Arithmetic computation

I think this name captures the essence of this project.

We often use certain mathematical functions without knowing how they can be derived mathematically. When looking up definitions for functions, they often either have circular dependencies, meaning that they are implicitly defined, or that they use notation that seems to skip or obfuscate important steps which one would need to derive them. An example can be how the piecewise notation can be expressed algebraically.

Why would this be interesting? Well, it's interesting to me because it answers questions that I've had for a while.

But for most of you, it might be more interesting if we talk about it in the context of neural networks. So I have 3 questions below that I will try to answer in this project:

1. If you'd think of creating a neural network that, in simple terms, use additive and multiplicative operations in higher dimensions, wanting it to learn how to algorithmically produce mathematical functions; how would one represent them using, for example, "pure" mathematical notation?

2. We know about the Universal approximation theorem that states that for any function $f(x)$ there exists a subset of neural networks that can approximate that function. So how could one explicitally express a higher mathematical function through "pure" addition/subtraction/multiplication/division operations?

3. If we take a look at how computers have implemented these functions, we see that they are defined in a closed system where logical gates are combined using to combinational logic circuits that can perform arithmetic calculations on binary sequences. So if we instead start with decimal digit sequences, and only use the tools that I've specified below, how would functions like $max(a, b)$, $a \lt x$, piecewise notation or $a\ mod\ b$ be written such that they can be expressed algebraically?

So the goal of this project is to see if it's possible to define mathematical functions with Arithmetic operations, Algebraic expressions, and Finite Series and Sequences.

I'm almost certain that there is literature about this in Number theory, Discrete Mathematics, Combinatorics or Theoretical Computer Science, but I'm not sure it has been posed in this way before.

Toolkit

So we will be using arithmetic additive and multiplicative operations, powers, roots and logarithms, finite summation and product series notation $\Sigma$ and $\Pi$, and limits $\lim_{x \to a}$ to derive integration and derivation $\int$ and $\frac{d}{dx}$. They need to be expressed algebraically.

Elements

We use Reals and Complex numbers in decimal base system.

Introduction

I would like to begin by introducing alternative ways to define functions that are implicitly defined, meaning in this case that they are circular, e.g. floor is defined by modulo, modulo is defined by floor. That includes ceil and fraction functions too.

Trigonometrical implementation

We can actually define these functions trigonometrically, and we only need to define the modulo operation because the rest can be derived from it.

Lets define it as:

$$mod(x, y) = \frac{y\times\cot^{-1}(\cot(\frac{\pi x}{y}))}{\pi}$$

I couldn't find this formula anywhere on the internet. But the gist of it is that it's built on taylor series. I also want to mention that the trigonometric functions are defined at the bottom of the document.

Note that these functions are implemented using range reductions, which are very useful when dealing with periodic functions, because they can scale a big number down such that it gives the same result as we would get if we never scaled it down.

Note that these trigonometric functions are defined with piecewise notation, keep that in mind.

Note that there are alternative ways to define this function too. See further down.

We can now define:

$$ \lbrace x \rbrace = frac(x) = mod(x, 1)$$

This will give us the faction part.

We can also define $trunc(x)$:

$$trunc(x) = x - sign(x) \cdot mod(|x|, 1)$$

$sign(x)$ is defined in the logical expressions section.

So with this, we can define the $\lfloor x \rfloor$ function.

Lets define it as:

$$\lfloor x \rfloor = floor(x) = x - \lbrace x \rbrace$$

We can also define the $\lceil x \rceil$ function now.

Lets define it as:

$$\lceil x \rceil = ceil(x) = x + \lbrace -x \rbrace$$

We can also define $round(x)$, both up and down.

Let us define it as:

$$roundUp(x) = \lfloor x + 0.5 \rfloor$$

Let us define it as:

$$roundDown(x) = \lceil x - 0.5 \rceil$$

We can also define the $isFraction(x)$ function.

Lets define it as:

$$isFraction(x) = \lceil x \rceil - \lfloor x \rfloor$$

Decimal Extraction Implementation

So it's kind of hard to choose a good title because now we are moving away from arithmetical operations a bit and moving into the territory of meta arithmetics. I don't know if that is a thing.

We first want to define a function that can in a sense check how long a sequence is.

We define the $length(x)$ function, which gives us the integer length of a number:

$$length(x) = \lfloor \log_{10}(|x|) \rfloor + 1$$

Note that this only works for the integer part of the number, not the fractional part of the number. I also want to add that this function depends on a max integer size also, so even if this function gives us a length, it only gives use the length if it's below the max interger size.

If building a arithmetic circuit with a max size, we can substitute this function with it.

