On the consistency problem for modular lattices and related structures

Abstract

Abstract. The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and Rabin (1958) it is known unsolvable for (the class of) groups and, recently, by Bridson and Wilton (2015) for finite groups. We derive unsolvability for (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite dimensional vector spaces over a fixed or arbitrary field of characteristic 0. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann-Cayley algebra and to functional and embedded multivalued dependencies in databases.

On the complexity of equational decision problems for finite height complemented and orthocomplemented modular lattices

“…We mention that V(R) contains all finite Rickart C * -algebras [14,Theorem 2]. According to [17,Theorem 22], SAT R is undecidable. Our main result is as follows.…”

“…The decision problem for Eq(L) was shown solvable in [9,12] with REF L ∈ BPN P 0 R in [16,Theorem 4.4]. On the other hand, SAT L was shown undecidable in [17], as well as SAT C for any class C of (expansions of) modular lattices containing some subspace lattice L(V F ) of a vector space V F where dim V F is infinite or contains all L(V F ) where F is of characteristic 0 and dim(V F ) finite.…”

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

“…We mention that V(R) contains all finite Rickart C * -algebras [11,Theorem 2]. According to [14,Theorem 22], SAT R is undecidable. Our main result is as follows.…”

“…The decision problem for Eq(L) was shown solvable in [6,9], REF L ∈ BPN P 0 R in [13,Theorem 4.4]. On the other hand, SAT L was shown undecidable in [14], as well as SAT C for any class of (expansions of) modular lattices containing some subspace lattice L(V F ) of a vector space V F where dim V F is infinite or contains all L(V F ) where F is of characteristic 0 and dim(V F ) finite.…”

A Note on the "Third Life of Quantum Logic"

“…For each H, with 3 ≤ dim H < ω, as well as for L(R) the set of tautologies is decidable [13,12]; in all cases, the complexity of the complementary problem is complete for the same class in the Bum-Shub-Smale model of non-deterministic real computation [19]. Satisfiability is complete for this class, given fixed H [19], undecidable for the class of all L(H) as well as for L(R) and CG(C) [17].…”