This paper presents experiments on common knowledge logic, conducted with the help of the proof assistant Coq. The main feature of common knowledge logic is the eponymous modality that says that a group of agents shares a knowledge about a certain proposition in a inductive way. This modality is specified by using a fixpoint approach. Furthermore, from these experiments, we discuss and compare the structure of theorems that can be proved in specific theories that use common knowledge logic. Those structures manifests the interplay between the theory (as implemented in the proof assistant Coq) and the metatheory.
We present an approach for testing student learning outcomes in a course on automated reasoning using the Isabelle proof assistant. The approach allows us to test both general understanding of formal proofs in various logical proof systems and understanding of proofs in the higher-order logic of Isabelle/HOL in particular. The use of Isabelle enables almost automatic grading of large parts of the exam. We explain our approach through a number of example problems, and explain why we believe that each of the kinds of problems we have selected are adequate measures of our intended learning outcom…
We present a system that utilizes machine learning for tactic proof search in the Coq Proof Assistant. In a similar vein as the TacticToe project for HOL4, our system predicts appropriate tactics and finds proofs in the form of tactic scripts. To do this, it learns from previous tactic scripts and how they are applied to proof states. The performance of the system is evaluated on the Coq Standard Library. Currently, our predictor can identify the correct tactic to be applied to a proof state 23.4% of the time. Our proof searcher can fully automatically prove 39.3% of the lemmas. When combined …
Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects deve…
An Encoding of Abstract Dialectical Frameworks into Higher-Order Logic
University of Luxembourg, Department of Computer Science（卢森堡大学计算机科学系）；Universität Greifswald, Institut für Mathematik und Informatik（格赖夫斯瓦尔德大学数学与信息学研究所）
Explanations for description logic (DL) entailments provide important support for the maintenance of large ontologies. The "justifications" usually employed for this purpose in ontology editors pinpoint the parts of the ontology responsible for a given entailment. Proofs for entailments make the intermediate reasoning steps explicit, and thus explain how a consequence can actually be derived. We present an interactive system for exploring description logic proofs, called Evonne, which visualizes proofs of consequences for ontologies written in expressive DLs. We describe the methods used for c…
The CIFF Proof Procedure for Abductive Logic Programming with Constraints: Theory, Implementation and Experiments
Università di Pisa（比萨大学）；Imperial College London（帝国理工学院）；University of Amsterdam（阿姆斯特丹大学）
给出了带约束的溯因逻辑程序设计的CIFF证明过程，并证明了其正确性。CIFF是溯因逻辑编程的IFF证明过程的扩展，放松了对变量量化（允许条件）的原始限制，并结合了约束求解器来处理约束逻辑编程中的数值约束。最后，我们描述了CIFF系统，将其与最先进的溯因系统和答案集求解器进行比较，并展示了如何使用它来编程一些应用程序。(To《逻辑程序设计理论与实践》（Theory and Practice of Logic Programming，TPLP）
The AI Research Assistant: Promise, Peril, and a Proof of Concept
Department of Aerospace Engineering & Engineering Mechanics, and Oden Institute for Computational Engineering & Sciences, University of Texas at Austin（航空航天工程与工程力学系，奥登计算工程与科学研究所，德克萨斯大学奥斯汀分校）
Microsoft Research（微软研究院）；Indian Institute of Science（印度科学学院）；London School of Geometry and Number Theory（伦敦几何与数论学院）；Indian Institute of Science Education and Research（印度科学教育与研究院）
Mathematics formalisation is the task of writing mathematics (i.e., definitions, theorem statements, proofs) in natural language, as found in books and papers, into a formal language that can then be checked for correctness by a program. It is a thriving activity today, however formalisation remains cumbersome. In this paper, we explore the abilities of a large language model (Codex) to help with formalisation in the Lean theorem prover. We find that with careful input-dependent prompt selection and postprocessing, Codex is able to formalise short mathematical statements at undergrad level wit…
InsightGUIDE: An Opinionated AI Assistant for Guided Critical Reading of Scientific Literature
Information Management Systems Institute（信息管理系统研究所）；ATHENA Research Centre（ATHENA研究中心）；Department of Informatics and Telecommunications（信息与电信系）；University of the Peloponnese（希腊皮埃里亚大学）
