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On the Consistency Problem for Modular Lattices and related structures

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Martin Ziegler

The weak/strong satisfiability problem for Quantum Logic over a fixed
inner product space $V$ asks for whether a given Boolean expression
$\varphi$ admits an assignment of subspaces of $V$ that makes $\varphi$
evaluate to a non-trivial subspace of/the entire space $V$. Its
computational complexity is known to climb from trivial $d=0$ via the
Boolean case $d=1$ and $d=2$ both NP-complete to every fixed higher
dimension being polynomial-time equivalent to the Existential Theory of
the Reals (Herrmann&Ziegler'11+2016).

We prove that strong satisfiability over SOME (i.e. additionally
unbounded existentially quantified) dimension $d>0$ is undecidable.

More generally we consider, for a fixed class of algebraic structures
of joint signature, the problem of whether a given conjunction of
equations admits a satisfying assignment in some non-singleton member
of the class; and, building on work by Bridson and Wilton (2015),
show it to be undecidable for (i) the class of all subspace lattices
of finite dimensional vector spaces over a fixed or arbitrary field of
characteristic $0$, and for (ii) classes of (finite) rings with unit.

This joint work with Christian Herrmann (TU Darmstadt) and Yasuyuki
Tsukamoto (Hakuryo High School) has recently appeared in the
International Journal of Algebra and Computation.