Oleg V. Kudinov

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.

Towards Computability over Effectively Enumerable Topological Spaces

In this paper we study different approaches to computability over effectively enumerable topological spaces. We introduce and investigate the notions of computable function, strongly-computable function and weakly-computable function. Under natural assumptions on effectively enumerable topological spaces the notions of computability and weakly-computability coincide.

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.

Spectra of highn and non-lown degrees

We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of non-lown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order ℚ. New results include realizations of the set of non-lown Turing degrees as the spectrum of a relation on ℚ for all n≥1, and a realization of the set of all non-lown Turing degrees as the spectrum of a linear order whenever n≥2. The state of current knowledge is summarized in a table in the concluding section.

Towards computability of higher type continuous data

This paper extends the logical approach to computable analysis via Σ–definability to higher type continuous data such as functionals and operators. We employ definability theory to introduce computability of functionals from arbitrary domain to the real numbers. We show how this concept works in particular cases.

Positive predicate structures for continuous data

In this paper, we develop a general framework for continuous data representations using positive predicate structures. We first show that basic principles of Σ-definability which are used to investigate computability, i.e., existence of a universal Σ-predicate and an algorithmic characterization of Σ-definability hold on all predicate structures without equality. Then we introduce positive predicate structures and show connections between these structures and effectively enumerable topological spaces. These links allow us to study computability over continuous data using logical and topological tools.

A Gandy Theorem for Abstract Structures and Applications to First-Order Definability

We establish a Gandy theorem for a class of abstract structures and deduce some corollaries, in particular the maximal definability result for arithmetical structures in the class. We also show that the arithmetical structures under consideration are biinterpretable (without parameters) with the standard model of arithmetic. As an example we show that for any k *** 3 a predicate on the quotient structure of the h -quasiorder of finite k -labeled forests is definable iff it is arithmetical and invariant under automorphisms.

Definability in the h-quasiorder of labeled forests

Abstract We prove that for any k ≥ 3 each element of the h -quasiorder of finite k -labeled forests is definable in the ordinary first order language and, respectively, each element of the h -quasiorder of (at most) countable k -labeled forests is definable in the language L ω 1 ω , in both cases provided that the minimal non-smallest elements are allowed as parameters. As corollaries, we characterize the automorphism groups of both structures and show that the structure of finite k -forests is atomic. Similar results hold true for two other relevant structures: the h -quasiorder of finite (resp. countable) k -labeled trees and of finite (resp. countable) k -labeled trees with a fixed label of the root element.

The Uniformity Principle for Σ-Definability with Applications to Computable Analysis

In this paper we prove the Uniformity Principle for Σ---definability over the real numbers extended by open predicates. Using this principle we show that if we have a Σ K -formula, i.e. a formula with quantifier alternations where universal quantifiers are bounded by computable compact sets, then we can eliminate all universal quantifiers obtaining a Σ-formula equivalent to the initial one. We also illustrate how the Uniformity Principle can be employed for reasoning about computability over continuous data in an elegant way.

The Uniformity Principle for Σ-definability

This article is an extended version of the paper published in Korovina and Kudinov (2007, Lecture Notes in Computer Science, Vol. 4497, pp. 416–425). The main goal of this research is to develop logical tools and techniques for effective reasoning about continuous data based on Σ-definability. In this article we invent the Uniformity Principleand prove it for Σ-definability over the real numbers extended by open predicates. Using the Uniformity Principle, we investigate different approaches to enrich the language of Σ-formulas in such a way that simplifies reasoning about computable continuous data without enlarging the class of Σ-definable sets. In order to do reasoning about computability of certain continuous data we have to pick up an appropriate language of a structure representing these continuous data. We formulate several major conditions how to do that in a right direction. We also employ the Uniformity Principleto argue that our logical approach is a good way for formalization of computable continuous data in logical terms.