There have been two main reductionist strategies to deal with mental states, and they both -- to say the least -- have stalled. The two strategies are:
1. Eliminative materialism 2. Identity theory
Well, that is the two *materialist* strategies. Buddhism, Hinduism, Plato, Berkeley and many others have proposed what we could
call Eliminative spiritualism or immaterialism, sometimes called "objective idealism" (sometimes confused with subjective idealism, i.e. solipsism). I think the 20th century (perhaps by the dazzling success of physics) is the first (and probably the last too!) during which
immaterialism has no more been defended (with some exceptions).
My work shows that comp entails a form of Eliminative immaterialism, and also that comp (but actually this works for Everett formulation of QM too) is incompatible with identity theory. We can still ascribe a mind (a first person with her private life) to a body (a third person describable object), but we cannot ascribe a body to a mind, we can only ascribe an infinity of "possible bodies" to a mind (like our counterparts in the QM parallel "universes": i.e. the mind body relation is not one-one.
In this setting the word "reductionism" is misleading: comp *reduces* both minds and bodies to arithmetical relations between numbers, but
it leads also to a machine psychology (or theology) which is everything but reductionist, for that psychology/theology is a sort of
"negation theology" (to use the theologians expression) in the sense that the machine self is able to "diagonalize" against any proposed and well defined definition of the self. The real self being necessarily undefinable by anyself. This looks weird and perhaps even somehow mystical, but can be related to the incompleteness phenomena, even easily so once we work a little bit on the modal logic formalization of those incompleteness phenomenon (cf Solovay's work).
A lot of problems in the "modern" philosophy of mind are due to a reductionist conception of machines which is just impossible since Godel&Al.
In some sense Godel's theorem is the big killer of any reductionist conception of machines. And as I said, some quasi dogmatic attachment to materialism (even in the weaker sense of a doctrine reifying matter) will not help.
Of course, just to mention them, there are also the dualist approaches (cf "the brain and its self" by Popper and Eccles, or even Chalmers).
But I don't think a lot in this list adhere to dualist positions, but please correct me if I'm wrong).