Comparing expression usage in mathematics and C source

Why does a particular expression appear in source code?

One reason is that the expression is the coded form of a formula from the application domain, e.g., E=mc^2.

Another reason is that the expression calculates an algorithm/housekeeping related address, or offset, to where a value of interest is held.

Most people (including me, many years ago) think that the majority of source code expressions relate to the application domain, in one-way or another.

Work on a compiler related optimizer, and you will soon learn the truth; most expressions are simple and calculate addresses/offsets. Optimizing compilers would not have much to do, if they only relied on expressions from the application domain (my numbers tool throws something up every now and again).

What are the characteristics of application domain expression?

I like to think of them as being complicated, but that’s because it used to be in my interest for them to be complicated (I used to work on optimizers, which have the potential to make big savings if things are complicated).

Measurements of expressions in scientific papers is needed, but who is going to be interested in measuring the characteristics of mathematical expressions appearing in papers? I’m interested, but not enough to do the work. Then, a few weeks ago I discovered: An Analysis of Mathematical Expressions Used in Practice, by Clare So; an analysis of 20,000 mathematical papers submitted to arXiv between 2000 and 2004.

The following discussion uses the measurements made for my C book, as the representative source code (I keep suggesting that detailed measurements of other languages is needed, but nobody has jumped in and made them, yet).

The table below shows percentage occurrence of operators in expressions. Minus is much more common than plus in mathematical expressions, the opposite of C source; the ‘popularity’ of the relational operators is also reversed.

Operator  Mathematics   C source
=         0.39          3.08
-         0.35          0.19 
+         0.24          0.38
<=        0.06          0.04
>         0.041         0.11
<         0.037         0.22

The most common single binary operator expression in mathematics is n-1 (the data counts expressions using different variable names as different expressions; yes, n is the most popular variable name, and adding up other uses does not change relative frequency by much). In C source var+int_constant is around twice as common as var-int_constant

The plot below shows the percentage of expressions containing a given number of operators (I've made a big assumption about exactly what Clare So is counting; code+data). The operator count starts at two because that is where the count starts for the mathematics data. In C source, around 99% of expressions have less than two operators, so the simple case completely dominates.

Percentage of expressions containing a given number of operators.

For expressions containing between two and five operators, frequency of occurrence is sort of about the same in mathematics and C, with C frequency decreasing more rapidly. The data disagrees with me again...

Mathematical proofs contain faults, just like software

The idea of proving programs correct, like mathematical proofs, is appealing, but is based on an incorrect assumption often made by non-mathematicians, e.g., mathematical proofs are fault free. In practice, mathematicians make mistakes and create proofs that contain serious errors; those of us who are taught mathematical techniques, but are not mathematicians, only get to see the good stuff that has been checked over many years.

An appreciation that published proofs contain mistakes is starting to grow, but Magnificent mistakes in mathematics is an odd choice for a book title on the topic. Quotes from De Millo’s article on “Social Processes and Proofs of Theorems and Programs” now appear regularly; On proof and progress in mathematics is worth a read.

Are there patterns to the faults that appear in claimed mathematical proofs?

A surprisingly common approach, used by mathematicians to avoid faults in their proofs, is to state theorems without giving a formal proof (giving an informal one is given instead). There are plenty of mathematicians who don’t think proofs are a big part of mathematics (various papers from the linked-to book are available as pdfs).

Next time you encounter an advocate of proving programs correct using mathematics, ask them what they think about the uncertainty about claimed mathematical proofs and all the mistakes that have been found in published proofs.