Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation
This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if is a monoid and ≤ is its submonoid, then flat rational sets of relative to are finite unions of the form 0 1 1 · · · where all s are rational subsets of and ∈ . We give quite general sufficient
... s under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if is a f.g. group such that GL(2, Z) < ≤ GL(2, Q), then either GL(2, Z) × Z , for some ≥ 1, or contains an extension of the Baumslag-Solitar group BS(1, ), with ≥ 2, of infinite index. It turns out that in the first case the membership problem for is decidable but the equality problem for rational subsets of is undecidable. In the second case, decidability of the membership problem is open for every such . In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of (2, Q) relative to the submonoid that is generated by the matrices from (2, Z) with determinants 0, ±1 and the central rational matrices. CCS CONCEPTS • Theory of computation → Formal languages and automata theory; • Computing methodologies → Symbolic and algebraic algorithms.