The concept of degrees of freedom looks so relevant to software development that I am wondering why it is not considered more often. Fortunately Michael L. Perry dedicates a full section of his blog to that concept. In this post I will quote a lot, please consider that as a sign of enthusiasm.

# A common concept in maths

The concept of DOF is central in solving systems of linear equations, and in the main post of his series, Michael L. Perry starts by focusing on mathematical system of linear equations:

A mathematical model of a problem is written with equations and unknowns. You can think of the number of unknowns as representing the number of dimensions in the problem.

Depending on the number m of equations compared to the number n of unknowns (variables), there are several cases:

- m > n: there is no solution, the problem is over-constrained
- m = n: there is only one solution
- m < n: the system is under-determined system, and the dimension of the solution set is usually equal to
*n*?*m*, where*n*is the number of variables and*m*is the number of equations.

# A common concept in mechanics

The concept of DOF is prevalent in mechanics. In particular, a system with more internal constraints than the total possible number of DOFs has no solution. However in practice it can still work, provided some of the bodies are not absolutely rigid.

The table is often given as an example, because it only needs three legs to be stable on the ground, but usually has 4 legs. This only works because the table is not fully rigid, and can accommodate the small imperfection of the ground.

# Not yet common in software

In his post, Michael L. Perry explains in practice how to analyse software using DOF. First find the unknowns:

To identify the degrees of freedom in software, start by defining the unknowns. These are usually pretty simple to spot. These are the things that can change. In a checkbook program, for example, each transaction amount is an unknown, as are the the account balance and the color used to display it (black or red).

Then find out the constraints between the DOFs.

Next, define the equations. These are the relationships between the unknowns that the software has to enforce. In the checkbook, the balance is the sum of all transaction amounts. And the color is red if the balance is negative or black otherwise.

Finally:

Subtract to find your degrees of freedom. One amount per transaction (n), one balance, and one color gives n+2 unknowns. The balance sum and the color rule give us two equations. n+2-2 = n degrees of freedom, one per transaction.

# What for?

Quoting again Michael L. Perry (across various posts in the DOF category):

Understanding the degrees of freedom in the software helps to create a maintainable design.

Adding independent data to a system increases its degrees of freedom. Adding dependent data does not. Adding an immutable field does not.

You want no more degrees of freedom in the system than the problem calls for.

The concept of degree of freedom is remarkably useful to help distilling the domain down to the essential variable parts and the constraints between them. Any extra independent data can only create opportunities for bugs.

Thanks for the great summary. I’ve started collecting ideas around mathematical concepts that can be applied to software. I’m posting them at http://qedcode.com.