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1. Model-Finding for Externally Verifying FOL Ontologies: A Study of Spatial Ontologies

   https://doi.org/10.3233/FAIA200675  

   Stephen, S.; Hahmann, T. (January 2020, Proc. of the International Conference on Formal Ontology in Information System (FOIS-2020))

   Use and reuse of an ontology requires prior ontology verification which encompasses, at least, proving that the ontology is internally consistent and consistent with representative datasets. First-order logic (FOL) model finders are among the only available tools to aid us in this undertaking, but proving consistency of FOL ontologies is theoretically intractable while also rarely succeeding in practice, with FOL model finders scaling even worse than FOL theorem provers. This issue is further exacerbated when verifying FOL ontologies against datasets, which requires constructing models with larger domain sizes. This paper presents a first systematic study of the general feasibility of SAT-based model finding with FOL ontologies. We use select spatial ontologies and carefully controlled synthetic datasets to identify key measures that determine the size and difficulty of the resulting SAT problems. We experimentally show that these measures are closely correlated with the runtimes of Vampire and Paradox, two state-of-the-art model finders. We propose a definition elimination technique and demonstrate that it can be a highly effective measure for reducing the problem size and improving the runtime and scalability of model finding. 

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2. The 10th IJCAR automated theorem proving system competition – CASC-J10

   https://doi.org/10.3233/AIC-201566  

   Sutcliffe, Geoff (March 2021, AI Communications)

   null (Ed.)

   The CADE ATP System Competition (CASC) is the annual evaluation of fully automatic, classical logic Automated Theorem Proving (ATP) systems. CASC-J10 was the twenty-fifth competition in the CASC series. Twenty-four ATP systems and system variants competed in the various competition divisions. This paper presents an outline of the competition design, and a commentated summary of the results. 

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3. Beta Embeddings for Multi-Hop Logical Reasoning in Knowledge Graphs

   Ren, Hongyu; Leskovec, Jure. (January 2020, Neural Information Processing Systems (NeurIPS))

   One of the fundamental problems in Artificial Intelligence is to perform complex multi-hop logical reasoning over the facts captured by a knowledge graph (KG). This problem is challenging, because KGs can be massive and incomplete. Recent approaches embed KG entities in a low dimensional space and then use these embeddings to find the answer entities. However, it has been an outstanding challenge of how to handle arbitrary first-order logic (FOL) queries as present methods are limited to only a subset of FOL operators. In particular, the negation operator is not supported. An additional limitation of present methods is also that they cannot naturally model uncertainty. Here, we present BETAE, a probabilistic embedding framework for answering arbitrary FOL queries over KGs. BETAE is the first method that can handle a complete set of first-order logical operations: conjunction (∧), disjunction (∨), and negation (¬). A key insight of BETAE is to use probabilistic distributions with bounded support, specifically the Beta distribution, and embed queries/entities as distributions, which as a consequence allows us to also faithfully model uncertainty. Logical operations are performed in the embedding space by neural operators over the probabilistic embeddings. We demonstrate the performance of BETAE on answering arbitrary FOL queries on three large, incomplete KGs. While being more general, BETAE also increases relative performance by up to 25.4% over the current state-of-the-art KG reasoning methods that can only handle conjunctive queries without negation. 

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4. Fuzzy Logic Based Logical Query Answering on Knowledge Graphs

   https://doi.org/10.1609/aaai.v36i4.20310  

   Chen, Xuelu; Hu, Ziniu; Sun, Yizhou (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   Answering complex First-Order Logical (FOL) queries on large-scale incomplete knowledge graphs (KGs) is an important yet challenging task. Recent advances embed logical queries and KG entities in the same space and conduct query answering via dense similarity search. However, most logical operators designed in previous studies do not satisfy the axiomatic system of classical logic, limiting their performance. Moreover, these logical operators are parameterized and thus require many complex FOL queries as training data, which are often arduous to collect or even inaccessible in most real-world KGs. We thus present FuzzQE, a fuzzy logic based logical query embedding framework for answering FOL queries over KGs. FuzzQE follows fuzzy logic to define logical operators in a principled and learning-free manner, where only entity and relation embeddings require learning. FuzzQE can further benefit from labeled complex logical queries for training. Extensive experiments on two benchmark datasets demonstrate that FuzzQE provides significantly better performance in answering FOL queries compared to state-of-the-art methods. In addition, FuzzQE trained with only KG link prediction can achieve comparable performance to those trained with extra complex query data. 

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5. Automatically Extracting OWL Versions of FOL Ontologies

   Hahmann, Torsten; Powell, Robert (October 2021, International Semantic Web Conference (ISWC 2021))

   null (Ed.)

   While OWL and RDF are by far the most popular logic-based languages for Semantic Web Ontologies, some well-designed ontologies are only available in languages with a much richer expressivity, such as first-order logic (FOL) or the ISO standard Common Logic. This inhibits reuse of these ontologies by the wider Semantic Web Community. While converting OWL ontologies to FOL is straightforward, the reverse problem of finding the closest OWL approximation of an FOL ontology is undecidable. However, for most practical purposes, a ``good enough'' OWL approximation need not be perfect to enable wider reuse by the Semantic Web Community. This paper outlines such a conversion approach by first normalizing FOL sentences into a function-free prenex conjunctive normal (FF-PCNF) that strips away minor syntactic differences and then applying a pattern-based approach to identify common OWL axioms. It is tested on the over 2,000 FOL ontologies from the Common Logic Ontology Repository. 

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