10.2.6 Computability, Universality  Summary and Q&A
TL;DR
There are different models of computation, but they all have the same power as Turing Machines, which can compute any computable function. The Universal Turing Machine is the foundation for modern generalpurpose computers.
Questions & Answers
Q: What were some of the different models of computation explored by early computer scientists?
Early computer scientists explored various models of computation, including recursive functions, rulebased systems for string rewriting, and the lambda calculus.
Q: What does Church's Thesis state?
Church's Thesis states that every discrete function computable by any realizable machine is computable by some Turing Machine.
Q: Is there a Universal Turing Machine?
Yes, a Universal Turing Machine (T_U) exists, capable of emulating any other Turing Machine and performing any computable function.
Q: How does the Universal Turing Machine work?
The Universal Turing Machine interprets a coded representation of a computation. It takes a program description (encoded by k) and an input (encoded by j) and emulates the steps of a specific Turing Machine to process the input and produce an output.
Summary & Key Takeaways

Early computer scientists explored different models for computation, including recursive functions, rulebased systems, and the lambda calculus, all of which could compute the same set of integer functions.

Church's Thesis states that every discrete function computable by any realizable machine can be computed by a Turing Machine.

A Universal Turing Machine exists, capable of emulating any other Turing Machine and performing any computable function.