Editing Turing machine (section)

== Comparison with the arithmetic model of computation ==
The [[arithmetic model of computation]] differs from the Turing model in two aspects:<ref name=":0">{{Cite Geometric Algorithms and Combinatorial Optimization}}</ref>{{Rp|page=32}}

* In the arithmetic model, every real number requires a single memory cell, whereas in the Turing model the storage size of a real number depends on the number of bits required to represent it. 
* In the arithmetic model, every basic arithmetic operation on real numbers (addition, subtraction, multiplication and division) can be done in a single step, whereas in the Turing model the run-time of each arithmetic operation depends on the length of the operands.

Some algorithms run in polynomial time in one model but not in the other one. For example:

* The [[Euclidean algorithm]] runs in polynomial time in the Turing model, but not in the arithmetic model.
* The algorithm that reads ''n'' numbers and then computes <math>2^{2^n}</math> by [[repeated squaring]] runs in polynomial time in the Arithmetic model, but not in the Turing model. This is because the number of bits required to represent the outcome is exponential in the input size.

However, if an algorithm runs in polynomial time in the arithmetic model, and in addition, the binary length of all involved numbers is polynomial in the length of the input, then it is always polynomial-time in the Turing model. Such an algorithm is said to run in [[Strongly polynomial|strongly polynomial time]].