A STRUCTURED WIKIPEDIA FOR MATHEMATICS
Mathematics in a Web 2.0 World
Henry Lin
∗
Institute for Theoretical Computer Science, FIT Building 1-208, Tsinghua University, 100084, Beijing, China
Keywords:
Online collaboration, Mathematics, Organization, Web 2.0 technologies.
Abstract:
In this paper, we propose a new idea for developing a collaborative online system for storing mathematical
work similar to Wikipedia, but much more suitable for storing mathematical results and concepts. The main
idea proposed in this paper is to design a system that would allow users to store mathematics in a structured
manner, which would make related work easier to find. The proposed system would have users use indentation
to add a hierarchical structure to mathematical results and concepts entered into the system. The hierarchi-
cal structure provided by the indentation of results and concepts would provide users with additional search
functionality useful for finding related work. Additionally, the system would automatically link related results
by using the structure provided by users, and also provide other useful functionality. The system would be
flexible in terms of letting users decide how much structure to add to each mathematical result or concept to
ensure that contributors are not overly burdened with having to add too much structure to each result. The
system proposed in this paper serves as a starting point for discussion on new ideas to organize mathematical
results and concepts, and many open questions remain for new research.
1 INTRODUCTION
As the amount of research in the mathematical sci-
ences continues to grow, it is becoming increasingly
difficult to stay up to date and find research that is rel-
evant to one’s line of work. A single conference or
journal can produce hundreds of pages of mathemat-
ical work each year, and many disciplines have mul-
tiple conferences and journals devoted to them occur-
ring each year. Moreover, as different research areas
continue to expand and become more interconnected,
a researcher working on some topics may have to
stay familiar with research from many different sub-
ject areas and disciplines. For example, a theoreti-
cal computer scientist working on a topic like algo-
rithmic game theory may have to stay up to date on
papers appearing in conferences and journals in the-
oretical computer science, operations research, eco-
nomics, and mathematics. As a result of the large
amount of new research produced every year, mathe-
matical results are often forgotten or overlooked, only
to be rediscovered later with much additional effort.
∗
This work was supported in part by the National
Natural Science Foundation of China Grant 60553001,
and the National Basic Research Program of China Grant
2007CB807900,2007CB807901.
To address the challenge of making related math-
ematical work easier to find, in this paper, we pro-
pose developing a new online system, which would
store mathematical results in a simple structured man-
ner and would make related work easier to find. Our
new system would be a collaborative website like
Wikipedia, but it would have additional structure and
functionality to make finding related work easier. In
order for our system to be useful, we would want our
system to be simple to understand and easy to use, yet
still have the functionality to link related results auto-
matically, and provide additional search capabilities
to find related work.
In line with the goal of making the system easy
to use and understand, we simply plan to ask users
to add structure to their results by indenting their re-
sults in a natural way, forming a hierarchical structure
(see Figure 1 for a concrete example). The indenta-
tion would be done so that if a line of text is indented
further below another line of text, it would mean that
the later, more indented line of text describes or mod-
ifies the less indented line of text, closest above it. For
example in Figure 1, we can see that the line contain-
ing “binary random variables” modifies and refines
the less indented line above it defining the variables
x
1
,x
2
,...,x
n
. The additional structure provided by
279
Lin H. (2010).
A STRUCTURED WIKIPEDIA FOR MATHEMATICS - Mathematics in a Web 2.0 World.
In Proceedings of the 5th International Conference on Software and Data Technologies, pages 279-284
DOI: 10.5220/0002922602790284
Copyright
c
 SciTePress

A special case of Chernoff’s bound
• Given Conditions:
– Let x
1
, x
2
, ..., x
n
be
∗ independent random variables
∗ binary random variables
· where each x
i
variable has probability
1
2
of being 0 or 1
– Let X =
∑
n
i=1
x
i
– Let µ = E[X]
• Conclusion:
– P[X ≥ (1 − δ)µ] ≤ e
−δ
2
µ
, for δ ∈ (0,1)
Figure 1: An example of how indentation would be used to
add structure to each result.
users in this indented manner could allow for some
very useful search functionality, and by analyzing the
text and indentation structure of each mathematical
result, the system could identify and link related re-
sults by looking for similar text and indentation struc-
ture in different results.
Before we describe our proposed system in more
detail, we first describe some existing resources on
the Internet in Section 2. Then, in Sections 3 and 4,
we describe how our system might automatically link
related theorems and mathematical objects like com-
plexity classes and algorithmic problems in computer
science. In Section 5, we describe some additional
search functionality that could be implemented to find
related work in our system, and finally, we conclude
with some open problems in Section 6.
