相关论文: Verified Real Number Calculations: A Library for I…

Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a…

A real number is a rule that, when provided with a rational interval, answers Yes or No depending on if the real number ought to be considered to be in the given interval. Since the goal is to define the real numbers, this can only motivate…

We introduce a method to derive theorems from Elementary Number Theory by means of relationships among formal languages. Using $\sigma$-algebras, we define what a proof of a number-theoretical statement by Language Theory means. We prove…

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…

Automatic and efficient verification of multiplier designs, especially through a provably correct method, is a difficult problem. We show how to utilize a theorem prover, ACL2, to implement an efficient rewriting algorithm for multiplier…

In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce…

Despite the success of large language models (LLMs), the task of theorem proving still remains one of the hardest reasoning tasks that is far from being fully solved. Prior methods using language models have demonstrated promising results,…

In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x…

Mathematical theorem proving is an important testbed for large language models' deep and abstract reasoning capability. This paper focuses on improving LLMs' ability to write proofs in formal languages that permit automated proof…

We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically…

Recent work by Clark et al. (2020) shows that transformers can act as 'soft theorem provers' by answering questions over explicitly provided knowledge in natural language. In our work, we take a step closer to emulating formal theorem…

Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…

How difficult are interactive theorem provers to use? We respond by reviewing the formalization of Hilbert's tenth problem in Isabelle/HOL carried out by an undergraduate research group at Jacobs University Bremen. We argue that, as…

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…

Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…

The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained…

The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…

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…

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the…

Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely.…