Now, given the importance of extracting parts from a whole, we will define the $digitAt(x, pos) = number_{position}$, which gives us the integer number at a position.

Lets define it as:

$$digitAt(x, pos) = x_{pos} = \lfloor \frac{\lfloor x - \lfloor \frac{x}{10^{pos}} \rfloor \cdot 10^{pos} \rfloor}{10^{pos - 1}} \rfloor$$

There are definitions that say that you can use $\lfloor \frac{x}{10^{pos - 1}} \rfloor \bmod 10$. But that one does not give accurate digits when working with contiguous binary sequences, so use my version instead.

Intermediate arithmetical logic gates

So I've noticed that there are some operations and functions that are necessary for other things to be derivable.

1. The first one is the absolute function. It has two properties (there are more though), idempotence and symmetric. Note that if we have a complex number $|z|$ would be interpreted as $\sqrt{a^2 + b^2}$. But here we say it's $\sqrt{x^2}$, which then would mean that $\sqrt{(-(real + imaginary))^2} = real + imaginary$.

$$|x|=\sqrt{x^2}$$

3. The second one uses uses the $0^{0}=1$ interpretation, which is important because it gives rise to $(|x| - x)^{|x|}(|x| - x)^{x}$ and $(|x| + x)^{|x|}(|x| + x)^{-x}$, which simplifies to $0^{|x| + x}$ and $0^{|x| - x}$. The power rule states that $0^{x > 0} = 0$, and $0^{x = 0} = 1$. It's important to note that $0^{0}$ is actually indeterminate, and has different definitions depending on the context. In combinatorics, it is defined as being equal to one, but using limits, one could argue for it being equal to either 1 or 0.

$$(|x| - x)^{|x|}(|x| - x)^{x}=(|x| - x)^{x}(|x| - x)^{|x|}=(|x| - x)^{|x|\ +\ x}=0^{|x|\ +\ x}$$

$$\text{\textit{off}}\ (x) = 0^{|x|+x} = \begin{cases} 0 & \text{if } x > 0 \\ 1 & \text{if } x \leq 0\end{cases}$$

$$on(x) = 1 - \text{\textit{off}}\ (x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x \leq 0\end{cases}$$

I've later discovered that these functions I'm defining here are often called step function, heaviside function, boxcar function, but they define it differently.

They are often defined using piecewise notation. The main difference here is that I'm saying that 0 is part of the negative number line and that I have an explicit definition of it.

So I also want to mention that these are also defined as a variation of the Dirac delta function.

$${\displaystyle \delta [n]= 0^{|n|} ={\begin{cases}1&n=0 \\ 0&n{\text{ is another integer}}\end{cases}}}$$

Irreducible expressions

I want to quickly mention that when starting to formalise these expressions you sometimes discover cases where some expressions cannot be simplified (reduced) anymore, because then they'd lose their unique property. So those expressions often get their own math notation.

We can also define it as this:

$$ \text{\textit{off}}\ (-x) = 0.5 + \frac{|x - mod(x, 1) + 0.5|}{2x - 2 \times mod(x, 1) + 1}$$

This would actually require $(|x|\ -\ x)^{|x|\ +\ x}$ because mod (arccot to be specific) uses piecewise notation.

We can also define the $sign(x)$ function. Its codomain looks like this:

$$sign(x) = (-1)^{on(-x)} = \begin{cases} +1 & \text{if } x \geq 0 \\ -1 & \text{if } x < 0 \end{cases}$$

Usually there is a third case here for zero, but I interpret that as being positive instead of just zero.

Arithmetical logic gates

So next up is to define the basic $not(x)$, $and(x, y)$, and $or(x, y)$.

Let us define:

$$not(x) = 1 - on(x) = on(\text{\textit{off}}\ (x))$$

$$and(x, y) = on(x) \times on(y)$$

$$or(x, y) = not(and(not(x), not(y)))$$

$$xor(x, y) = or(and(not(x), y), and(x, not(y)))$$

The $max(a, b)$ function can easily be defined using the previous logic operations.

Let us define it as:

$$max(a, b) = a\cdot on\left(a\ -\ b\right)\ +\ b\cdot\left(1\ -\ on\left(a\ -\ b\right)\right)$$

and the $min(a, b)$ function can then just switch the arguments around.

$$min(a, b) = b\cdot on\left(a\ -\ b\right)\ +\ a\cdot\left(1\ -\ on\left(a\ -\ b\right)\right)$$

Let's do Greater than, Lesser than / or equal.