2 EXISTING MATHEMATICS
RESOURCES ONLINE
Fortunately, for researchers working in the mathemat-
ical sciences, there are a variety of resources avail-
able online containing a large amount of mathemat-
ical work. Unfortunately, the problem is that find-
ing related work in many of the existing resources
on the Internet can still be a challenge. For exam-
ple, Wikipedia (Wales et al., 2010), PlanetMath (Egge
et al., 2010), and Wolfram’s Mathworld (Weisstein
et al., 2010) all contain a great deal of work on math-
ematics, but finding the precise result one is looking
for can still be a challenge. The mathematical results
in these systems are typically listed alphabetically by
name, are assigned keywords to facilitate searching,
and/or are organized into broad categories. How-
ever, many theorems often have arbitrary names based
on the mathematician(s) who discovered them (e.g.,
the Cook-Levin theorem or Hoeffding’s inequality),
which can make it difficult to search for a particular
theorem, if one does not know the name of the mathe-
matician who discovered it. Furthermore, attempting
to find a result by category or keyword search can re-
quire a researcher to browse through a large number
of results.
Similar problems exist for other mathematical re-
sources on the web, such as the Open Problem Gar-
den (DeVos et al., 2010), which stores open problems
in mathematics, and the Complexity Zoo (Aaronson
et al., 2010), which stores complexity classes in the-
oretical computer science. For similar reasons, it can
also be difficult to search for related algorithmic prob-
lems in the Complexity Garden (Monroe et al., 2010)
and the NP Compendium (Crescenzi et al., 2010),
which store results on various algorithmic problems
in theoretical computer science. (We use 2010 for the
citation year because many of the above resources are
still being developed and improved upon, although
many were initially created earlier).
The only resource on the web (as far as the author
knows) that does a very good job of organizing related
results appears to be the Scheduling Zoo (Brucker and
Knust, 2010), which stores results known about var-
ious scheduling problems. In order to search for re-
sults known about a particular scheduling problem, a
user is allowed to select various parameters and con-
ditions which define a scheduling problem, and then
query the system to see if anything is known about
the problem selected. The only limitation of this re-
source is that it was built specifically to store results
related to job scheduling problems. It currently can-
not be used to store other types of results, and outside
users cannot contribute new knowledge to the system.
In contrast, the system we propose to develop would
seek to match the organization and search capability
as provided by the Scheduling Zoo, but it would be
capable of storing a wider variety of results. More-
over, any user would be allowed to contribute to it.
It is our hope that our new system would help
organize and link related mathematical results and
mathematical objects provided in the resources men-
tioned above. Our resource would not necessarily
replace the resources mentioned above, but would
complement them. For example, our system might
store and link related mathematical theorems, but it
might not list any proofs if they are already pro-
vided by other websites. For those proofs, our system
might just provide links to proofs listed on existing
resources, like Wikipedia and Planetmath. Similarly,
our system might only store and link the definitions of
related complexity classes, and provide links to rele-
vant entries in the Complexity Zoo for users to find
ICSOFT 2010 - 5th International Conference on Software and Data Technologies
280

out more about those complexity classes on the Com-
plexity Zoo website itself.
3 ORGANIZING
MATHEMATICAL THEOREMS
In this section, we illustrate how our system might
work to link related mathematical theorems by imag-
ining what would happen as related theorems are en-
tered into the system. As we will see, our system will
have three main mechanisms for linking related re-
sults. The first mechanism simply creates a list of re-
lated work for each entry in the system by scanning
entries for similar structure and text. Results with the
most similarity in terms of structure and text, with the
result being examined, would be displayed highest on
the list of related results for the entry being examined.
Furthermore, if any two results are similar enough,
the system may also opt to display both results on the
same page, or otherwise, it may also create a drop
down box so that users can traverse directly between
two related results. We will illustrate all three of these
mechanisms with some examples below.
Let us first imagine that our first theorem shown
in Figure 1 is added into the system, and then the
second theorem shown in Figure 2 is also added to
the system. Assuming that this second theorem is en-
tered into the system as shown in Figure 2, note that
this theorem has exactly the same indentation struc-
ture and text as the first theorem in Figure 1, except
for the last line in the conclusion. As our system au-
tomatically searches for related results, it would look
at each existing entry in the system, and check how
closely its structure and text matches the theorem be-
ing examined. Indeed, if the two results were entered
as shown, it would be easy for the system to detect
that these two results were related and list them highly
in each other’s related results list.
