On the Lattices of Effectively Open Sets
Oleg Kudinov
S.L. Sobolev Institute of Mathematics SB RAS
Novosibirsk, Russia
and
Victor Selivanov
A.P. Ershov Institute of Informatics Systems SB RAS
Novosibirsk, Russia
The lattice E of c.e. sets is a popular object of study in Computability Theory [So87]. A principal fact
about this lattice is the undecidability of its first-order theory T h(E). Moreover, T h(E) is known [HN98]
to be m-equivalent to the first-order arithmetic T h(N) where N := (ω; +, ×).
The lattice E may be considered as the lattice of effectively open subsets of the discrete topological space
ω. The lattices of effectively open sets Σ01 (X) of various “effective” topological spaces X are of interest
in Computable Analysis [We00] and Effective Descriptive Set Theory, hence it is natural to ask which
results about E hold true for the lattices of effectively open sets of “natural” effective topological spaces.
Here we attempt to make apparently first steps in this direction.
First we show that for many natural effective spaces X the theory T h(Σ01 (X)) is undecidable. By strongly
computable ϕ-space (SCPS) we mean an effective (not necessarily complete) ϕ-space [Se06] X such that the
specialization order is computable on the compact elements, and there is a computable infinite sequence
of pairwise incomparable compact elements. Many popular domains are SCPSs. In particular, P ω and
the domains of partial continuous functionals of finite types are SCPSs.
Theorem 1 Let X be a SCPS or a non-singleton computable metric space [We00] without isolated points.
Then T h(Σ01 (X)) is hereditarily undecidable.
Next we discuss the complexity of T h(Σ01 (X)). First we establish a natural upper bound that applies to
many locally compact spaces. By arithmetically locally compact space (ALCS) we mean a triple (X, β, κ)
consisting of a countably based space X, a numbering β of a base in X, and a numbering κ of compact
sets in X such that any set βn is a union of some sets in {κi | i < ω}, and the relation “κi ⊆ βn0 ∪· · ·∪βnk ”
is arithmetical. Many popular locally compact spaces are ALCSs. In particular, the effective ϕ-spaces
and the finite dimensional Euclidean spaces are ALCSs.
Proposition 2 If X is an ALCS then T h(Σ01 (X)) ≤m T h(N).
The precise estimation of the complexity of T h(Σ01 (X)) is very subtle and strongly depends on the
topology of X. So far we were able to precisely characterize the complexity of T h(Σ01 (X)) only for small
classes of spaces, e.g. we have the following result:
Theorem 3 Let X be the space of reals R, or the domain P ω, or the domain of partial continuous
functionals of a given finite type. Then T h(Σ01 (X)) ≡m T h(N).
For many natural spaces X we still have a big gap between the known (to us) lower and the upper bound
for T h(Σ01 (X)). In particular, this is the case even for the Baire space.
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References
[HN98]
L. Harrington and A. Nies. Coding in the lattice of enumerable sets. Advances in Mathematics,
133 (1998), 133-162.
[Se06]
V.L. Selivanov. Towards a descriptive set theory for domain-like structures. Theoretical Computer Science, 365 (2006), 258–282.
[So87]
R.I. Soare. Recursively Enumerable Sets and Degrees. Berlin, Springer, 1987.
[We00]
K. Weihrauch. Computable Analysis. Berlin, Springer, 2000.
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