Computing with Classical Real Numbers

There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way interfacing these two libraries by showing that their real number structures are isomorphic assuming the classical axioms already present in the standard library reals. This allows us to use O'Connor's decision procedure for solving ground inequalities present in CoRN to solve inequalities about the reals from the Coq standard library, and it allows theorems from the Coq standard library to apply to problem about the CoRN reals

Computing with Exact Real Numbers in a Radix-r System

This paper investigates an arithmetic based upon the representation of computable exact real numbers by lazy infinite sequences of signed digits in a positional radix-r system. We discuss advantages and problems associated with this representation, and develop well-behaved algorithms for a comprehensive range of numeric operations, including the four basic operations of arithmetic

New formats for computing with real-numbers under round-to-nearest

An edited version of this work was accepted in IEEE Transactions on computers, DOI 10.1109/TC.2015.2479623In this paper, a new family of formats to deal with real number for applications requiring round to nearest is proposed. They are based on shifting the set of exactly represented numbers which are used in conventional radix-R number systems. This technique allows performing radix complement and round to nearest without carry propagation with negligible time and hardware cost. Furthermore, the proposed formats have the same storage cost and precision as standard ones. Since conversion to conventional formats simply require appending one extra-digit to the operands, standard circuits may be used to perform arithmetic operations with operands under the new format. We also extend the features of the RN-representation system and carry out a thorough comparison between both representation systems. We conclude that the proposed representation system is generally more adequate to implement systems for computation with real number under round-to-nearest.Ministry of Education and Science of Spain under contracts TIN2013-42253-P

Simulating FRSN P Systems with Real Numbers in P-Lingua on sequential and CUDA platforms

Fuzzy Reasoning Spiking Neural P systems (FRSN P systems, for short) is a variant of Spiking Neural P systems incorporating fuzzy logic elements that make it suitable to model fuzzy diagnosis knowledge and reasoning required for fault diagnosis applications. In this sense, several FRSN P system variants have been proposed, dealing with real numbers, trapezoidal numbers, weights, etc. The model incorporating real numbers was the first introduced [13], presenting promising applications in the field of fault diagnosis of electrical systems. For this variant, a matrix-based algorithm was provided which, when executed on parallel computing platforms, fully exploits the model maximally parallel capacities. In this paper we introduce a P-Lingua framework extension to parse and simulate FRSN P systems with real numbers. Two simulators, implementing a variant of the original matrix-based simulation algorithm, are provided: a sequential one (written in Java), intended to run on traditional CPUs, and a parallel one, intended to run on CUDAenabled devices.Ministerio de Economía y Competitividad TIN2012-3743

Bypassing dynamical systems : A simple way to get the box-counting dimension of the graph of the Weierstrass function

In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by~${\cal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left ( 2\, \pi\,N_b^n\,x \right)$, where~$\lambda$ and~$N_b$ are two real numbers such that~\mbox{$0 1$, using a sequence a graphs that approximate the studied one.Comment: arXiv admin note: substantial text overlap with arXiv:1703.06839, arXiv:1703.0337

Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.Comment: 40 page

Stiefel-Whitney Numbers for Singular Varieties

This paper determines which Stiefel-Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying the F_2-vector space of Stiefel-Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces of RP^3 bundles. These Stiefel-Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of twisted(CP^3) bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.Comment: 17 pages, final revised versio