Although one might wonder if we can really ex-
pect a second user to enter this second theorem in
exactly the same format as in the first theorem, this
might not be too hard to imagine, if we assume that
the second user first tried to search the system for re-
sults similar to his/her result before entering his/her
new result. By using a standard keyword search to
look for results containing the words “independent,”
“binary,” and “random variable” it might not be un-
reasonable to assume that our second user would have
found our first theorem. Assuming that he was able
to find our first result shown in Figure 1, it would
not be hard to imagine that this second user would
just copy the text used to describe the first result, and
only change the conclusion line. In this manner, both
Another special case of Chernoff’s bound
• Given Conditions:
– Let x
1
, x
2
, ..., x
n
be
∗ independent random variables
∗ binary random variables
· where each random variable x
i
has prob-
ability
1
2
of being 0 or 1
– Let X =
∑
n
i=1
x
i
– Let µ = E[X]
• Conclusion:
– P[X ≥ (1 + δ)µ] ≤ e
−δ
2
µ
, for δ > 0
Figure 2: A second theorem added to our system.
results would be closely linked as described. In the
worst case, where two users created two very differ-
ent entries which were not similar at first, we would
hope that some user would discover this later and help
put both entries in a similar format, which would link
them automatically.
Additionally, note that the two results have ex-
actly the same text and structure listed in terms of
their “Given Conditions.” For convenience, we might
imagine that our system could be designed to auto-
matically list these two results together on the same
page, so that one does not have to navigate between
pages to learn about these two very related results.
The special conditions that would cause two results to
be automatically listed on the same page might have
to be fixed and specifically implemented by the sys-
tem designer, but in case the defined rules do not give
the best results, we would also add functionality to
allow users to decide for themselves whether or not
two results should be placed on the same page, or be
placed on two separate pages.
Now, as we imagine more results being added to
the system, we might also imagine that the theorem
shown in Figure 3 would be closely linked to the first
two theorems as well, if added to the system. This
new theorem only differs from the first two theorems
in terms of the conclusion and in terms of the line that
defines the probability with which each random vari-
able is 0 or 1 (shown in italics), so this result would
also be listed highly among the related results of the
first two theorems. Also, since this result only dif-
fers from the previous two results at one point in the
“Given Conditions” section, we might imagine that
our system would take this opportunity to link these
results with a drop down box at their point of differ-
ence. For example, the third theorem would have a
drop down box to switch the condition that defines the
random variable x
i
to have probability p
i
of being 1, to
A STRUCTURED WIKIPEDIA FOR MATHEMATICS - Mathematics in a Web 2.0 World
281

A general case of Chernoff’s bound
• Given Conditions:
– Let x
1
, x
2
, ..., x
n
be
∗ independent random variables
∗ binary random variables
· where each x
i
variable has probability p
i
of being 1, and probability (1− p
i
) of be-
ing 0, for i = 1,... , n
– Let X =
∑
n
i=1
x
i
– Let µ = E[X]
• Conclusion:
– P[X ≥ (1 + δ)µ] ≤ (e
−δ
/(1+ δ)
1+δ
)
µ
,
for δ > 0
Figure 3: A third theorem added to our system.
a condition that defines each x
i
variable to have prob-
ability 1/2 of being 1. Some thought might be needed
to decide when the system should automatically link
two or more related results with a drop down box, but
in case the defined rules do not give the best results,
we would also make sure to give users the power to
link and unlink results with a drop down box, if the
system does not produce good results.
Finally, we show two more results in Figure 4 that
would be listed as results related to our previous three
results. The first result shown in Figure 4, Hoeff-
ding’s inequality, might also be linked to the first three
results with a drop down box because it would only
involve switching one line (shown in italics) into two
adjacent lines in the previousthree theorems. The sec-
ond result shown in Figure 4, the Azuma-Hoeffding
inequality, might not be automatically linked with the
prior four results with a drop down box, however, be-
cause it may contain too many line differences with
the four prior results. It would however most likely
still be displayed very highly on the related results
list of each of the prior four results, and users would
be allowed to manually link these results with a drop
down box that switches the relevant lines of text.
By linking these five results as described, we
would hope that users would have a much easier time
of finding these related results. If someone was un-
familiar with the Azuma-Hoeffding inequality, but at
least knew of one of the first four related results, like
the special case of Chernoff’s bound, then they could
first browse for Chernoff’s bound. Then upon reach-
ing the Chernoff bound entry, they could then browse
the related works listed to find the Azuma-Hoeffding
inequality.
