Artificial intelligence (AI) is a term used for computational systems that attempt to mimic aspects of human intelligence (e.g., see [17]). Russell and Norvig [56] describe AI as the study of general principles of rational agents and components for constructing these agents. More broadly, the field of AI involves building intelligent entities that mimic'cognitive' functions we intuitively associate with human minds, such as'learning' and'problem solving.' We interpret the term AI broadly to include closely-related areas such as machine learning [43]. Systems that heavily use AI, henceforth referred to as AIbased systems, have had a significant impact in society in domains that include healthcare, transportation, social networking, e-commerce, education, etc.

In order to reason about probabilistic knowledge, we must reason about time and actions as well. When we say, for example, that "the probability of'heads' after a coin toss is 50% and that of'tails' is 50%", we implicitly assume that there is an action (in this example, tossing a coin) which can bring the world to different states in the next moment in time. The uncertainty lies in the state transition: the world may end up in a state where the coin shows heads or in a state where it shows tails. Despite the evident dependence of our informal notion of probability on the notions of action and time, the formal mathematical languages that we use to talk about probabilities rarely support mentioning action and time explicitly. Kolmogorov's probability theory, for example, merely defines probability as the measure function in a measure space with total measure 1 [9]. The task of modeling time-dependent actions and their possible outcomes in terms of events in a probabilistic space remains informal. While this informality is not problematic in the simplest situations (e.g. when we are interested in the possible outcomes of a single action, or when multiple actions are independent of each other), slightly more complex situations may already lead to confusion and difficulty. A famous example is the Monty Hall problem [10]. Another inconvenience of dealing with probabilities just in terms of a measure space is that its set-theoretic language (where events are represented as subsets of the sample space) is rather limited.

Recently, a number of noteworthy results have been achieved in various fields of artificial intelligence, and many aspects of the problem solving process have received significant attention by the scientific community. In this context, the extraction of comprehensive knowledge suitable for problem solving and reasoning, from textual and pictorial problem descriptions, has been less investigated, but recognized as essential for autonomous thinking in Artificial Intelligence. In this work we present a challenge where methods and tools for deep understanding are strongly needed for enabling problem solving: we propose to solve mathematical puzzles by means of computers, starting from text and diagrams describing them, without any human intervention. We are aware that the proposed challenge is hard and of difficult solution nowadays (and in the foreseeable future), but even studying and solving only single parts of the proposed challenge would represent an important step forward for artificial intelligence.

Proofdoku is an AI-based game design that extends Sudoku. In addition to playing by the rules of the traditional logic puzzle, players must explain their reasoning. An AI system checks this reasoning and provides hints that guide the player to discover new reasoning patterns for themselves. Co-developing the game design and the AI system, implemented using the technology of Answer Set Programming (ASP), guided us to include features that depend on high-complexity combinatorial search and optimization. We developed Proofdoku to better understand the implications of designing and deploying game systems that depend on ASP for live interaction. This paper offers design tradeoffs and makes suggestions for future deployments of ASP-backed game designs.

We examine the meaning and the complexity of probabilistic logic programs that consist of a set of rules and a set of independent probabilistic facts (that is, programs based on Sato's distribution semantics). We focus on two semantics, respectively based on stable and on well-founded models. We show that the semantics based on stable models (referred to as the "credal semantics") produces sets of probability measures that dominate infinitely monotone Choquet capacities; we describe several useful consequences of this result. We then examine the complexity of inference with probabilistic logic programs. We distinguish between the complexity of inference when a probabilistic program and a query are given (the inferential complexity), and the complexity of inference when the probabilistic program is fixed and the query is given (the query complexity, akin to data complexity as used in database theory). We obtain results on the inferential and query complexity for acyclic, stratified, and normal propositional and relational programs; complexity reaches various levels of the counting hierarchy and even exponential levels.

Recently proposed models which learn to write computer programs from data use either input/output examples or rich execution traces. Instead, we argue that a novel alternative is to use a glass-box loss function, given as a program itself that can be directly inspected. Glass-box optimization covers a wide range of problems, from computing the greatest common divisor of two integers, to learning-to-learn problems. In this paper, we present an intelligent search system which learns, given the partial program and the glass-box problem, the probabilities over the space of programs. We empirically demonstrate that our informed search procedure leads to significant improvements compared to brute-force program search, both in terms of accuracy and time. For our experiments we use rich context free grammars inspired by number theory, text processing, and algebra. Our results show that (i) performing 4 rounds of our framework typically solves about 70% of the target problems, (ii) our framework can improve itself even in domain agnostic scenarios, and (iii) it can solve problems that would be otherwise too slow to solve with brute-force search.

Latent factor models are increasingly popular for modeling multi-relational knowledge graphs. By their vectorial nature, it is not only hard to interpret why this class of models works so well, but also to understand where they fail and how they might be improved. We conduct an experimental survey of state-of-the-art models, not towards a purely comparative end, but as a means to get insight about their inductive abilities. To assess the strengths and weaknesses of each model, we create simple tasks that exhibit first, atomic properties of binary relations, and then, common inter-relational inference through synthetic genealogies. Based on these experimental results, we propose new research directions to improve on existing models.

Answer Set Programming (ASP) is a well-known paradigm of declarative programming with roots in logic programming and non-monotonic reasoning. Similar to other closely related problem-solving technologies, such as SAT/SMT, QBF, Planning and Scheduling, advancements in ASP solving are assessed in competition events. In this paper, we report about the design and results of the Sixth ASP Competition, which was jointly organized by the University of Calabria (Italy), Aalto University (Finland), and the University of Genoa (Italy), in affiliation with the 13th International Conference on Logic Programming and Non-Monotonic Reasoning. This edition maintained some of the design decisions introduced in 2014, e.g., the conception of sub-tracks, the scoring scheme, and the adherence to a fixed modeling language in order to push the adoption of the ASP-Core-2 standard. On the other hand, it featured also some novelties, like a benchmark selection stage classifying instances according to their empirical hardness, and a "Marathon" track where the top-performing systems are given more time for solving hard benchmarks.

G\"odel's ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. The following has been done (and in this order): A detailed natural deduction proof. A formalization of the axioms, definitions and theorems in the TPTP THF syntax. Automatic verification of the consistency of the axioms and definitions with Nitpick. Automatic demonstration of the theorems with the provers LEO-II and Satallax. A step-by-step formalization using the Coq proof assistant. A formalization using the Isabelle proof assistant, where the theorems (and some additional lemmata) have been automated with Sledgehammer and Metis.

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, then instantiate the abstract setting to sets and relations and to finite dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.