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Patterns of regular expression usage: duplicate regexs
Regular expressions are widely used, but until recently they were rarely studied empirically (i.e., just theory research).
This week I discovered two groups studying regular expression usage in source code. The VTLeeLab has various papers analysing 500K distinct regular expressions, from programs written in eight languages and StackOverflow; Carl Chapman and Peipei Wang have been looking at testing of regular expressions, and also ran an interesting experiment (I will write about this when I have decoded the data).
Regular expressions are interesting, in that their use is likely to be purely driven by an application requirement; the use of an integer literals may be driven by internal housekeeping requirements. The number of times the same regular expression appears in source code provides an insight (I claim) into the number of times different programs are having to solve the same application problem.
The data made available by the VTLeeLab group provides lots of information about each distinct regular expression, but not a count of occurrences in source. My email request for count data received a reply from James Davis within the hour 🙂
The plot below (code+data; crates.io has not been included because the number of regexs extracted is much smaller than the other repos) shows the number of unique patterns (y-axis) against the number of identical occurrences of each unique pattern (x-axis), e.g., far left shows number of distinct patterns that occurred once, then the number of distinct patterns that each occur twice, etc; colors show the repositories (language) from which the source was obtained (to extract the regexs), and lines are fitted regression models of the form: 
, where: 
is driven by the total amount of source processed and the frequency of occurrence of regexs in source, and 
is the rate at which duplicates occur.
So most patterns occur once, and a few patterns occur lots of times (there is a long tail off to the unplotted right).
The following table shows values of for the various repositories (languages):
StackOverflow cpan godoc maven npm packagist pypi rubygems
-1.8 -2.5 -2.5 -2.4 -1.9 -2.6 -2.7 -2.4 |
The lower (i.e., closer to zero) the value of , the more often the same regex will appear.
The values are in the region of -2.5, with two exceptions; why might StackOverflow and npm be different? I can imagine lots of duplicates on StackOverflow, but npm (I’m not really familiar with this package ecosystem).
I am pleased to see such good regression fits, and close power law exponents (I would have been happy with an exponential fit, or any other equation; I am interested in a consistent pattern across languages, not the pattern itself).
Some of the code is likely to be cloned, i.e., cut-and-pasted from a function in another package/program. Copy rates as high as 70% have been found. In this case, I don’t think cloned code matters. If a particular regex is needed, what difference does it make whether the code was cloned or written from scratch?
If the same regex appears in source because of the same application requirement, the number of reuses should be correlated across languages (unless different languages are being used to solve different kinds of problems). The plot below shows the correlation between number of occurrences of distinct regexs, for each pair of languages (or rather repos for particular languages; top left is StackOverflow).
Why is there a mix of strong and weakly correlated pairs? Is it because similar application problems tend to be solved using different languages? Or perhaps there are different habits for cut-and-pasted source for developers using different repositories (which will cause some patterns to occur more often, but not others, and have an impact on correlation but not the regression fit).
There are lot of other interesting things that can be done with this data, when connected to the results of the analysis of distinct regexs, but these look like hard work, and I have a book to finish.
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., 
.
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.
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…
Amount of end-user usage of code in Firefox
How much end-user usage does the code in Firefox receive over time?
Short answer: The available data is very sparse and lots of hand waving is needed to concoct something.
The longer answer is below as another draft section from my book Empirical software engineering with R. As always comments and pointers to more data welcome. R code and data here.
Suggestions for alternative methods of calculation also welcome.
Amount of end-user usage of code in Firefox
Source code that is never executed will not have any faults reported against it while code that is very frequently executed is more likely to have a fault reported against it than less frequently executed code.
The Firefox browser has been the subject of several fault related studies. The study by Massacci, Neuhaus and Nguyen <book Massacci_11> is of interest here because it provides the information needed to attempt to build a fault model that takes account of the total amount of usage that code experiences from all end-users of a program. The data used by the study applies to 899 Mozilla Firefox-related Security Advisories (MFSA, a particular kind of fault), noting the earliest and latest versions of Firefox that exhibits each fault; six major releases (i.e., versions 1.0, 1.5, 2.0, 3.0, 3.5 and 3.6) were analysed; the amount of code in each version that originated in earlier versions was measured (see plot below).
Massacci et al make their raw data available under an agreement that does not permit your author to directly distribute it to readers <book ???>; the raw data for the following analysis was reverse engineered from the Massacci et al paper <book Massacci_11> or obtained from other sources.
