Shared benchmarks and datasets have historically played a crucial role in driving advances in large-scale applications of deep learning, e.g. in computer vision ([6]) and natural language processing ([19, 13, 11]). Neural theorem proving is a rapidly developing area which aims to apply techniques from deep learning to interactive theorem proving. To date, most contributions in this area have focused on individual theorem proving systems, each with a separately-implemented mathematics library and with results reported on a dataset-specific test split; examples include the HOList [2], CoqGym [24] and LeanStep [7] theorem proving environments and benchmarks. However, benchmarks from this paradigm are not ideal for measuring the mathematical reasoning ability of neural theorem provers for several reasons. Library-specific train/test splits are siloed by construction, dependent on how theorems and lemmas are split in these libraries, and as such are not directly comparable across systems. Moreover, formal mathematics libraries are closer to software repositories than informal mathematical exposition, and many lemmas are implementation-specific artifacts without precise informal mathematical or cross-system translations. To date, the neural theorem proving community has not organized its efforts around a cross-system benchmark. To address this need and to provide a common resource to research groups working on formal theorem proving, we present miniF2F, a unified cross-system benchmark of formal mathematics of progressively increasing difficulty, centering around Olympiad-level problem statements (AMC, AIME, IMO) as well as high-school and undergraduate maths classes.

The Shapes Constraint Language (SHACL) is the recent W3C recommendation language for validating RDF data, by verifying certain shapes on graphs. Previous work has largely focused on the validation problem and the standard decision problems of satisfiability and containment, crucial for design and optimisation purposes, have only been investigated for simplified versions of SHACL. Moreover, the SHACL specification does not define the semantics of recursively-defined constraints, which led to several alternative recursive semantics being proposed in the literature. The interaction between these different semantics and important decision problems has not been investigated yet. In this article we provide a comprehensive study of the different features of SHACL, by providing a translation to a new first-order language, called SCL, that precisely captures the semantics of SHACL. We also present MSCL, a second-order extension of SCL, which allows us to define, in a single formal logic framework, the main recursive semantics of SHACL. Within this language we also provide an effective treatment of filter constraints which are often neglected in the related literature. Using this logic we provide a detailed map of (un)decidability and complexity results for the satisfiability and containment decision problems for different SHACL fragments. Notably, we prove that both problems are undecidable for the full language, but we present decidable combinations of interesting features, even in the face of recursion.

In this paper, we address the problem of handling inconsistent data in Temporal Description Logic (TDL) knowledge bases. Considering the data part of the knowledge base as the source of inconsistency over time, we propose an ABox repair approach. This is the first work handling the repair in TDL Knowledge bases. To do so, our goal is twofold: 1) detect temporal inconsistencies and 2) propose a data temporal reparation. For the inconsistency detection, we propose a reduction approach from TDL to DL which allows to provide a tight NP-complete upper bound for TDL concept satisfiability and to use highly optimised DL reasoners that can bring precise explanation (the set of inconsistent data assertions). Thereafter, from the obtained explanation, we propose a method for automatically computing the best repair in the temporal setting based on the allowed rigid predicates and the time order of assertions.

The integration of learning and reasoning is one of the key challenges in artificial intelligence and machine learning today, and various communities have been addressing it. That is especially true for the field of neural-symbolic computation (NeSy) [10, 21], where the goal is to integrate symbolic reasoning and neural networks. NeSy already has a long tradition, and it has recently attracted a lot of attention from various communities (cf. the keynotes of Y. Bengio and H. Kautz on this topic at AAAI 2020, the AI Debate [9] between Y. Bengio and G. Marcus). Another domain that has a rich tradition in integrating learning and reasoning is that of statistical relational learning and artificial intelligence (StarAI) [39, 85]. But rather than focusing on integrating logic and neural networks, it is centred around the question of integrating logic with probabilistic reasoning, more specifically probabilistic graphical models. Despite the common interest in combining symbolic reasoning with a basic paradigm for learning, i.e., probabilistic graphical models or neural networks, it is surprising that there are not more interactions between these two fields.

There are some recent approaches and results about the use of answer-set programming for specifying counterfactual interventions on entities under classification, and reasoning about them. These approaches are flexible and modular in that they allow the seamless addition of domain knowledge. Reasoning is enabled by query answering from the answer-set program. The programs can be used to specify and compute responsibility-based numerical scores as attributive explanations for classification results.

This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification. We further identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former. This leads to a refinement of quantified Boolean logic with literal quantification as its primitive.

The language of epistemic specifications and epistemic logic programs extends disjunctive logic programs under the stable model semantics with modal constructs called subjective literals. Using subjective literals, it is possible to check whether a regular literal is true in every or some stable models of the program, those models, in this context also called belief sets, being collected in a set called world view. This allows for representing, within the language, whether some proposition should be understood accordingly to the open or the closed world assumption. Several attempts for capturing the intuitions underlying the language by means of a formal semantics were given, resulting in a multitude of proposals that makes it difficult to understand the current state of the art. In this paper, we provide an overview of the inception of the field and the knowledge representation and reasoning tasks it is suitable for.

Consistent answers to a query from a possibly inconsistent database are answers that are simultaneously retrieved from every possible repair of the database. Repairs are consistent instances that minimally differ from the original inconsistent instance. It has been shown before that database repairs can be specified as the stable models of a disjunctive logic program. In this paper we show how to use the repair programs to transform the problem of consistent query answering into a problem of reasoning w.r.t. a theory written in second-order predicate logic. It also investigated how a first-order theory can be obtained instead by applying second-order quantifier elimination techniques.

Fox News Flash top headlines are here. Check out what's clicking on Foxnews.com. House Speaker Nancy Pelosi and her leadership team are mounting a pressure campaign on centrist Democrats to get them to support their party's budget resolution without a vote on the bipartisan infrastructure bill – and they now have the White House's support for a procedural move to advance them together next week. "Today, President Biden endorsed the House Rule which will allow us to consider the budget resolution, H.R. 4 and the bipartisan infrastructure bill next week," Pelosi, D-Calif., said in a Tuesday letter. "[A]ny delay in passing the budget resolution could threaten our ability to pass this essential legislation through reconciliation. This jeopardizes the once-in-a-generation opportunity we face to enact initiatives that meet the needs of working families at this crucial time."

The language of epistemic specifications and epistemic logic programs extends disjunctive logic programs under the stable model semantics with modal constructs called subjective literals. Using subjective literals, it is possible to check whether a regular literal is true in every or some stable models of the program, those models, in this context also called \emph{belief sets}, being collected in a set called world view. This allows for representing, within the language, whether some proposition should be understood accordingly to the open or the closed world assumption. Several attempts for capturing the intuitions underlying the language by means of a formal semantics were given, resulting in a multitude of proposals that makes it difficult to understand the current state of the art. In this paper, we provide an overview of the inception of the field and the knowledge representation and reasoning tasks it is suitable for. We also provide a detailed analysis of properties of proposed semantics, and an outlook of challenges to be tackled by future research in the area. Under consideration in Theory and Practice of Logic Programming (TPLP)