We can try to assume that a horcrux is literally a fragment of a soul, in the Hofstadterian sense. It is then indeed an abstract algorithm (or a set of them), and it need not include memory and separate intelligence, although it would include personality and feelings.

Extrapolating on what we know about how the Source of Magic interprets things, we should expect inanimate object horcruxes to be generally passive, while alive horcruxes to incorporate the algorithms into their own minds, although still somewhat separate.

[It'd be cool to read about how a horcruxed software would behave.]