I've always been intrigued by, but never delved deeply enough into the depths of the more mathematical sides of programming. I've always wondered is there a good mathematical representation of an algorithm that is useful for algebraic manipulation? For example:
Take an ugly equation like: x^2 + 3x+y+2+(−x)^2 = 3+(−x)^2 . It can be transformed to either (x+1)(x+2)-3 = y or y=−x^2−3x+1 fairly cleanly and usefully with just a few algebraic theorems. I remove redundancy among other things.
Can I take "bubble sort" work my way in steps to produce a "quick sort" or a "merge sort"? Sure you could prove they produce identical results but it doesn't feel like you're bridging that gap. Maybe this is too vague a question.
Take an ugly equation like: x^2 + 3x+y+2+(−x)^2 = 3+(−x)^2 . It can be transformed to either (x+1)(x+2)-3 = y or y=−x^2−3x+1 fairly cleanly and usefully with just a few algebraic theorems. I remove redundancy among other things.
Can I take "bubble sort" work my way in steps to produce a "quick sort" or a "merge sort"? Sure you could prove they produce identical results but it doesn't feel like you're bridging that gap. Maybe this is too vague a question.