# 8.3 Summary

The focus of Part II has been on predicting properties of procedures, in particular how their running time scales with the size of their input. This involved many encounters with the three powerful ideas introduced in Section 1.4: recursive definitions, universality, and abstraction. The simple Turing Machine model is a useful abstraction for modeling nearly all conceivable computing machines, and the few simple operations it defines are enough for a universal computer. Actual machines use the digital abstraction to allow a continuous range of voltages to represent just two values. The asymptotic operators used to describe running times are also a kind of abstraction---they allow us to represent the set of infinitely many different functions with a compact notation.

In Part III, we will see many more recursive definitions, and extend the notion of recursive definitions to the language interpreter itself. We change the language evaluation rules themselves, and see how different evaluation rules enable different ways of expressing computations.