Greater than or equal:

$$x \geq a = gte(x, a) = 0^{\left|a\ -\ \min\left(x,\ a\right)\right|}$$

Lesser than or equal:

$$x \leq a = lte(x, a) = 0^{\left|a\ -\ \max\left(x,\ a\right)\right|}$$

Greater than:

$$x \gt a = gt(x, a) = 1-lte(x, a)$$

Lesser than:

$$x < a = lt(x, a) = 1-gte(x, a)$$

Equal to:

$$x = a \to gte(x, a) \times lte(x, a) $$

This can also be expressed with the Kronecker delta function. $$\delta_{ij}=\delta_{i}^{j} = \begin{cases} 1 & j = i \ 0 & j \neq i\end{cases}$$

Open interval:

$$between(x, a, b) = 0^{|on(x - a) - on(-x - b)|} =\begin{cases} 1 & \text{if } a \lt x \lt b \\ 0 & \text{else }\end{cases}$$

Half-open interval:

$$ a \leq x \lt b = \lbrack\ a,\ b\ \rparen (x) = gte(x, a) \times lt(x, b) \times x$$

Arithmetic ALU

Now we are getting to the fun parts. The title might be weird because I tried to encapsulate the essence of what this section will be about, and that is; how we represent arithmetic operations using arithmetical expressions algebraically.

$$Add(x, y) = mod(x + y, 10)$$

$$Carry(x, y) = mod(x + y, 1)$$

$$S = max(length(x), length(y))$$

$$Addition(x, y) = \sum_{n=1}^{S}10^{n-1} \cdot Add(x_n, y_n) + Carry(x_{n-1}, y_{n-1})$$

$$Addition(120, 25) = 145$$

Binary to decimal representation

So when we want to show the data, we need to map it into a character set.

$$fromDecimalToBinary(x) = \sum_{n=0}^{\lfloor\log_{2}\left(x\right)\rfloor}10^{n}mod\left(\lfloor\frac{x}{2^{n}}\rfloor,\ 2\right)$$

$$fromDecimalToBinary(85) = 1010101$$

We can do a binary counting function: $$fromBinaryToDecimal(x)=\sum_{n=0}^{length(x)} 2^{n}\cdot on(digitAt(x, n + 1)) $$

$$fromBinaryToDecimal(1010101) = 85$$

We can do a binary16 (IEEE Standard for Floating-Point Arithmetic) to decimal:

$$ fromBinary16ToDecimal(x) = \left(-1\right)^{digitAt\left(x,\ 16\right)} \cdot 2^{^+\langle digitAt\left(x,\ 11\ +\ n\right)\cdot2^{n} \rangle_{n=0}^{4}-15} \cdot \left(1\ +\ \sum_{n=0}^{9}digitAt\left(x,\ 10\ -\ n\right)\cdot2^{-\left(n\ +\ 1\right)}\right)$$

$$fromBinary16ToDecimal(0100001100000000) = 3.5$$

This is just a proof of concept for turning a binary sequence number only containing 1 and 0 into a decimal number.

$$extractDecimalDigitFromBinary(x, pos) = mod(\lfloor \frac{x}{10^{pos - 1}} \rfloor, 10)$$

$$extractDecimalDigitFromBinary(1010101, 1) = 5$$

$$extractDecimalDigitFromBinary(1010101, 2) = 8$$

I know this is not fully binary, but it gives an idea of the process. For example, when we write a program that uses numbers, and then we want to display them in text, the computers needs to first decode each digit from the binary sequence. Such an operation divides up an integer number into decimal parts that are still represented in binary. So a binary number will be split into $\lfloor length(1010101) \cdot \log_{10}2\rfloor = 2$ binary sequences.

Piecewise functions

But now we can do Piecewise functions with "pure" math.

$$between(x, 0, 4) \times x^{2} = \begin{cases} x^2 & 3 \lt x \lt 6 \\ 0 & \text{else }\end{cases}$$

TODO, give more examples. ....

Special Fourier series

Rectangular function can be defined using the logical operations above. But it's also possible to define it using a logistic function IIF we have defined $\frac{x}{0} = \infty$, and $\frac{x}{\infty} = 0$.

$$rect(x) = \frac{1}{1 + 0^{x + 0.5}} - \frac{1}{1 + 0^{x - 0.5}} = \sum_{|x|}^{0}1$$

The summation and product notation are OP, e.g. $\sum_{|x|}^{0}1$

TODO, give more examples.

Left over functions

$$rect(x) = on(x + 0.5) \cdot off(x - 0.5)$$

$$round(x) = \sum_{k=-\infty}^{\infty}k\ \cdot\ rect\left(x-k\right)$$

$$ precision(x, p) = on(x + p) \cdot off(x - (1 - p)) $$

$$floor(x) = \sum_{k=-\infty}^{\infty}k\ \cdot\ precision\left(x-k, 0.001\right)$$

$$w^{x} \equiv 0^{|x| - x}$$

$$floor(x) = \sum_{k=-\infty}^{\infty}k[w^{x - k} - w^{x - k - 1}]$$

This is my own notation for sigma notation.

