Using regression tests

Overview

Questions

• Test-driven development

Objectives

• Be able to create and run test files
• Understand how test discrepancies and runtime regressions can be identified and interpreted
• Understand how to adjust tests to check randomised algorithms
• Learn the ‘Make it right, then make it fast’ concept

The code of AvgOrdOfGroup is very simple, and nothing could possibly go wrong with it. By iterating over the group instead of creating a list of its elements, it avoids running out of memory (calling AsList(SymmetricGroup(11)) already results in exceeding the permitted memory). That said, the computation still takes time, with several minutes needed to calculate the average order of an element of SymmetricGroup(11). But at least we are confident that it is correct.

Now we would like to write a better version of this function using some theoretical facts we know from Group Theory. We may put avgord.g under version control to revert changes if need be; we may create a new function to keep the old one around and compare the results of both; but this could be made even more efficient if we use regression testing: this is the term for testing based on rerunning previously completed tests to check that new changes do not impact their correctness or worsen their performance.

To start with, we need to create a test file. The test file looks exactly like a GAP session, so it is easy to create it by copying and pasting a GAP session with all GAP prompts, inputs and outputs into a text file (a test file could be also created from a log file with a GAP session recorded with the help of LogTo). During the test, GAP will run all inputs from the test file, compare the outputs with those in the test file and report any differences.

GAP test files are just text files, but the common practice is to name them with the extension .tst. Now create the file avgord.tst in the current directory (to avoid typing the full path) with the following content:

GAP

# tests for average order of a group element

# permutation group
gap> S:=SymmetricGroup(9);
Sym( [ 1 .. 9 ] )
gap> AvgOrdOfGroup(S);
3291487/362880

As you see, the test file may include comments, with certain rules specifying where they may be placed, because one should be able to distinguish comments in the test file from GAP output started with #. For that purpose, lines at the beginning of the test file that start with #, and one empty line together with one or more lines starting with #, are considered as comments. All other lines are interpreted as GAP output from the preceding GAP input.

To run the test, one should use the function Test (see documentation. For example (assuming that the function AvgOrdOfGroup is already loaded):

GAP

Test("avgord.tst");

OUTPUT

true

In this case, Test reported no discrepancies and returned true, so we conclude that the test has passed.

We will not cover the topic of writing a good and comprehensive test suite here, nor will we cover the various options of the Test function, allowing us, for example, to ignore differences in the output formatting, or to display the progress of the test, as these are described in its documentation.

Instead, we will now add more groups to avgord.tst, to demonstrate that the code also works with other kinds of groups, and to show various ways of combining commands in the test file:

GAP

# tests for average order of a group element

# permutation group
gap> S:=SymmetricGroup(9);
Sym( [ 1 .. 9 ] )
gap> AvgOrdOfGroup(S);
3291487/362880

# pc group
gap> D:=DihedralGroup(512);
<pc group of size 512 with 9 generators>
gap> AvgOrdOfGroup(D);
44203/512
gap> G:=TrivialGroup();; # suppress output
gap> AvgOrdOfGroup(G);
1

# fp group
gap> F:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^8=b^2=1, b^-1ab=a^-1");
<fp group on the generators [ a, b ]>
gap> IsFinite(G);
true
gap> AvgOrdOfGroup(G);
59/16

# finite matrix group over integers
gap> AvgOrdOfGroup( Group( [[0,-1],[1,0]] ) );
11/4

# matrix group over a finite field
gap> AvgOrdOfGroup(SL(2,5));
221/40

Let us test the extended version of the test again and check that it works:

GAP

Test("avgord.tst");

OUTPUT

true

Now we will work on a better implementation. Of course, the order of an element is an invariant of a conjugacy class of elements of a group, so we need only to know the orders of conjugacy classes of elements and their representatives. The following code, which we add into avgord.g, reads into GAP and redefines AvgOrdOfGroup without any syntax errors:

GAP

AvgOrdOfGroup := function(G)
local cc, sum, c;
cc:=ConjugacyClasses(G);
sum:=0;
for c in cc do
  sum := sum + Order( Representative(c) ) * Size(cc);
od;
return sum/Size(G);
end;

but when we run the test, here comes a surprise!

GAP

Read("avgord.g");
Test("avgord.tst");

OUTPUT

########> Diff in avgord.tst, line 6:
# Input is:
AvgOrdOfGroup(S);
# Expected output:
3291487/362880
# But found:
11/672
########
########> Diff in avgord.tst, line 12:
# Input is:
AvgOrdOfGroup(D);
# Expected output:
44203/512
# But found:
2862481/512
########
########> Diff in avgord.tst, line 23:
# Input is:
AvgOrdOfGroup(G);
# Expected output:
59/16
# But found:
189/16
########
########> Diff in avgord.tst, line 29:
# Input is:
AvgOrdOfGroup(SL(2,5));
# Expected output:
221/40
# But found:
69/20
########
false

Indeed, we made a typo (deliberately) and replaced Size(c) by Size(cc). The correct version of course should look as follows:

GAP

AvgOrdOfGroup := function(G)
local cc, sum, c;
cc:=ConjugacyClasses(G);
sum:=0;
for c in cc do
  sum := sum + Order( Representative(c) ) * Size(c);
od;
return sum/Size(G);
end;

Now we will fix this in avgord.g, and read and test it again to check that the tests run correctly.

GAP

Read("avgord.g");
Test("avgord.tst");

OUTPUT

true

Thus, the approach ‘Make it right, then make it fast’ helped detect a bug immediately after it has been introduced.

Key Points

• It is easy to create a test file by copying and pasting a GAP session.
• Writing a good and comprehensive test suite requires some effort.
• Make it right, then make it fast!