Alexander G. Melnikov

Limitwise monotonic sequences and degree spectra of structures

We study effective monotonic approximations of sets and sequences of sets. We show that there is a sequence of sets which has no uniform computable monotonic approximation, but has an xcomputable monotonic approximation, for every hyperimmune degree x. We also construct a Σ2 set which is not limitwise monotonic, but is x-limitwise monotonic relative to every nonzero ∆2 degree x. We show that if a sequence of sets is uniformly limitwise monotonic in x for all, except countably many, degrees x, then it has to be uniformly limitwise monotonic. Finally, we apply these results to investigate degree spectra of abelian groups, equivalence relations, and א1-categorical structures.

Enumerations and Completely Decomposable Torsion-Free Abelian Groups

We study possible degree spectra of completely decomposable torsion-free Abelian groups.

Computable functors and effective interpretability

Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theory, and the other one is a computational notion which is a strengthening of the well-known Medvedev reducibility. We extend our result to effective bi-interpretability and also to effective reductions between classes of structures.

Computable completely decomposable groups

A completely decomposable group is an abelian group of the form ⊕ i Hi, where Hi ≤ (Q,+). We show that every computable completely decomposable group is Δ5-categorical. We construct a computable completely decomposable group which is not Δ4-categorical, and give an example of a computable completely decomposable group G which is Δ4-categorical but not Δ3-categorical. We also prove that the index set of computable completely decomposable groups is arithmetical.

K-triviality in computable metric spaces

A point x in a computable metric space is called K-trivial if for each positive rational δ there is an approximation p at distance at most δ from x such that the pair p, δ is highly compressible in the sense that K(p, δ) ≤ K(δ) + O(1). We show that this local definition is equivalent to the point having a Cauchy name that is K-trivial when viewed as a function from N to N. We use this to transfer known results on K-triviality for functions to the more general setting of metric spaces. For instance, we show that each computable Polish space without isolated points contains an incomputable K-trivial point.

Computable ordered abelian groups and fields

We present transformations of linearly ordered sets into ordered abelian groups and ordered fields. We study effective properties of the transformations. In particular, we show that a linear order L has a Δ20 copy if and only if the corresponding ordered group (ordered field) has a computable copy. We apply these codings to study the effective categoricity of linear ordered groups and fields.

Decidability and Computability of Certain Torsion-Free Abelian Groups

We study completely decomposable torsion-free abelian groups of the form GS := ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ2 and has a computable copy if and only if S is Σ 0 3.

The Classification Problem for Compact Computable Metric Spaces

We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classification-type results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to isometry, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that deciding if two compact computable metric spaces are isometric is a \(\Pi^0_2\) complete problem within the class of compact computable spaces, which in itself is \(\Pi^0_3\). On the other hand, if there is an isometry, then ∅ ′′ can compute one. In fact, there is a set low relative to ∅ ′ which can compute an isometry. We show that the result can not be improved to ∅ ′. We also give further results for special classes of compact spaces, and for other related classes of Polish spaces.

0˝-Categorical Completely Decomposable Torsion-Free Abelian Groups

We show that every homogeneous completely decomposable torsion-free abelian group is 0˝-categorical. We give a description of effective categoricity for some natural class of torsion-free abelian groups. In particular, we give examples of 0˝-categorical but not 0˝-categorical torsion-free abelian groups.

COMPUTABLE POLISH GROUP ACTIONS

Using methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. Among other results, we provide a new recursion-theoretic characterization of Σ3-orbits in Gspaces, and we also give a sufficient condition for an orbit under effective G-action to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.