factorization yields more efficient inference. FOR example, IN junction tree propagation it helps TO avoid large cliques. IN ORDER TO keep potentials small, the number OF states OF the hidden variable should be minimized.We transform this problem INTO a combinatorial problem OF minimal base IN a particular space.We present an example OF a computerized adaptive test, IN which the factorization method IS significantly more efficient than previous inference methods.

Robust optimization is one of the fundamental approaches to deal with uncertainty in combinatorial optimization. This paper considers the robust spanning tree problem with interval data, which arises in a variety of telecommunication applications. It proposes a constraint satisfaction approach using a combinatorial lower bound, a pruning component that removes infeasible and suboptimal edges, as well as a search strategy exploring the most uncertain edges first. The resulting algorithm is shown to produce very dramatic improvements over the mathematical programming approach of Yaman et al. and to enlarge considerably the class of problems amenable to effective solutions

Recently, it has been emphasized that the possibility theory framework allows us to distinguish between i) what is possible because it is not ruled out by the available knowledge, and ii) what is possible for sure. This distinction may be useful when representing knowledge, for modelling values which are not impossible because they are consistent with the available knowledge on the one hand, and values guaranteed to be possible because reported from observations on the other hand. It is also of interest when expressing preferences, to point out values which are positively desired among those which are not rejected. This distinction can be encoded by two types of constraints expressed in terms of necessity measures and in terms of guaranteed possibility functions, which induce a pair of possibility distributions at the semantic level. A consistency condition should ensure that what is claimed to be guaranteed as possible is indeed not impossible. The present paper investigates the representation of this bipolar view, including the case when it is stated by means of conditional measures, or by means of comparative context-dependent constraints. The interest of this bipolar framework, which has been recently stressed for expressing preferences, is also pointed out in the representation of diagnostic knowledge.

Ancestral graphs are a class of graphs that encode conditional independence relations arising in DAG models with latent and selection variables, corresponding to marginalization and conditioning. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We introduce a simple representation of a Markov equivalence class of ancestral graphs, thereby facilitating model search. \ More specifically, we define a join operation on ancestral graphs which will associate a unique graph with a Markov equivalence class. We also extend the separation criterion for ancestral graphs (which is an extension of d-separation) and provide a proof of the pairwise Markov property for joined ancestral graphs.

The ability to make decisions and to assess potential courses of action is a corner-stone of many AI applications, and usually this requires explicit information about the decision-maker s preferences. IN many applications, preference elicitation IS a serious bottleneck.The USER either does NOT have the time, the knowledge, OR the expert support required TO specify complex multi - attribute utility functions. IN such cases, a method that IS based ON intuitive, yet expressive, preference statements IS required. IN this paper we suggest the USE OF TCP - nets, an enhancement OF CP - nets, AS a tool FOR representing, AND reasoning about qualitative preference statements.We present AND motivate this framework, define its semantics, AND show how it can be used TO perform constrained optimization.

We present new algorithms for inference in credal networks --- directed acyclic graphs associated with sets of probabilities. Credal networks are here interpreted as encoding strong independence relations among variables. We first present a theory of credal networks based on separately specified sets of probabilities. We also show that inference with polytrees is NP-hard in this setting. We then introduce new techniques that reduce the computational effort demanded by inference, particularly in polytrees, by exploring separability of credal sets.