Hoeffding’s Inequality
• Given Conditions:
– Let x
1
, x
2
, ..., x
n
be
∗ independent random variables
∗ such that x
i
∈ [a
i
,b
i
] almost surely,
for i = 1,... , n
– Let X =
∑
n
i=1
x
i
– Let µ = E[X]
• Conclusion:
– P[X ≥ µ + δ] ≤ e
−2δ
2
/
∑
n
i=1
(a
i
−b
i
)
2
, for δ > 0
Azuma-Hoeffding Inequality
• Given Conditions:
– Let x
1
, x
2
, ..., x
n
be
∗ such that Y
i
= x
1
+ . . . + x
i
forms a martin-
gale, for i = 1,...,n
∗ such that x
i
< c
i
almost surely,
for i = 1,... , n
– Let X =
∑
n
i=1
x
i
– Let µ = E[X]
• Conclusion:
– P[X ≥ δ] ≤ e
−δ
2
/(2
∑
n
i=1
c
i
2
)
, for δ > 0
Figure 4: Two more theorems that would be linked as re-
lated results.
4 ORGANIZING ALGORITHMIC
PROBLEMS AND
COMPLEXITY CLASSES
In this section, we provide some additional examples
on how our system could be used to organize and
link related complexity classes and algorithmic prob-
lems. These examples show how our system might
be used to make it easier to find complexity classes
listed in the Complexity Zoo (Aaronson et al., 2010)
and the algorithmic problems listed in the Complexity
Garden (Monroe et al., 2010) and NP Compendium
(Crescenzi et al., 2010).
4.1 Using the System to Organize
Algorithmic Problems
In Figure 5, we illustrate how the vertex cover prob-
lem could be linked to the vertex dominating set prob-
lem. Note that the problems are defined in very sim-
ilar ways, except the vertex cover problem requires
ICSOFT 2010 - 5th International Conference on Software and Data Technologies
282

Vertex Cover
• Given Input:
– A graph G = (V,E)
• Required Output:
– A subset of nodes V
′
⊆ V of minimum size
such that
∗ Each edge in E is adjacent to a node in V
′
(a) Vertex Cover Problem Definition
Vertex Dominating Set
• Given Input:
– A graph G = (V,E)
• Required Output:
– A subset of nodes V
′
⊆ V of minimum size
such that
∗ Each node in V is adjacent to a node in V
′
(b) Vertex Dominating Set Problem Definition
Figure 5: An example showing how the vertex cover prob-
lem is related to the vertex dominating set problem.
that each edge in E is adjacent to a node in V
′
, while
the vertex dominating set problem requires that each
vertex in V is adjacent to a node in V
′
. Our system
would recognize that these two problems as very sim-
ilar and link the two results together by creating a drop
down box for the last line of each definition. The drop
down box would allow users visiting the vertex cover
page for example to switch the last line of the problem
definition to require that each vertex in V is adjacent
to a node in V
′
instead, and thus allow the user to
reach the related problem of finding a minimum ver-
tex dominating set.
Furthermore, it is not hard to see that many other
algorithmic problems could be linked in a similar
way. For example, the minimum cut, minimum k-cut,
minimum multi-cut, and minimum multiway cut all
have very similar definitions which could be linked
as well. Moreover, each of the problems mentioned
above has a variant where the objective is to be maxi-
mized, and those versions could also be linked.
4.2 Using the System to Organize
Complexity Classes
In Figure 6, we illustrate how the complexity classes
NP, RP, and BPP can all be linked together. Note that
each complexity class has a very similar definition,
except for the quantifiers which specify how many
computationpaths must accept on ’yes’ instances, and
NP: Nondeterministic Polynomial-Time
• The class of decision problems solvable by an
NP machine such that:
– If the answer is ’yes,’ at least one of the
computation paths accept.
– If the answer is ’no,’ all of the computation
paths reject.
RP: Randomized Polynomial-Time
• The class of decision problems solvable by an
NP machine such that:
– If the answer is ’yes,’ at least 1/2 of the
computation paths accept.
– If the answer is ’no,’ all of the computation
paths reject.
BPP: Bounded-Error Probabilistic Polynomial-
Time
• The class of decision problems solvable by an
NP machine such that:
– If the answer is ’yes,’ at least 2/3 of the
computation paths accept.
– If the answer is ’no,’ at least 2/3 of the
computation paths reject.
