Knuth answers this question in chapter 0.
It's also a bridge too far to make a blanket statement of Knuth "already answered this" (it's also not accurate to attribute that idea to Knuth; e.g. Church did work exactly on this decades before). Meaningful discourse and mathematical analysis requires nuance and degrees of equivalence and mixing them up makes it needlessly confusing and difficult.
Sameness as "same input same output" is NOT the same as the "isomorphic equivalence" which is more difficult, strict, and abstract. In this sense they are the same if they follow the same "steps" (also isomorphic between programs), i.e. they are different representations of the same algorithm.
He doesn’t say this.
> NOT the same as the "isomorphic equivalence" which is more difficult, strict, and abstract.
This is what the exercise is about. So I would recommend reading before making assumptions.
Of course there are more than one definition of equivalence and isomorphism, but the one explored there is just as interesting as the comment I’m replying to.
drpossum you stop with these gotcha posts? This is our 3rd encounter.
If I’m hitting this issue at random, others are hitting it too.