Related papers: Certifying rings of integers in number fields
Chain-of-Thought (CoT) prompting has become the de facto method to elicit reasoning capabilities from large language models (LLMs). However, to mitigate hallucinations in CoT that are notoriously difficult to detect, current methods such as…
In this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a…
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…
Authentication is a process by which an entity,which could be a person or intended computer,establishes its identity to another entity.In private and public computer networks including the Internet,authentication is commonly done through…
Neural networks are increasingly deployed in scientific, safety critical, and mission critical pipelines, yet verification and analysis are often performed outside the programming environment that defines and runs the model. This creates a…
Chain-of-Thought reasoning has emerged as a powerful approach for solving complex mathematical and logical problems. However, it can often veer off track through incorrect or unsubstantiated inferences. Formal mathematical reasoning, which…
Despite the vast body of research literature proposing algorithms with formal guarantees, the amount of verifiable code in today's systems remains minimal. This discrepancy stems from the inherent difficulty of verifying code, particularly…
For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal…
In two-party secret sharing scheme, values are typically encoded as unsigned integers $\mathsf{uint}(x)$, whereas real-world applications often require computations on signed real numbers $\mathsf{Real}(x)$. To enable secure evaluation of…
In this paper, we study the employment of $\Sigma_1$-sentences with certificates, i.e., $\Sigma_1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some…
We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems…
We introduce MLFMF, a collection of data sets for benchmarking recommendation systems used to support formalization of mathematics with proof assistants. These systems help humans identify which previous entries (theorems, constructions,…
The fusion rules and modular matrix of a rational conformal field theory obey a list of properties. We use these properties to classify rational conformal field theories with not more than six primary fields and small values of the fusion…
Optimization is used extensively in engineering, industry, and finance, and various methods are used to transform problems to the point where they are amenable to solution by numerical methods. We describe progress towards developing a…
We present a framework for verifying the deterministic structured computations surrounding a large language model rather than the model itself, extending a Lean 4 trust-boundary architecture to the generic interfaces of modern LLM…
In the digital age, ensuring the correctness, safety, and reliability of software through formal verification is paramount, particularly as software increasingly underpins critical infrastructure. Formal verification, split into theorem…
We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.
The precise control of complex quantum systems promises numerous technological applications including digital quantum computing. The complexity of such devices renders the certification of their correct functioning a challenge. To address…
We consider the signatures $\Sigma_m=(0,1,-,+, \cdot, \ ^{-1})$ of meadows and $(\Sigma_m, {\mathbf s})$ of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these…
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…