Punctual categoricity and universality
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 -categorical, and we show that this upper bound is tight. We also construct an...
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| Main Authors: | , , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/159279 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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 -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. We also prove that, with a bit of work, the latter result can be pushed beyond, 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. |
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