RAM BSS model based (or its variant) computer recognizing Boolean languages
Christine Gaßsner studied the question you are asking. These papers seem relevant, but you should have a look at her list of publications (DBLP):


The Separation of Relativized Versions of P and DNP for the Ring of the Reals. J. UCS 16(18): 2563-2568 (2010)
Oracles and Relativizations of the P =? NP Question for Several Structures. J. UCS 15(6): 1186-1205 (2009)
Relativizations of the P =? DNP Question for the BSS Model


You seem to be asking a basic question, namely: can BSS RAM machines recognize any boolean languages? Well, a BSS machine always has some kind of comparison operator, at the very least $&lt;$ on real numbers. We can use it to decide equality of $0$ and $1$, from which it follows that a BSS machine can recognize at least all the languages that an ordinary RAM machine can. The more interesting question is whether it can do more than that, or perhaps more efficiently.