Paper detail

On the complexity of equational decision problems for finite height(ortho)complemented modular lattices

We study the computational complexity of satisfiability problems for classes of simple finite height (ortho)complemented modular lattices $L$. For single finite $L$, these problems are shown tobe $\mc{NP}$-complete; for $L$ of height at least $3$, equivalent to a feasibility problem for the division ring associated with $L$. Moreover, it is shown that the equational theory of the class of subspace ortholattices as well as endomorphism *-rings (with pseudo-inversion) of finite dimensional Hilbert spaces is complete for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This results extends to the category of finite dimensional Hilbert spaces, enriched by pseudo-inversion.