Figure 6: An example showing how the complexity classes
NP, RP, BPP, and P all have very similar definitions.
how many computation paths must reject on ’no’ in-
stances. Thus the complexity classes mentioned could
all be linked by allowing the user to select how many
computation paths must be accepted by a ’yes’ in-
stance (at least one, at least 1/2, at least 2/3, or all)
and how many computation paths must be rejected by
a ’no’ instance, by using two drop down boxes. Al-
though each combination of acceptance and rejection
probability requirements might not yield a standard
well-known complexity class, our system would in-
form users when they select a combination of require-
ments, which does not yield a standard complexity
class. Note that the complexity classes co-NP and co-
RP could also be linked with the above results. Al-
though it may not be trivial to get the system to au-
tomatically recognize and link the above complexity
classes with drop downboxes, it would not be difficult
to implement functionality to allow users to manually
link results with drop down boxes.
A STRUCTURED WIKIPEDIA FOR MATHEMATICS - Mathematics in a Web 2.0 World
283

5 EXTENDED SEARCH
FUNCTIONALITY
If we look at the indented structure provided in our
examples in Section 3, we see that the structure pro-
vided could be useful for providing additional search
functionality. For example, if we ask users to struc-
ture each theorem with a “Given Conditions” section
and a “Conclusion” section, we could make queries to
the system that could ask for all theorems that contain
certain objects in the “Given Conditions” section or
the “Conclusion” section. Moreover, we might allow
users to query for certain keywords or indented struc-
tures appearing in the “Given Conditions” or “Con-
clusion” section of a theorem. Similarly when search-
ing for objects like algorithmic problems, we might
want to search for all problems which contain a cer-
tain keyword like “graph” in the “Given Input” sec-
tion of the problem, and/or “subset of nodes” in the
“Required Output” section of the problem. There may
also be other functionality which would be useful to
have, although we leave this as an open question.
6 CONCLUSIONS
In this paper, we have presented one idea for de-
veloping a system to store mathematical results and
concepts in a structured manner, which would serve
to help users find related work more easily. How-
ever, there are still many open questions that could
be asked. For example, the system described in this
paper is very simple, but is it too simple? Should
other rules be added to ensure that entries are con-
sistent and related concepts are linked properly? How
do we handle results which can be written in many
different ways? One way would be to add special
functionality to allow users to mark various entries
as equivalent, and searches would take into account
equivalent representations when searching for related
results. However, it may become confusing if a result
has too many equivalent representations, so it may be
good to create some guidelines to ensure that each en-
try does not have too many redundant and useless rep-
resentations. Besides the search functionality men-
tioned above, what other functionality should be im-
plemented so that users can find related work? The
system presented here is just one idea for organizing
mathematical results and there may be room for re-
finement and improvement. There may also be other
good ideas for storing mathematical results, and we
hope that this paper can serve as a starting point for
further thought and discussion.
Additionally, there are a few other projects related
to designing systems to help mathematicians conduct
research. For example, there is the Tricki website
(Gowers et al., 2010b), which serves to store common
tricks useful for solving mathematical problems, and
the polymath project (Gowers et al., 2010a), which
seeks to enable many mathematicians to get together
to solve the same mathematical problem. These two
projects have the same problem of finding good ways
to organize and link related proof techniques and
ideas for proofs. Can a system be described to help
organize techniques for proving mathematical results,
and/or organize different ideas for proving a theorem?
Lastly, one might ask, are there other tools that could
be developed to help researchers find related work or
conduct research more effectively in general?
REFERENCES
Aaronson, S., Kuperberg, G., Granade, C., et al. (2010).
Complexity zoo. http://qwiki.stanford.edu/wiki/com-
plexity zoo.
Brucker, P. and Knust, S. (2010). The scheduling zoo.
http://www.lix.polytechnique.fr/˜durr/query/.
Crescenzi, P., Kann, V., Halldorsson, M., Karpin-
ski, M., and Woeginger, G. (2010). A
compendium of NP optimization problems.
http://www.csc.kth.se/∼viggo/problemlist/.
DeVos, M., Samal, R., et al. (2010). Open problem garden.
http://garden.irmacs.sfu.ca.
Egge, N., Krowne, A., et al. (2010). Planetmath.
http://www.planetmath.org.
Gowers, T. et al. (2010a). Polymath.
http://www.polymathprojects.org.
Gowers, T. et al. (2010b). Tricki. http://www.tricki.org.
Monroe, H. et al. (2010). Complexity garden.
http://qwiki.stanford.edu/wiki/complexity garden.
Wales, J., Sanger, L., et al. (2010). Wikipedia.
http://www.wikipedia.org.
Weisstein, E., Wolfram, S., et al. (2010). Mathworld.
http://mathworld.wolfram.com/.
ICSOFT 2010 - 5th International Conference on Software and Data Technologies
284