Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reasoning owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in propositional SAT solving, the past two decades have witnessed the emergence of well-engineered systems for solving the answer set satisfiability problem, i.e., finding models or answer sets for a given answer set program. In recent years, there has been growing interest in problems beyond satisfiability, such as model counting, in the context of ASP. Akin to the early days of propositional model counting, state-of-the-art exact answer set counters do not scale well beyond small instances. Exact ASP counters struggle with handling larger input formulas. The primary contribution of this paper is a new ASP counting framework, called sharpASP, which counts answer sets avoiding larger input formulas. This relies on an alternative way of defining answer sets that allows for the lifting of key techniques developed in the context of propositional model counting. Our extensive empirical analysis over 1470 benchmarks demonstrates significant performance gain over current state-of-the-art exact answer set counters. Specifically, by using sharpASP, we were able to solve 1062 benchmarks with PAR2 score of 3082 whereas using prior state-of-the-art, we could only solve 895 benchmarks with a PAR2 score of 4205, all other experimental conditions being the same.

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

Logical reasoning is central to complex human activities, such as thinking, debating, and planning; it is also a central component of many AI systems as well. In this paper, we investigate the extent to which encoder-only transformer language models (LMs) can reason according to logical rules. We ask whether those LMs can deduce theorems in propositional calculus and first-order logic; if their relative success in these problems reflects general logical capabilities; and which layers contribute the most to the task. First, we show for several encoder-only LMs that they can be trained, to a reasonable degree, to determine logical validity on various datasets. Next, by cross-probing fine-tuned models on these datasets, we show that LMs have difficulty in transferring their putative logical reasoning ability, which suggests that they may have learned dataset-specific features, instead of a general capability. Finally, we conduct a layerwise probing experiment, which shows that the hypothesis classification task is mostly solved through higher layers.

Neural-symbolic learning, an intersection of neural networks and symbolic reasoning, aims to blend neural networks' learning capabilities with symbolic AI's interpretability and reasoning. This paper introduces an approach designed to improve the performance of neural models in learning reasoning tasks. It achieves this by integrating Answer Set Programming (ASP) solvers and domain-specific expertise, which is an approach that diverges from traditional complex neural-symbolic models. In this paper, a shallow artificial neural network (ANN) is specifically trained to solve Sudoku puzzles with minimal training data. The model has a unique loss function that integrates losses calculated using the ASP solver outputs, effectively enhancing its training efficiency. Most notably, the model shows a significant improvement in solving Sudoku puzzles using only 12 puzzles for training and testing without hyperparameter tuning. This advancement indicates that the model's enhanced reasoning capabilities have practical applications, extending well beyond Sudoku puzzles to potentially include a variety of other domains. The code can be found on GitHub: https://github.com/Fadi2200/ASPEN.

Substantial efforts have been made in developing various Decision Modeling formalisms, both from industry and academia. A challenging problem is that of expressing decision knowledge in the context of incomplete knowledge. In such contexts, decisions depend on what is known or not known. We argue that none of the existing formalisms for modeling decisions are capable of correctly capturing the epistemic nature of such decisions, inevitably causing issues in situations of uncertainty. This paper presents a new language for modeling decisions with incomplete knowledge. It combines three principles: stratification, autoepistemic logic, and definitions. A knowledge base in this language is a hierarchy of epistemic theories, where each component theory may epistemically reason on the knowledge in lower theories, and decisions are made using definitions with epistemic conditions.

In the article, within the framework of the Boolean Satisfiability problem (SAT), the problem of estimating the hardness of specific Boolean formulas w.r.t. a specific complete SAT solving algorithm is considered. Based on the well-known Strong Backdoor Set (SBS) concept, we introduce the notion of decomposition hardness (d-hardness). If $B$ is an arbitrary subset of the set of variables occurring in a SAT formula $C$, and $A$ is an arbitrary complete SAT solver , then the d-hardness expresses an estimate of the hardness of $C$ w.r.t. $A$ and $B$. We show that the d-hardness of $C$ w.r.t. a particular $B$ can be expressed in terms of the expected value of a special random variable associated with $A$, $B$, and $C$. For its computational evaluation, algorithms based on the Monte Carlo method can be used. The problem of finding $B$ with the minimum value of d-hardness is formulated as an optimization problem for a pseudo-Boolean function whose values are calculated as a result of a probabilistic experiment. To minimize this function, we use evolutionary algorithms. In the experimental part, we demonstrate the applicability of the concept of d-hardness and the methods of its estimation to solving hard unsatisfiable SAT instances.

