tool to calculate patterns from different set assuming every set uses the same rules

Question

I am looking for a tool that takes in sets, for example:

$200 = 300$

$600 = 700$

$44400 = 44500$

And i want the program to tell me the rules of the set, for example 'add $100$'

I need this to test complicated results to find out how it was calculated.

There are tools to calculate the continuation of sets out there. But i have not found one that fits my needs.

Answer 1

The book A=B is about checking identities in general (how do we verify that $A$ is indeed equal to $B$?) and recognizing certain integer sequences (which can be viewed as functions with the natural numbers as domain).

On interesting definition is the one about "closed form":

Definition 8.1.1: A function $f(n)$ is said to be of closed form if it is equal to a linear combination of a fixed number, $r$, say, of hypergeometric terms. The number $r$ must be an absolute constant, i.e., it must be independent of all variables and parameters of the problem.

This is taylored for their pet pet problem, hypergeometric series, but the interesting bit seems to be the finiteness of the term, not unlike the Wikipedia definition closed-form expression.

You seem to expect something along this line, entering some set of pairs $P_i = (x_i, y_i)$ and as answer expecting some function $f$ given by a finite sized expression, which fulfills $(x_i, f(x_i))$ for all the indices $i$.

The naive solution is to provide a list of expected operations and basic functions of which the solution candidate functions can be built from. Then one generates all possible expressions up to a certain size, and checks if the candidate expression matches the given set of data points $P_i$.

In practice there might be a lot of possible expressions to check and the task fails due to this combinatorial explosion.

The A=B book seems to show a problem, where the authors managed to get that sheer size under control by clever reasoning.