Artifical intelligence and inherent mathematical difficulty

• URL: http://arxiv.org/abs/2408.03345v1
• Date: Thu, 1 Aug 2024 20:08:31 GMT
• Title: Artifical intelligence and inherent mathematical difficulty
• Authors: Walter Dean, Alberto Naibo,
• Abstract summary: We first present an updated version of a traditional argument that limitative results from computability and complexity theory show that proof discovery is an inherently difficult problem. We then illustrate how several recent applications of artificial intelligence-inspired methods do indeed raise novel questions about the nature of mathematical proof.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper explores the relationship of artificial intelligence to the task of resolving open questions in mathematics. We first present an updated version of a traditional argument that limitative results from computability and complexity theory show that proof discovery is an inherently difficult problem. We then illustrate how several recent applications of artificial intelligence-inspired methods -- respectively involving automated theorem proving, SAT-solvers, and large language models -- do indeed raise novel questions about the nature of mathematical proof. We also argue that the results obtained by such techniques do not tell against our basic argument. This is so because they are embodiments of brute force search and are thus capable of deciding only statements of low logical complexity.

Related papers

• Counterfactual and Semifactual Explanations in Abstract Argumentation: Formal Foundations, Complexity and Computation [19.799266797193344]
  Argumentation-based systems often lack explainability while supporting decision-making processes. Counterfactual and semifactual explanations are interpretability techniques. We show that counterfactual and semifactual queries can be encoded in weak-constrained Argumentation Framework.
  arXiv  Detail & Related papers  (2024-05-07T07:27:27Z)
• Apriori Knowledge in an Era of Computational Opacity: The Role of AI in Mathematical Discovery [0.7673339435080445]
  We argue that if a proof-checker is attached to such machines, then we can obtain apriori mathematical knowledge from them. Many accept that we can acquire genuine mathematical knowledge of the Four Color Theorem from Appel and Haken's program.
  arXiv  Detail & Related papers  (2024-03-15T21:38:26Z)
• Soft Reasoning on Uncertain Knowledge Graphs [85.1968214421899]
  We study the setting of soft queries on uncertain knowledge, which is motivated by the establishment of soft constraint programming. We propose an ML-based approach with both forward inference and backward calibration to answer soft queries on large-scale, incomplete, and uncertain knowledge graphs.
  arXiv  Detail & Related papers  (2024-03-03T13:13:53Z)
• A Hybrid System for Systematic Generalization in Simple Arithmetic Problems [70.91780996370326]
  We propose a hybrid system capable of solving arithmetic problems that require compositional and systematic reasoning over sequences of symbols. We show that the proposed system can accurately solve nested arithmetical expressions even when trained only on a subset including the simplest cases.
  arXiv  Detail & Related papers  (2023-06-29T18:35:41Z)
• A Survey of Deep Learning for Mathematical Reasoning [71.88150173381153]
  We review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. Recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning.
  arXiv  Detail & Related papers  (2022-12-20T18:46:16Z)
• Pushing the Limits of Rule Reasoning in Transformers through Natural Language Satisfiability [30.01308882849197]
  We propose a new methodology for creating challenging algorithmic reasoning datasets. Key idea is to draw insights from empirical sampling of hard propositional SAT problems and from complexity-theoretic studies of language. We find that current transformers, given sufficient training data, are surprisingly robust at solving the resulting NLSat problems.
  arXiv  Detail & Related papers  (2021-12-16T17:47:20Z)
• Design of quantum optical experiments with logic artificial intelligence [1.6114012813668934]
  We propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state. We find that the use of logic AI improves significantly the resolution of this problem.
  arXiv  Detail & Related papers  (2021-09-27T18:01:08Z)
• Recognizing and Verifying Mathematical Equations using Multiplicative Differential Neural Units [86.9207811656179]
  We show that memory-augmented neural networks (NNs) can achieve higher-order, memory-augmented extrapolation, stable performance, and faster convergence. Our models achieve a 1.53% average improvement over current state-of-the-art methods in equation verification and achieve a 2.22% Top-1 average accuracy and 2.96% Top-5 average accuracy for equation completion.
  arXiv  Detail & Related papers  (2021-04-07T03:50:11Z)
• NaturalProofs: Mathematical Theorem Proving in Natural Language [132.99913141409968]
  We develop NaturalProofs, a multi-domain corpus of mathematical statements and their proofs. NaturalProofs unifies broad coverage, deep coverage, and low-resource mathematical sources. We benchmark strong neural methods on mathematical reference retrieval and generation tasks.
  arXiv  Detail & Related papers  (2021-03-24T03:14:48Z)
• Machine Number Sense: A Dataset of Visual Arithmetic Problems for Abstract and Relational Reasoning [95.18337034090648]
  We propose a dataset, Machine Number Sense (MNS), consisting of visual arithmetic problems automatically generated using a grammar model--And-Or Graph (AOG) These visual arithmetic problems are in the form of geometric figures. We benchmark the MNS dataset using four predominant neural network models as baselines in this visual reasoning task.
  arXiv  Detail & Related papers  (2020-04-25T17:14:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.