The square of opposition is a structure involving two involutive negations and relating quantified statements, invented in Aristotle time. Rediscovered in the second half of the XXth century, and advocated as being of interest for understanding conceptual structures and solving problems in paraconsistent logics, the square of opposition has been recently completed into a cube, which corresponds to the introduction of a third negation. Such a cube can be encountered in very different knowledge representation formalisms, such as modal logic, possibility theory in its all-or-nothing version, formal concept analysis, rough set theory and abstract argumentation. After restating these results in a unified perspective, the paper proposes a graded extension of the cube and shows that several qualitative, as well as quantitative formalisms, such as Sugeno integrals used in multiple criteria aggregation and qualitative decision theory, or yet belief functions and Choquet integrals, are amenable to transformations that form graded cubes of opposition. This discovery leads to a new perspective on many knowledge representation formalisms, laying bare their underlying common features. The cube of opposition exhibits fruitful parallelisms between different formalisms, which leads to highlight some missing components present in one formalism and currently absent from another.

Recent years have witnessed a great deal of interest in extending answer set programming with function symbols. Since the evaluation of a program with function symbols might not terminate and checking termination is undecidable, several classes of logic programs have been proposed where the use of function symbols is limited but the program evaluation is guaranteed to terminate. In this paper, we propose a novel class of logic programs whose evaluation always terminates. The proposed technique identifies terminating programs that are not captured by any of the current approaches. Our technique is based on the idea of measuring the size of terms and atoms to check whether the rule head size is bounded by the body, and performs a more fine-grained analysis than previous work. Rather than adopting an all-or-nothing approach (either we can say that the program is terminating or we cannot say anything), our technique can identify arguments that are "limited'' (i.e., where there is no infinite propagation of terms) even when the program is not entirely recognized as terminating. Identifying arguments that are limited can support the user in the problem formulation and help other techniques that use limited arguments as a starting point. Another useful feature of our approach is that it is able to leverage external information about limited arguments. We also provide results on the correctness, the complexity, and the expressivity of our technique.

The study of groundedness was limited to exact lattice points; in this paper, we extend it to the bilattice: for an approximator A of O, we define A-groundedness. We show that all partial A-stable fixpoints are A-grounded and that the A-well-founded fixpoint is uniquely characterised as the least precise A-grounded fixpoint. We apply our theory to logic programming and study complexity.

The use of expressive logical axioms to specify derived predicates often allows planning domains to be formulated more compactly and naturally. We consider axioms in the form of a logic program with recursively defined predicates and negation-as-failure, as in PDDL 2.2. We show that problem formulations with axioms are not only more elegant, but can also be easier to solve, because specifying indirect action effects via axioms removes unnecessary choices from the search space of the planner. Despite their potential, however, axioms are not widely supported, particularly by cost-optimal planners. We draw on the connection between planning axioms and answer set programming to derive a consistency-based relaxation, from which we obtain axiom-aware versions of several admissible planning heuristics, such as hmax and pattern database heuristics.

Probabilistic planners are very flexible tools that can provide good solutions for difficult tasks. However, they rely on a model of the domain, which may be costly to either hand code or automatically learn for complex tasks. We propose a new learning approach that (a) requires only a set of state transitions to learn the model; (b) can cope with uncertainty in the effects; (c) uses a relational representation to generalize over different objects; and (d) in addition to action effects, it can also learn exogenous effects that are not related to any action, e.g., moving objects, endogenous growth and natural development. The proposed learning approach combines a multi-valued variant of inductive logic programming for the generation of candidate models, with an optimization method to select the best set of planning operators to model a problem. Finally, experimental validation is provided that shows improvements over previous work.

Recent work in proof theory has shed some light on the possibility of modeling reasoning while avoiding undesirable formal paradoxes. Based on category theory and inspired by the seminal work of J. Lambek, monoidal logics were introduced as a foundational framework that allows to treat a wide range of formal systems, including substructural logics (e.g., the syntactic calculus, linear logic, relevant logic, etc.), algebras (e.g., Kleene algebra) as well as intuitionistic, intermediate, and classical logic. This framework has been extended to modal logics and has been used to model normative reasoning, actions and knowledge, and it has been shown that non-classical logics better deal with the formal problems that are usually related to these notions. As such, non-classical systems of modal logics were proposed to model reasoning, actions and knowledge, but unresolved problems remained as to how to deal with conflicting obligations when facing normative inconsistencies. In this paper, we expose this problem and sketch an avenue for future research that might overcome this limitation.

There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we present MATR, a new framework for automated theorem proving explicitly designed to easily adapt to unusual logics or integrate new reasoning processes. MATR is formalism-agnostic, highly modular, and programmer-friendly. We explain the high-level design of MATR as well as some details of its implementation. To demonstrate MATR's utility, we then describe a formalized metalogic suitable for proofs of Gödel's Incompleteness Theorems, and report on our progress using our metalogic in MATR to semi-autonomously generate proofs of both the First and Second Incompleteness Theorems.

Combining Answer Set Programming (ASP) and Constraint Logic Programming (CLP) can create a more powerful language for knowledge representation and reasoning. The language AC(C) is designed to integrate ASP and CLP. Compared with existing integration of ASP and CSP, AC(C) allows representing user-defined constraints. Such integration provides great power for applications requiring logical reasoning involving constraints, e.g., temporal planning. In AC(C), user-defined and primitive constraints can be solved by a CLP inference engine while the logical reasoning over those constraints and regular logic literals is solved by an ASP inference engine (i.e., solver). My PhD work includes improving the language AC(C), implementing its faster inference engine and investigating how effective the new system can be used to solve a challenging application, temporal planning.

Game generation and analysis has commonly relied on hand authored rules and heuristics. This authoring task comes with a high authorial burden, both in the amount of rules and heuristics that need to be authored for decent coverage and in the complexity of authoring these rules. In this paper I present early work on \textit{Leda} and inductive logic programming system designed to learn these rules, so as to support further generation and analysis. I present Leda, describe its process, and finally show a sample set of the rules that it learns.

Morals are an important part of many stories, and central to why storytelling developed in the first place as a means of communication. They have the potential to provide a framework for developing story structure, which could be utilised by modern storytelling systems. To achieve this we need a general representation for morals. We propose patterns of character emotion as a suitable foundation. In this paper, we categorise Aesop's fables based on the morals they convey, and use them as a source of emotion data corresponding to those morals. We use inductive logic programming to identify relationships between particular patterns of emotion and the morals of the stories in which they arise.