相关论文: Finding Proofs in Tarskian Geometry

We give a method to describe all congruence images of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if…

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…

In this paper I present a kind of proof for classical Euclidean geometric problems which relies on both synthetic and analytic geometry. Using the elementary tools of polynomial algebra and multivariate calculus we manage to reduce the…

Introduction to the special issue of Phil. Trans. R. Soc. A 376, 2018, `Hilbert's Sixth Problem'. The essence of the Sixth Problem is discussed and the content of this issue is introduced. In 1900, David Hilbert presented 23 problems for…

Theorem proving is a fundamental aspect of mathematics, spanning from informal reasoning in natural language to rigorous derivations in formal systems. In recent years, the advancement of deep learning, especially the emergence of large…

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous…

The Axiom-Based Atlas is a novel framework that structurally represents mathematical theorems as proof vectors over foundational axiom systems. By mapping the logical dependencies of theorems onto vectors indexed by axioms - such as those…

Large vision language models exhibit notable limitations on Geometry Problem Solving (GPS) because of their unreliable diagram interpretation and pure natural-language reasoning. A recent line of work mitigates this by using symbolic…

The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $\delta$ on the space $\mathcal{S}$ of coarse similarity classes defined by letting…

Proving geometric theorems constitutes a hallmark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level…

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order…

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…

In this paper we present efficient algorithms for the computation of several invariant objects for Hamiltonian dynamics. More precisely, we consider KAM tori (i.e diffeomorphic copies of the torus such that the motion on them is conjugated…

The Gromov-Wasserstein (GW) framework adapts ideas from optimal transport to allow for the comparison of probability distributions defined on different metric spaces. Scalable computation of GW distances and associated matchings on graphs…

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element…

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two…

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the s.c. "gravitational theories with covariant and contravariant connection and metrics", it is…

The Tarskian classical relevant logic TR arises from Tarski's work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of…

An approach is shown that proves various theorems of plane geometry in an algorithmic manner. The approach affords transparent proofs of a generalization of the Theorem of Morley and other well known results by casting them in terms of…

We present a benchmark of 29687 problems derived from the On-Line Encyclopedia of Integer Sequences (OEIS). Each problem expresses the equivalence of two syntactically different programs generating the same OEIS sequence. Such programs were…