The programmer at wit’s end for lack of space can often do best by disentangling himself from his code, rearing back, and contemplating his data. Representation is the essence of programming. – Fred Brooks, The Mythical Man-Month

As years pass with newer projects, the one thing I notice more is that data is more important than algorithms. You can alleviate bad algorithms with better ones during the course of a project, but to fix bad data design, a full rewrite is usually necessary.

Kolmogorov

Complexity of a sequence = The shortest algorithm that produces it.

Martin-Löf

• A sequence is random if it passes all statistical tests.
• It cannot be produced by a program shorter than itself.

The digits of $\pi$ are not random in this sense.

Not just “difficult to compute”, there is no consistent way to define the shortest algorithm.

It’s impossible to find a way to ensure that a sequence is random. It’s similar to the halting problem in a way.

My Idea

It might be possible to define an algorithm in terms of a Turing machine. If the instructions of a Turing machine are shorter than the sequence itself, it’s not random.

Another idea

It says $P(b_n | s_{n-1}) = \frac{1}{2}$ for a binary sequence if the random process is ideal. But expecting this might be against randomness.

Pseudorandomness

Producing seemingly random sequences by algorithms, given a secret key.

The sequence is determined and can be inferred with high computing power. But for practical purposes, it’s random.

Uses of Randomness

• Randomized Algorithms: When a definite algorithm is too costly, it’s possible to run randomized algorithms for e.g. checking whether a number is prime.
• Cryptography: When a message needs to be encrypted with a secure key, the most secure way to produce a key is randomness.