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Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.Each number in a sequence is referred to as a term. Typically, the terms are denoted as a1, a2, a3,…, where the...

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Rice sequences of relations.

Antonio Montalbán¹

¹Department of Mathematics, University of Chicago, IL 60637, USA. antonio@math.uchicago.edu

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|June 20, 2012

Summary

This study introduces a new framework for analyzing the computational complexity of definable relations. It unifies existing concepts like computably enumerable (c.e.)-ness and reducibility within a novel structure.

Area of Science:

* Computability Theory
* Logic
* Theoretical Computer Science

Background:

* Existing research on definable relations and their computational properties.
* The need for a unified framework to understand these concepts.

Purpose of the Study:

* To propose a novel framework for studying the computational complexity of definable relations.
* To unify and re-examine existing notions like c.e.-ness, reducibility, join, and jump.
* To survey and connect related results from different research settings.

Main Methods:

* Development of a framework for analyzing sequences of relations over a given structure.
* Introduction of new notions of c.e.-ness, reducibility, join, and jump within this framework.
* Comparative analysis of these notions with existing concepts in other settings.

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Main Results:

* A coherent framework is established for studying the computational complexity of definable relations.
* Equivalence is demonstrated between the newly developed notions and previously studied concepts.
* Differences between these notions are clarified.

Conclusions:

* The proposed framework offers a new and unified perspective on the computational complexity of definable relations.
* The study bridges different areas of research by highlighting equivalences and distinctions.
* This work provides a foundation for further exploration in computability and logic.