Victor L. Selivanov

Fine hierarchies and Boolean terms

We consider fine hierarchies in recursion theory, descriptive set theory, logic and complexity theory. The main results state that the sets of values of different Boolean terms coincide with the levels of suitable fine hierarchies. This gives new short descriptions of these hierarchies and shows that collections of sets of values of Boolean terms are almost well ordered by inclusion. For the sake of completeness we mention also some earlier results demonstrating the usefulness of fine hierarchies.

Towards a descriptive set theory for domain-like structures

This is a survey of results in descriptive set theory for domains and similar spaces, with the emphasis on the ω-algebraic domains. We try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science. Since the subject is still in its beginning, we discuss in detail several open questions and possible future development. We also mention some relevant facts of (effective) descriptive set theory.

Two Refinements of the Polynomial Hierarcht

We introduce and study two classifications refining the polynomial hierarchy. Both extend the difference hierarchy over NP and are analogs of some hierarchies from recursion theory. We answer some natural questions on the introduced classifications, e.g. we extend the result of J.Kadin that the difference hierarchy over NP does not collapse (if the polynomial hierarchy does not collapse).

Wadge-like reducibilities on arbitrary quasi-Polish spaces

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well-ordered), but for many other natural non-zero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called \Delta^0_\alpha-reductions, and try to find for various natural topological spaces X the least ordinal \alpha_X such that for every \alpha_X \leq \beta < \omega_1 the degree-structure induced on X by the \Delta^0_\beta-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that \alpha_X \leq {\omega} for every quasi-Polish space X, that \alpha_X \leq 3 for quasi-Polish spaces of dimension different from \infty, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

Undecidability in the Homomorphic Quasiorder of Finite Labelled Forests

We prove that the homomorphic quasiorder of finite k-labelled forests has a hereditary undecidable first-order theory for k ≥ 3, in contrast to the known decidability result for k = 2. We establish also hereditary undecidability (again for every k ≥ 3) of first-order theories of two other relevant structures: the homomorphic quasiorder of finite k-labelled trees, and of finite k-labelled trees with a fixed label of the root element. Finally, all three first-order theories are shown to be computably isomorphic to the first-order arithmetic.

Hierarchies of Δ02-measurable k -partitions

A vehicle level regulator control system controls the vehicle level regulator to improve the response characteristics of the suspension member in response to the vehicle level detector. The control system includes a means for resetting the preset in the delay means for directly activating the suspension member responsive to the vehicle level detector signal when the starter switch is turned on.

Undecidability in the homomorphic quasiorder of finite labeled forests

We prove that the homomorphic quasiorder of finite k-labeled forests has undecidable elementary theory for k ≥3, in contrast to the known decidability result for k=2. We establish also undecidablity (again for every k ≥3) of elementary theories of two other relevant structures: the homomorphic quasiorder of finite k-labeled trees, and of finite k-labeled trees with a fixed label of the root element.

Hierarchies in φ-spaces and applications

We establish some results on the Borel and difference hierarchies in φ-spaces. Such spaces are the topological counterpart of the algebraic directed-complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non-collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω-ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Some hierarchies of QCB 0 -spaces

We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain P\omega, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.

Fine hierarchies and m-reducibilities in theoretical computer science

This is a survey of results about versions of fine hierarchies and many-one reducibilities that appear in different parts of theoretical computer science. These notions and related techniques play a crucial role in understanding complexity of finite and infinite computations. We try not only to present the corresponding notions and facts from the particular fields but also to identify the unifying notions, techniques and ideas.