This work presents insights gained by investigating the relationship between algorithmic fairness and the concept of secure information flow. The problem of enforcing secure information flow is well-studied in the context of information security: If secret information may "flow" through an algorithm or program in such a way that it can influence the program's output, then that is considered insecure information flow as attackers could potentially observe (parts of) the secret. There is a strong correspondence between secure information flow and algorithmic fairness: if protected attributes such as race, gender, or age are treated as secret program inputs, then secure information flow means that these ``secret'' attributes cannot influence the result of a program. While most research in algorithmic fairness evaluation concentrates on studying the impact of algorithms (often treating the algorithm as a black-box), the concepts derived from information flow can be used both for the analysis of disparate treatment as well as disparate impact w.r.t. a structural causal model. In this paper, we examine the relationship between quantitative as well as qualitative information-flow properties and fairness. Moreover, based on this duality, we derive a new quantitative notion of fairness called fairness spread, which can be easily analyzed using quantitative information flow and which strongly relates to counterfactual fairness. We demonstrate that off-the-shelf tools for information-flow properties can be used in order to formally analyze a program's algorithmic fairness properties, including the new notion of fairness spread as well as established notions such as demographic parity.

The rise of AI-based and autonomous systems is raising concerns and apprehension due to potential negative repercussions stemming from their behavior or decisions. These systems must be designed to comply with the human contexts in which they will operate. To this extent, Townsend et al. (2022) introduce the concept of SLEEC (social, legal, ethical, empathetic, or cultural) rules that aim to facilitate the formulation, verification, and enforcement of the rules AI-based and autonomous systems should obey. They lay out a methodology to elicit them and to let philosophers, lawyers, domain experts, and others to formulate them in natural language. To enable their effective use in AI systems, it is necessary to translate these rules systematically into a formal language that supports automated reasoning. In this study, we first conduct a linguistic analysis of the SLEEC rules pattern, which justifies the translation of SLEEC rules into classical logic. Then we investigate the computational complexity of reasoning about SLEEC rules and show how logical programming frameworks can be employed to implement SLEEC rules in practical scenarios. The result is a readily applicable strategy for implementing AI systems that conform to norms expressed as SLEEC rules.

Deep neural learning uses an increasing amount of computation and data to solve very specific problems. By stark contrast, human minds solve a wide range of problems using a fixed amount of computation and limited experience. One ability that seems crucial to this kind of general intelligence is meta-reasoning, i.e., our ability to reason about reasoning. To make deep learning do more from less, we propose the first neural meta-symbolic system (NEMESYS) for reasoning and learning: meta programming using differentiable forward-chaining reasoning in first-order logic. Differentiable meta programming naturally allows NEMESYS to reason and learn several tasks efficiently. This is different from performing object-level deep reasoning and learning, which refers in some way to entities external to the system. In contrast, NEMESYS enables self-introspection, lifting from object- to meta-level reasoning and vice versa. In our extensive experiments, we demonstrate that NEMESYS can solve different kinds of tasks by adapting the meta-level programs without modifying the internal reasoning system. Moreover, we show that NEMESYS can learn meta-level programs given examples. This is difficult, if not impossible, for standard differentiable logic programming

The synthesis of string transformation programs from input-output examples utilizes various techniques, all based on an inductive bias that comprises a restricted set of basic operators to be combined. A new algorithm, Transduce, is proposed, which is founded on the construction of abstract transduction grammars and their generalization. We experimentally demonstrate that Transduce can learn positional transformations efficiently from one or two positive examples without inductive bias, achieving a success rate higher than the current state of the art.