$^+\langle k[w^{x - k} - w^{x - k - 1}] \rangle _{k\ =\ -\infty}^{\infty}$

$^+\overset{k\ =\ \overset{-}{\infty}\ \to\ \overset{+}{\infty}}{\langle k[w^{x - k} - w^{x - k - 1}] \rangle}$

From Zero-Product Rule to Iverson Brackets

So while writing all of this, I've noticed that in when formalising these expressions, there are times where we do a lot of unnecessary calculations. In fact, many expressions will evaluate to zero which makes any other expression multiplied by that zero by default.

I was thinking that we might be able to indicate when an expression is more likely to be zero such that we then can evaluate it first, and then determine if we need to continue down that branch. So I wanted to introduce a new notaion to express this idea ((expression)), which when evaluated to be zero, any expression multiplied by it would automatically resolve to zero, even if that expression was undefined. It would in a sense act as a halt for that evaluation branch.

I tried to look up definitions for zero-product rule and null factor law, to see if there was any explicit rule for this, but they didn't mention anything about it directly directly, it was just something one could deduce from it. So I wrote a little about it, thinking I could define this myself only to discover by accident that these things have already been studied and defined.

Apparently, Iverson bracket notation is this idea, which is a generalization of Kronecker delta function. I will get into it what they are further down, but I just found it to be very interesting given that this is what I intended to write about.

Generalized functions and test functions, all of which I never really knew existed, nor claim to understand. But I've come to understand a little of what these ideas can be used for, so I have found it quite amusing to first have thought of them my self, and tried to formalise them for this project, only to later discover that they have already been formalised.

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Base formulas

NOTE: There're certain Analytical continuations for certain functions, but that's not the point of this.

$$ln(x) = 2 \times \sum_{n=0}^{\infty} \frac{1}{2n + 1}(\frac{x - 1}{x + 1})^{2n+1}$$

$$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$$

$$b^{x} = e^{x \times ln(b)}$$

$$x! = \lim_{N \to \infty} N^x\prod^{N}_{k=1} \frac{k}{k + x}$$

$$sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n + 1)!} x^{2n + 1} = \frac{x}{(\frac{x}{\pi})!(-\frac{x}{\pi})!}$$

$$cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!} x^{2n} = sin(x + \frac{\pi}{2})$$

$$\pi cot(\pi x)= \lim_{N \to \infty} \sum_{n=-N}^{N} \frac{1}{x + n} = \pi\frac{\sin\left(x\pi\ -\frac{\pi}{2}\right)}{\cos\left(x\pi+\frac{\pi}{2}\right)}$$

$$tan^{-1}(x)=\sum_{n = 0}^{\infty} \frac{(-1)^n}{2n + 1} x^{2n + 1} = \int_{0}^{n}\frac{1}{1\ +\ x^{2}}dx$$

$$ cot^{-1}(x) = \begin{cases} \frac{\pi}{2} - tan^{-1}(x) & \text{if } -1 \leq x \leq 1 \\ tan^{-1}(\frac{1}{x}) & \text{if } x \geq 1 \\ \pi + tan^{-1}(\frac{1}{x}) & \text{if } x \leq -1 \end{cases} = \frac{\pi}{2} - \int_{0}^{n}\frac{1}{1\ +\ x^{2}}dx $$

$$\int^{\infty}_{0} t^{x-1}e^{-t}dt$$

$$N^x\prod^{\infty}_{k=1} \frac{k}{k + x}$$

Bernoulli number

Explicit formula

$$B_{n} = \sum^{n}{k=0}{\sum^{k}{j=0}{(-1)^j \binom{k}{j} \frac{j^n}{k + 1}}}$$

Recursive Formula

$$ B_{n} = -\sum^{n - 1}_{k=0} \binom{n}{k} \frac{B_k}{n - k + 1} $$

I want to expand on these later and use them to formulate alternative functions to modulo operation, etc.

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$$ {\begin{aligned}B_{m}^{+}&=-{\frac {1}{m+1}}\sum {k=0}^{m-1}{\binom {m+1}{k}}B{k}^{+}\end{aligned}} $$

etc..

Notes

Most of these operations and functions work because of infinite series. So to make this computable we should define a maximum number length so that we know how to specify the series so that they correctly converges.

For periodic functions we also would use range reduction.

The base formulas are defined at the bottom. TODO: Expand these subject

1. Numerical stability
2. Trigonometry functions
3. Gamma function
4. Numerical Integration