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 various new decidability results
for 2Γ2 matrices over Q. For that, we introduce the concept
of flat rational sets: if M is a monoid and N is a submonoid, then ``flat
rational sets of M over N'' are finite unions of the form L0βg1βL1ββ―gtβLtβ where all Liβ's are rational subsets of N and giββM.
We give quite general sufficient conditions 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 (in singly exponential
time). It is possible that such a strong decidability result cannot be pushed
any further inside GL(2,Q).
We also show a dichotomy for nontrivial group extension of GL(2,Z)
in GL(2,Q): if G is a f.g. group such that GL(2,Z)<Gβ€GL(2,Q), then either Gβ GL(2,Z)ΓZk, for
some kβ₯1, or G contains an extension of the Baumslag-Solitar group
BS(1,q), with qβ₯2, of infinite index. In the first case of the
dichotomy the membership problem for G is decidable but the equality problem
for rational subsets of G is undecidable. In the second case, decidability of
the membership problem for rational subsets in G is open.
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 (in doubly exponential time) for flat rational subsets of
Q2Γ2 over the submonoid that is generated by the matrices from
Z2Γ2 with determinants in {β1,0,1}.Comment: 30 pages + 2 pages appendi