The following analysis is an attempt to build a model of amount of Firefox code usage, by end-users, over time, i.e., number of lines of Firefox source code being executed per unit time summed over all end-users at a given moment in time. The intent is to couple this model with fault data, looking for a relationship of the form: an X% change in usage results in a Y% change in reported faults.
Figure 1. Amount of source (millions of lines) in each version broken down by the version in which it first appears. Data from Massacci, Neuhaus and Nguyen <book Massacci_11>.
As expected a large amount of code from previous versions appears in later versions.
Since we are interested in the relationship between end-user code usage and faults (MFSAs in this case) we are only interested in versions of Firefox that are actively maintained by Mozilla. Every version has a first official release date and an end-of-support date beyond which no faults reported against it are fixed; any usage of a version after the end-of-support date is not of interest in this analysis.
How many people are using each version of Firefox at any time?
A number of web sites list information on Firefox market share over time (as a percentage of all browsers measured), but only two known to your author break this information down by Firefox version. Massacci et al used url[netmarketshare.com] for Firefox version market share (data going back to November 2007), but your author found it easier to obtain information from url[www.w3schools.com] (data going back to May 2007). The W3schools data is obtained from the log of visitors to their site which will obviously be subject to fluctuations (of unknown magnitude).
For the period November 2004 to April 2007 the market share of each Firefox version was estimated as follows:
- total Firefox market share was based on that listed by url[marketshare.hitslink.com]
- during the period when only version 1.0 was available its market share was assumed to be the total Firefox market share,
- the market share for versions 1.5 and 2.0 was assumed to follow the trend of growth and decline seen in later releases for which data is available. Numbers were concocted that followed the version trend and summed to the known total market share.
The plot below shows the market share of the six versions of Firefox between official release and end-of-support. Estimated values appear to the left of the vertical red line, values from measurements to the right. It can be seen that at its end-of-support date version 2.0 still had a significant market share.
Figure 2. Market share of Firefox versions between official release and end-of-support. Data from url[www.w3schools.com].
The International Telecommunications Union publishers an estimate of the number of people per 100 head of population with Internet access for each year between 2003 and 2011 <book ITU_12>; the data is broken down by developed/developing countries and also by major world regions. Assuming that everybody who users the Internet uses a browser this information can be combined with market share and human population data to estimate the number of Firefox users.
The ITU do not provide much information about how the usage figure is calculated or even which month of the year it applies to (since we are interested in change over time knowing the month is not important and the start of the year is assumed). As the figure below shows the estimate over the period of interest can be accurately modeled by a straight line. A linear model was fitted to the data to predict usage between published estimates; over the period of interest the rate of growth in the Developed world has been almost twice as great as the rate in the whole world.
Figure 3. Number of people with Internet access per 100 head of population in the developed world and the whole world. Data from ITU <book <ITU_12>.
We are interested in relative change in total user population and this can be obtained by multiplying the per-head of population value by the change in population (a 0.8% yearly growth is assumed for the developed world).
Possible significant factors for why the formula 
might not accurately reflect the probability of a MFSA being reported include:
- the characteristics of people who started using the Internet in 2004 may be different from those who first started in 2010:
- there will be variation in the amount of time people spend browsing, does the distribution of time usage differ between early and late adopters?
- some people are more likely than others to report a fault (e.g., my mum is a late adopter and extremely unlikely to report a fault whereas I might report a fault),
- there may be significant regional differences, e.g., European users vs. Chinese users. These differences include the Internet sites visited (the behavior of Firefox will depend on the content of the web page visited) and may affect their propensity to report a problem (e.g., do the cultural stereotypes of Chinese acceptance of authority mean they are unlikely to report a fault while those noisy Americans complain about everything?)
The end-user usage for code originally written for a particular version, at a point in time, is calculated as follows:
- number of lines of code originally written for a particular version that is contained within the code used to build a later version, or that particular version; call this the build version,
- times the market share of the build version,
- times the number of Internet users of the build version (users in the Developed world was used).
The plot below is an example using the source code originally written for Firefox version 1.0. The green points are the code usage for version 1.0 code executing in Firefox build version 1.0, the orange points the code usage for version 1.0 code executing in build version 1.5 and so on to the yellow points which is the code usage for version 1.0 code executing in build version 3.6. The black points are the sum over all build versions.
Figure 4. Amount of end-user usage of code originally written for Firefox version 1.0 by various other versions.
Much of the overall growth comes from growth in Internet usage, and in the early years there is also substantial growth in browser market share.
An analysis that attempts to connect Firefox usage with reported MFSAs will appear shortly (it would be surprising if fault report rate scaled linearly with end-user usage).
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