相关论文: Computing with Classical Real Numbers

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…

Software certification aims at proving the correctness of programs but in many cases, the use of external libraries allows only a conditional proof: it depends on the assumption that the libraries meet their specifications. In particular, a…

We present several steps towards large formal mathematical wikis. The Coq proof assistant together with the CoRN repository are added to the pool of systems handled by the general wiki system described in \cite{DBLP:conf/aisc/UrbanARG10}. A…

Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique…

We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…

This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with…

An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and…

We describe the formalisation in Coq of a proof that the numbers e and $\pi$ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex…

Recent research into analog computing has introduced new notions of computing real numbers. Huang, Klinge, Lathrop, Li, and Lutz defined a notion of computing real numbers in real-time with chemical reaction networks (CRNs), introducing the…

Conventional canonical quantization procedures directly link various c-number and q-number quantities. Here, we advocate a different association of classical and quantum quantities that renders classical theory a natural subset of quantum…

Hybrid Quantum Neural Networks (HQNNs) have gained attention for their potential to enhance computational performance by incorporating quantum layers into classical neural network (NN) architectures. However, a key question remains: Do…

Verifying the functional correctness of programs with both classical and quantum constructs is a challenging task. The presence of probabilistic behaviour entailed by quantum measurements and unbounded while loops complicate the…

Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $\mathbb{Q}(a_1,\ldots,a_n)$ where the extensions numbers $a_k$ may be…

While quantum architectures are still under development, when available, they will only be able to process quantum data when machine learning algorithms can only process numerical data. Therefore, in the issues of classification or…

Quantum computing represents a paradigm shift for computation requiring an entirely new computer architecture. However, there is much that can be learned from traditional classical computer engineering. In this paper, we describe the…

The power of quantum computers is still somewhat speculative. While they are certainly faster than classical ones at some tasks, the class of problems they can efficiently solve has not been mapped definitively onto known classical…

The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bound than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that…

In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence…

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…

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…