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PUNCTUAL CATEGORICITY AND UNIVERSALITY

Part of: Computability and recursion theory

Published online by Cambridge University Press:  22 October 2020

ROD DOWNEY ,
NOAM GREENBERG ,
ALEXANDER MELNIKOV ,
KENG MENG NG  and
DANIEL TURETSKY
ROD DOWNEY
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTONWELLINGTON, PO BOX 600, NEW ZEALANDE-mail: rod.downey@vuw.ac.nzE-mail: greenberg@msor.vuw.ac.nz
NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTONWELLINGTON, PO BOX 600, NEW ZEALANDE-mail: rod.downey@vuw.ac.nzE-mail: greenberg@msor.vuw.ac.nz
ALEXANDER MELNIKOV
Affiliation:
MASSEY UNIVERSITY AUCKLAND PRIVATE BAG 102904, NORTH SHOREAUCKLAND0745, NEW ZEALANDE-mail: alexander.g.melnikov@gmail.com
KENG MENG NG
Affiliation:
SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES DIVISION OF MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPOREE-mail: selwyn.km.ng@gmail.com
DANIEL TURETSKY
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTONWELLINGTON, PO BOX 600, NEW ZEALANDE-mail: dan.turetsky@vuw.ac.nz
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Abstract

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.

As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

Keywords

primitive recursioncomputable structurespunctual computability

MSC classification

Primary: 03D45: Theory of numerations, effectively presented structures
Secondary: 03D20: Recursive functions and relations, subrecursive hierarchies 03D99: None of the above, but in this section
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1427 - 1466
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Copyright
© The Association for Symbolic Logic 2020

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