When Are Two Algorithms the Same? Andreas Blass, Nachum Dershowitz, Yuri Gurevich. February 2008

People usually regard algorithms as more abstract than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue that no such equivalence relation exists.

A bit more philosophical than usual, but the issue is quite relevant to discussions in the field.

It is possible to stipulate any equivalence relation that is considered useful (e.g., equivalence up to local transformations) but the notion of a universally applicable relation is indeed problematic.