There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

/pdf/infinite-abelian-groups-2dvcz0046a.pdf

Infinite Abelian Groups

On computable numbers

Descriptive Set Theory

Was sind und was sollen die Zahlen

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/69F36E5C8677C2BAFB91AAEC6C8D7F1E/S1079898619000209a.pdf/div-class-title-foundations-of-online-structure-theory-div.pdf

Foundations of online structure theory

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is -complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/3689C3974B80B6F47A575934EAA6CD03/S0022481219000264a.pdf/div-class-title-automatic-and-polynomial-time-algebraic-structures-div.pdf

Automatic and polynomial-time algebraic structures

Autostability spectra for Boolean algebras and their various enrichments are explored. The basic notions and definitions of computable structure theory used in the paper can be found in [1, 2]. As usual, we write “c.e.” as an abbreviation for “computably enumerable.” Recall that a computable structure M is said to be decidable if its full diagram FD(M) is computable. A structure M is strongly constructivizable if M has a decidable copy. Let d be a Turing degree. A constructivizable structure A is said to be d-autostable if, for any computable copies B and C of A, there exists a d-computable isomorphism f : B → C. A strongly constructivizable structure A is d-autostable relative to strong constructivizations if, for any decidable copies B and C for A, there exists a d-computable isomorphism f : B → C. If A is a constructivizable structure, then the autostability spectrum of A is the set of all Turing degrees d for which the structure A is d-autostable. The autostability spectrum relative to strong constructivizations for a strongly constructivizable structure A is the set of all degrees d such that A is d-autostable relative to strong constructivizations. Denote the autostability spectrum of a structure A by AutSpec (A), and the autostability spectrum relative to strong constructivizations for A by SCAutSpec (A). We say that a Turing degree d is the degree of autostability (degree of autostability relative to strong constructivizations) for A if d is least in AutSpec (A) (in SCAutSpec (A)). Note that in ∗Supported by RFBR (project No. 14-01-00376), by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1), and by the Russian Ministry of Education and Science (gov. contract No. 2014/139).

Autostability Spectra for Boolean Algebras

It is proved that every computable superatomic Boolean algebra has a strong degree of categoricity.

Degrees of categoricity for superatomic Boolean algebras

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure , the SC-autostability spectrum of is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of . The degree of SC-autostability for is the least degree in the spectrum (if such a degree exists). We prove that for a computable successor ordinal α, every Turing degree c.e. in and above 0 (α) is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above 0 (2β+1) is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative to n-constructivizations.

Nikolay Bazhenov

Papers

Foundations of online structure theory

Automatic and polynomial-time algebraic structures

Autostability Spectra for Boolean Algebras

Degrees of categoricity for superatomic Boolean algebras

Autostability spectra for decidable structures