How can I create an inverse of a function with two inputs?

I created a "distance" function, where I input two values and it finds the numerical distance between them. Is it possible to create a second function where I can input one of my original values and input a numerical value that used to serve as my output and get a new output of all the possible second "inputs" from my original function?

Answer 1

[Walter Roberson]

Is there a finite limited alphabet? Is there a finite limit on how many characters to search through? Is the "distance" reported an integer or rational number?

If the answer to all three of those is Yes, then the answer is Yes: you simply iterate through all the possible words of up to the maximum length, and you calculate the distance of each to the target word, and compare the result to the given distance, retaining the ones that match.

If the answer to the first two is Yes but the third is No, that is if the "distance" is a floating point number, then in the general case, No, the general case says that there is the possibility of words whose distance calculates arbitrarily close to the given distance and might accidentally match, or that words which should match have their distance calculation slightly different than the given distance due to floating point round-off and so do not compare as equal.

If the answer to either of the first two is No, that either the alphabet is not finite or the maximum word length is not finite, then in the general case it would take an infinite amount of time to make the calculation.

The above discusses the general case. Given any particular distance algorithm, it may be possible to reason about its properties to determine the set of words that would match without having to try all the candidates. (Sometimes, though, the computation of which words to try gets so complicated that it is more efficient just to try them all!)

But you didn't ask about any particular distance algorithm, only about the theoretic possibilities for all (presumably finite) distances algorithms.

[bh]

sorry about that, here is my function:

function [e] = letterdist(m,n)

s = double(n - 96);

r = double(m - 96);

d = abs(s - r);

e = sum(d);

It takes in two words and spits out the "distance" between them. What I want to do is be able to input one word and a number and get an output of all the other possible words that are that distance away from the first word. Is that possible?

[Walter Roberson]

For that particular function where everything is integer valued and you can assign definite maximum values, you can approach it by finding all vectors of non-negative integers of the given length whose sum is the given distance. Then, given any such vector, take all combinations of the positive and negative of each entry -- e.g.,

[1 7 2] -> [1 7 2], [1 7 -2], [1 -7 2], [1 -7 -2], [-1 7 2], [-1 7 -2], [-1 -7 2], [-1 -7 -2]

and take each such generated entry add add it to the target word, char() the result, and that gives you a word that is the given distance away.

Your results would be more plausible if you were to restrict your alphabet. For example, '[' is the same distance from 'a' as 'g' is away from 'a', but do you want to allow words with '[' in them?

[bh]

I think that would work, but how could I write the first part in a new program? (sorry i'm a bit new at this)

[Walter Roberson]

It would help if you could explicitly state the valid characters. The hint is there that a "word" should be comprised entirely of lower-case English alphabetic characters, but I do not want to make that assumption.

[bh]

yes, each word is inputted in lowercase English, and converted using ASCii.

[Walter Roberson]

Tagging myself to make it easier to find this again