In daviddoret/haricot: Algorithm Composition, Manipulation And Analysis

knitr::opts_chunk$set(echo = TRUE)

Notation

Notation | Example | Definition ---------|------------|-------- $(b_{1},b_{2}, \ldots ,b_{d})$ | $(1,0,1,1,0,0,0)$ | A modular binary number of dimension $d$, with $b_{1},b_{2},...,b_{d} \in \mathbb{B}^{1}$ and $d \in \mathbb{N}^{+}$. $b_{1}b_{2} \ldots b_{d}$ | $1011000$ | Simplified notation without the parenthesis and comma separators. ... | $b1011000$ | Constants may be prefixed with "$b$" when used nearby numbers of different bases to avoid confusion. $B^{d}$ | $B^{32}$ | Variables representing modular binary numbers of dimension $d \in \mathbb{N}, d > 0$. When several variables are used, subscript notation may be used and noted $B^{d}_{i}$.
$B[i]$ | $B[5]$ | Bracket notation may be used to reference a specific bit in a modular binary number, with $i \ in \mathbb{N}^{+}$

Definition

In the context of the haricot package, a modular binary number of dimension $d$ noted $B^{d}$ is defined as an ordered set of $d$ elements of $\mathbb{B}^1$. By definition, $B^{d} \in \mathbb{B}^{d}$.

Formal Definition

Let the modular binary number dimension $d$ be a natural number greater than 0: $d \in \mathbb{N}, d > 0$.
Let $B_{d}$ be an ordered n-tuple set $(b_{1}, b_{2}, ..., b_{d})$ where $b_{1}, b_{2}, ..., b_{d} \in \mathbb{B}^{1}$.
To designate such n-tuples, we will use the word modular binary number.
The modular property will be defined later.

Samples

$$ B^{1} = 0, B^{1} \in \mathbb{B}_{4} $$

$$ B^{1} = 1, B^{1} \in \mathbb{B}_{4} $$

$$ B^{4} = 0000, B^{4} \in \mathbb{B}{4} $$ $$ B^{4} = 1010, B^{4} \in \mathbb{B}{4} $$ $$ B^{6} = 000011, B^{6} \in \mathbb{B}{6} $$ $$ B^{6} = 101010, B^{6} \in \mathbb{B}{6} $$ $$ B^{9} = 111000111, B^{9} \in \mathbb{B}_{9} $$

Mapping of modular binary numbers to the natural numbers

Modular binary numbers can be TODO: COMPLETE THIS. Because binary numbers can be mapped to a subset of the natural numbers, we may also partially define modular binary numbers as \eqn{\mathbb{B}^{d} = \left{ n | n \in \mathbb{N}, 0 <= n <= (2^{d})-1 \right}}.

daviddoret/haricot documentation built on May 21, 2019, 1:42 a.m.