相关论文: Proof-checking Euclid
The standard cosmological model is based on the fundamental assumptions of a spatially homogeneous and isotropic universe on large scales. An observational detection of a violation of these assumptions at any redshift would immediately…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
To constrain models beyond $\Lambda$CDM, the development of the Euclid analysis pipeline requires simulations that capture the nonlinear phenomenology of such models. We present an overview of numerical methods and $N$-body simulation codes…
We present a novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking. The benefits of this format are demonstrated by implementing a proof checker in C, which outperforms…
We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic…
We present and analyze the employment of the Diproche system, a natural language proof checker, within a one-semester mathematics beginners lecture with 228 participants. The system is used to check the students' solution attempts to…
G\"odel's first and second incompleteness theorems are corner stones of modern mathematics. In this article we present a new proof of these theorems for ZFC and theories containing ZFC, using Chaitin's incompleteness theorem and a very…
Explicit theory axioms are added by a saturation-based theorem prover as one of the techniques for supporting theory reasoning. While simple and effective, adding theory axioms can also pollute the search space with many irrelevant…
We investigate how large language models can be used as research tools in scientific computing while preserving mathematical rigor. We propose a human-in-the-loop workflow for interactive theorem proving and discovery with LLMs. Human…
The 2-point correlation function of the galaxy spatial distribution is a major cosmological observable that enables constraints on the dynamics and geometry of the Universe. The Euclid mission aims at performing an extensive spectroscopic…
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
For formulas of the Implicational Propositional Calculus (IPC) that are theorems of the classical Propositional Calculus (PC) we show that PC proofs yield IPC proofs. As a consequence, completeness of PC yields completeness of IPC.
Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is…
Computational Logic is the use of computers to establish facts in a logical formalism. Originating in 19th-century attempts to understand the nature of mathematical reasoning, the subject now comprises a wide variety of formalisms,…
This paper describes the formal verification of two Turing machines using the program verifier Dafny. Both machines are deciders, so we prove total correctness. They are typical first examples of Turing machines used in any course of…
The study of propositional logic -- fundamental to the theory of computing -- is a cornerstone of the undergraduate computer science curriculum. Learning to solve logical proofs requires repeated guided practice, but undergraduate students…
We present a search for strong gravitational lenses in Euclid imaging with high stellar velocity dispersion ($\sigma_\nu > 180$ km/s) reported by SDSS and DESI. We performed expert visual inspection and classification of $11\,660$ \Euclid…
We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).
We present the results of a large-scale computational analysis of mathematical papers from the ArXiv repository, demonstrating a comprehensive system that not only detects mathematical errors but provides complete referee reports with…
Cyp (Check Your Proofs) (Durner and Noschinski 2013; Traytel 2019) verifies proofs about Haskell-like programs. We extended Cyp with a pattern matcher for programs and proof terms, and a type checker. This allows to use Cyp for auto-grading…