Sometimes a step in a calculation can be reversed, i.e., [imath]y/2 = x \implies y = 2x[/imath], and sometimes it cannot, i.e., [imath]√y = x \implies y = x^2[/imath] (where [imath]√[/imath] is the principle root). At least in this sort of case, the reversibility follows from the relation between the two variables involved being a bijection. Suppose I perform some calculations, and then designate a set of steps performed which subjectively strike me as important, as Step A, Step B, etc.
I want to understand which of these steps can be derived from which others. For example, if I arrived at Step E after Step B in some linear progression of steps, then obviously I can get from Step B to Step E, but may or may not be able to get from Step E back to Step B. If I can't get from Step E to Step B, then I'm thinking of the calculations from Step B to Step E as "losing information." More interestingly, if a progression of steps is nonlinear, then the information lost might have a nonlinear structure. By a nonlinear progression of steps, I mean something like the following...
Problem: [imath]100y = 100x[/imath] (Step A)
(I divide by 10): [imath]10y = 10x[/imath]
(I multiply by 5): [imath]50y = 50x[/imath] (I judge this result important and designate it as Step B)
(I divide by 2): [imath]25y = 25x[/imath]
Now I follow two branches...
(I add 2): [imath]25y + 2 = 25x + 2[/imath]
(I multiply by 2): [imath]50y + 4 = 50x + 4[/imath] (I designate this as Step C)
...and...
(I multiply by 9): [imath]225y = 225x[/imath]
(I divide by 3): [imath]75y = 75x[/imath] (I designate this as Step D)
(I subtract 75y): [imath]75x - 75y = 0[/imath]
(I multiply by 2): [imath]150x - 150y = 0[/imath] (I designate this as Step E)
No surprise that all steps here are entirely equivalent, but in more interesting calculations with this sort of structure, Step C may lose information from Step B independently or partially-dependently of how Step E loses information from Step B. My questions are...
1) Is there a more standard and concise mathematical framework used to represent what I'm talking about here, or is this not something commonly cared about? Is it accurate to say the calculations are discarding "information" as they go, and if so, is this related to Shannon Information, or a completely different concept of information?
2) Is it ever possible for this information to "come back" in a later step? For example, in the structure I exemplified above (but with nontrivial calculations), could it be that Step B and Step E are equivalent, but Step D contains less information than them? Or to come at it from another angle, it is possible that not every step in a sequence of calculations is reversible, but the first and last steps are still equivalent?
I want to understand which of these steps can be derived from which others. For example, if I arrived at Step E after Step B in some linear progression of steps, then obviously I can get from Step B to Step E, but may or may not be able to get from Step E back to Step B. If I can't get from Step E to Step B, then I'm thinking of the calculations from Step B to Step E as "losing information." More interestingly, if a progression of steps is nonlinear, then the information lost might have a nonlinear structure. By a nonlinear progression of steps, I mean something like the following...
Problem: [imath]100y = 100x[/imath] (Step A)
(I divide by 10): [imath]10y = 10x[/imath]
(I multiply by 5): [imath]50y = 50x[/imath] (I judge this result important and designate it as Step B)
(I divide by 2): [imath]25y = 25x[/imath]
Now I follow two branches...
(I add 2): [imath]25y + 2 = 25x + 2[/imath]
(I multiply by 2): [imath]50y + 4 = 50x + 4[/imath] (I designate this as Step C)
...and...
(I multiply by 9): [imath]225y = 225x[/imath]
(I divide by 3): [imath]75y = 75x[/imath] (I designate this as Step D)
(I subtract 75y): [imath]75x - 75y = 0[/imath]
(I multiply by 2): [imath]150x - 150y = 0[/imath] (I designate this as Step E)
No surprise that all steps here are entirely equivalent, but in more interesting calculations with this sort of structure, Step C may lose information from Step B independently or partially-dependently of how Step E loses information from Step B. My questions are...
1) Is there a more standard and concise mathematical framework used to represent what I'm talking about here, or is this not something commonly cared about? Is it accurate to say the calculations are discarding "information" as they go, and if so, is this related to Shannon Information, or a completely different concept of information?
2) Is it ever possible for this information to "come back" in a later step? For example, in the structure I exemplified above (but with nontrivial calculations), could it be that Step B and Step E are equivalent, but Step D contains less information than them? Or to come at it from another angle, it is possible that not every step in a sequence of calculations is reversible, but the first and last steps are still equivalent?