In Answer Set Programming (ASP), the user can define declaratively a problem and solve it with efficient solvers; practical applications of ASP are countless and several constraint problems have been successfully solved with ASP. On the other hand, solution time usually grows in a superlinear way (often, exponential) with respect to the size of the instance, which is impractical for large instances. A widely used approach is to split the optimization problem into sub-problems that are solved in sequence, some committing to the values assigned by others, and reconstructing a valid assignment for the whole problem by juxtaposing the solutions of the single sub-problems. On the one hand this approach is much faster, due to the superlinear behavior; on the other hand, it does not provide any guarantee of optimality: committing to the assignment of one sub-problem can rule out the optimal solution from the search space. In other research areas, Logic-Based Benders Decomposition (LBBD) proved effective; in LBBD, the problem is decomposed into a Master Problem (MP) and one or several Sub-Problems (SP). The solution of the MP is passed to the SPs, that can possibly fail. In case of failure, a no-good is returned to the MP, that is solved again with the addition of the new constraint. The solution process is iterated until a valid solution is obtained for all the sub-problems or the MP is proven infeasible. The obtained solution is provably optimal under very mild conditions. In this paper, we apply for the first time LBBD to ASP, exploiting an application in health care as case study. Experimental results show the effectiveness of the approach. We believe that the availability of LBBD can further increase the practical applicability of ASP technologies.

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, i.e. operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of nondeterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola, et al., and (3) we generalize the characterisations of disjunctive logic programs to disjunctive logic programs with aggregates. This is an extended version of our paper that will be presented at ICLP 2023 and will appear in the special issue of TPLP with the ICLP proceedings.

The Bar-Hillel construction is a classic result in formal language theory. It shows, by a simple construction, that the intersection of a context-free language and a regular language is itself context-free. In the construction, the regular language is specified by a finite-state automaton. However, neither the original construction (Bar-Hillel et al., 1961) nor its weighted extension (Nederhof and Satta, 2003) can handle finite-state automata with $\varepsilon$-arcs. While it is possible to remove $\varepsilon$-arcs from a finite-state automaton efficiently without modifying the language, such an operation modifies the automaton's set of paths. We give a construction that generalizes the Bar-Hillel in the case where the desired automaton has $\varepsilon$-arcs, and further prove that our generalized construction leads to a grammar that encodes the structure of both the input automaton and grammar while retaining the asymptotic size of the original construction.

As Artificial Intelligence, and Deep Learning in particular, reach impressive results, it gains also unprecedented popularity not only in academics and industry but also in popular culture and society in general. This increasingly ubiquitous AI presence has arisen several concerns about its impacts on humanity and the planet, with some well-known scientists like Stephen Hawking having spoken concerns about AI's accountability [1]. Despite achieving outstanding results in Computer Vision, Natural Language Processing and Game Playing [2, 3], tasks in which AIs formerly have poor performance compared to humans, those concerns about AI triggered debates among research communities, including those discussed by Gary Marcus [4] and on AAAI-2020 debate with Geoffrey Hinton, Yoshua Bengio and Yann LeCun [5].

Artificial Intelligence plays a main role in supporting and improving smart manufacturing and Industry 4.0, by enabling the automation of different types of tasks manually performed by domain experts. In particular, assessing the compliance of a product with the relative schematic is a time-consuming and prone-to-error process. In this paper, we address this problem in a specific industrial scenario. In particular, we define a Neuro-Symbolic approach for automating the compliance verification of the electrical control panels. Our approach is based on the combination of Deep Learning techniques with Answer Set Programming (ASP), and allows for identifying possible anomalies and errors in the final product even when a very limited amount of training data is available. The experiments conducted on a real test case provided by an Italian Company operating in electrical control panel production demonstrate the effectiveness of the proposed approach.

Answer Set Programming with Quantifiers ASP(Q) extends Answer Set Programming (ASP) to allow for declarative and modular modeling of problems from the entire polynomial hierarchy. The first implementation of ASP(Q), called qasp, was based on a translation to Quantified Boolean Formulae (QBF) with the aim of exploiting the well-developed and mature QBF-solving technology. However, the implementation of the QBF encoding employed in qasp is very general and might produce formulas that are hard to evaluate for existing QBF solvers because of the large number of symbols and sub-clauses. In this paper, we present a new implementation that builds on the ideas of qasp and features both a more efficient encoding procedure and new optimized encodings of ASP(Q) programs in QBF. The new encodings produce smaller formulas (in terms of the number of quantifiers, variables, and clauses) and result in a more efficient evaluation process. An algorithm selection strategy automatically combines several QBF-solving back-ends to further increase performance. An experimental analysis, conducted on known benchmarks, shows that the new system outperforms qasp.

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ$\exists$R, which can be viewed as a succinct variant of the well-studied class $\exists$R, and show that the problems we consider are complete for succ$\exists$R. Our results imply even stronger algorithmic limitations than were proven by Fagin, Halpern, and Megiddo (1990) and Moss\'{e}, Ibeling, and Icard (2022) for some variants of the standard languages used commonly in probabilistic and causal inference.

Using reinforcement learning for automated theorem proving has recently received much attention. Current approaches use representations of logical statements that often rely on the names used in these statements and, as a result, the models are generally not transferable from one domain to another. The size of these representations and whether to include the whole theory or part of it are other important decisions that affect the performance of these approaches as well as their runtime efficiency. In this paper, we present NIAGRA; an ensemble Name InvAriant Graph RepresentAtion. NIAGRA addresses this problem by using 1) improved Graph Neural Networks for learning name-invariant formula representations that is tailored for their unique characteristics and 2) an efficient ensemble approach for automated theorem proving. Our experimental evaluation shows state-of-the-art performance on multiple datasets from different domains with improvements up to 10% compared to the best learning-based approaches. Furthermore, transfer learning experiments show that our approach significantly outperforms other learning-based approaches by up to 28%.

This survey paper proposes a clearer view of natural language reasoning in the field of Natural Language Processing (NLP), both conceptually and practically. Conceptually, we provide a distinct definition for natural language reasoning in NLP, based on both philosophy and NLP scenarios, discuss what types of tasks require reasoning, and introduce a taxonomy of reasoning. Practically, we conduct a comprehensive literature review on natural language reasoning in NLP, mainly covering classical logical reasoning, natural language inference, multi-hop question answering, and commonsense reasoning. The paper also identifies and views backward reasoning, a powerful paradigm for multi-step reasoning, and introduces defeasible reasoning as one of the most important future directions in natural language reasoning research. We focus on single-modality unstructured natural language text, excluding neuro-symbolic techniques and mathematical reasoning.

The advances in Machine Learning (ML) in recent years have been both impressive and far-reaching. However, the deployment of ML models is still impaired by a lack of trust in how the best-performing ML models make predictions. The issue of lack of trust is even more acute in the uses of ML models in high-risk or safety-critical domains. eXplainable artificial intelligence (XAI) is at the core of ongoing efforts for delivering trustworthy AI. Unfortunately, XAI is riddled with critical misconceptions, that foster distrust instead of building trust. This paper details some of the most visible misconceptions in XAI, and shows how formal methods have been used, both to disprove those misconceptions, but also to devise practically effective alternatives.