Valentina Harizanov

R. Shore proved that every recursively enumerable (r. e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ϵ of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two (non) effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non‐nowhere simple sets. We show that every r. e. set is either the disjoint union of two effectively nowhere simple sets or two noneffectively nowhere simple sets. We characterize r. e. sets whose every nontrivial splitting is into nowhere simple sets, and r. e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, the lattice L* (A) is effectively isomorphic to ϵ*, and that there is a nowhere simple set A such that L*(A) is not effectively isomorphic to ϵ*. We prove that every nonzero r. e. Turing degree contains a noneffectively nowhere simple set A with the lattice L*(A) effectively isomorphic to ϵ*. Mathematics Subject Classification: 03D25, 03D10.

We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.A computable structure is categorical if for all computable isomorphic copies of , there is an isomorphism from onto , which is . We prove that for every computable successor ordinal α, there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.An additional relation R on the domain of a computable structure is intrinsically on if in all computable isomorphic copies of , the image of R is . We prove that for every computable successor ordinal α, there is an intrinsically relation on a computable structure, which not relatively intrinsically . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable.The dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal α and every n≥1, there is a computable structure with dimension n. This generalizes the result of Goncharov that there is a structure of computable dimension n for every n≥1.Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that relative to X is not . In particular, for every finite n, there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees

Mathematical logic is the study of reasoning about mathematical objects and the degree to which mathematical and scientific reasoning can be formalized and mechanized. Logic provides the foundations of mathematics and of theoretical computer science. Classical logic defined truth, developed the theory of infinite numbers, resolved paradoxes of naive set theory, defined what an algorithm is, and established that certain mathematical principles are independent from the rest of mathematics. Modern logic contributes to current developments in mathematics and its foundations, and continues to closely interact with various areas of mathematics. Essential and surprising connections are often found in order to solve big problems. To bring various resources into a single argument, common methods, uniform terminology, and concisely stated theorems are needed. This interaction of disciplines has not only led to new approaches and results but also to new areas of mathematical research, such as computable structure theory, reverse mathematics, and the emerging quantum computing theory.

Computability theory is the mathematical theory of algorithms, which explores the power and limitations of computation. Classical computability theory formalized the intuitive notion of an algorithm and provided a theoretical basis for digital computers. It also demonstrated the limitations of algorithms and showed that most sets of natural numbers and the problems they encode are not decidable (Turing computable). Important results of modern computability theory include the classification of the computational difficulty of sets and problems. Arithmetical and hyperarithmetical hierarchies and the Turing degrees provide important measures of the level of such difficulty. There are uncountably many Turing degrees and they are partially ordered. These classifications are used in contemporary research to investigate the complexity of problems in modern computable mathematics, for example, regarding structures and their properties.

In 1934, Skolem gave a remarkable construction of a countable nonstandard model of arithmetic. His construction contains ideas of the ultrapower construction which was introduced in model theory 20 years later. However, typical ultrapower constructions produce uncountable models. Skolem’s construction can also be connected with ideas from computability theory, formalized by Turing and others in 1936. The proof of one of Skolem’s key statements can be interpreted using computability-theoretic notions of indecomposable sets developed in the late 1950s. We favor the computability-theoretic approach and describe the connections of Skolem’s ideas with our current study of cohesive powers of models in computable model theory.

A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive set. Thus, unlike many classical ultrapowers, a cohesive power is a countable structure. In this paper we focus on cohesive powers of graphs, equivalence structures, and computable structures with a single unary function satisfying various properties, which can also be viewed as directed graphs. For these computable structures, we investigate the isomorphism types of their cohesive powers, as well as the properties of cohesive powers when they are not isomorphic to the original structure.

Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual presentation of$\omega $, then every cohesive power of$\mathcal {L}$has order-type$\omega + \zeta \eta $. However, there are computable copies of$\omega $, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$\omega + \zeta \eta $. For example, we show that there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \eta $. Our most general result is that if$X \subseteq \mathbb {N} \setminus \{0\}$is a Boolean combination of$\Sigma _2$sets, thought of as a set of finite order-types, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$, where$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$denotes the shuffle of the order-types inXand the order-type$\omega + \zeta \eta + \omega ^*$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X)$.

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering ${\mathcal{A}}$ has the feature that for every ${\mathcal{B} \cong \mathcal{A}}$ , there is an infinite chain or antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , then we have uniform dichotomy: either for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite chain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$ , or for all copies ${\mathcal{B}}$ of ${\mathcal{A}}$ , there is an infinite antichain that is ${\Delta _{2}^{0}}$ relative to ${\mathcal{B}}$

Fraïssé studied countable structures S through analysis of the age of S i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum

We show that for every computable limit ordinal α, there is a computable structure A that is $\Delta _\alpha ^0 $ categorical, but not relatively $\Delta _\alpha ^0 $ categorical (equivalently. it does not have a formally $\Sigma _\alpha ^0 $ Scott family). We also show that for every computable limit ordinal a, there is a computable structure A with an additional relation R that is intrinsically $\Sigma _\alpha ^0 $ on A. but not relatively intrinsically $\Sigma _\alpha ^0 $ on A (equivalently, it is not definable by a computable $\Sigma _\alpha $ formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.