Monotone Boolean Function (MBF)
===============================

Organization of the data
------------------------

This repository contains the MBFs for 0 <= n <= 6.
The directories contain the following data:
 - mbfs: list of all monotone boolean functions per profile
 - imbfs: inequivalent MBF and number of MBF associated
 - na-imbs: inequivalent MBF non-1additive and number of MBF associated
 - n2a-imbs: inequivalent MBF non-2additive and number of MBF associated
 - a-imbfs: inequivalent MBF 1additive and number of MBF associated
 - 2a-imbfs: inequivalent MBF 2additive and number of MBF associated
Each directory contains subfolders in which the (i)MBFs for n criteria are
stored per profile. The MBFs are stored in pickle format and bzipped.
To load the MBFs from these files, one can use the following Python code:
  > import pickle, bz2
  > f = bz2.BZ2File(filepath, 'rb')
  > imbfs = pickle.load(f)
  > f.close()
Except for the objects in the "mbfs" directory, the imbfs object obtained
after unpickling the file content is a dictionnary having as key the IMBF
and as value the number of isomorphic MBFs associated to the IMBF.
The format of the object after unpickling elements in the mbfs directory
is a set containing the different MBFs.
In each of these subdirectory, a file 'list.txt' contains the list of MBFs
that are stored in the .pkl.bz2 files.

Some results
------------

The first table below contains the number of different MBFs for n
variables, D(n), and the number of MBFs that are not 1-additive, N-1add D(n),
or 2-additive, N-2add R(n). For 0 <= n <= 6, all MBFs are 3-additive.

n |      D(n) |        N-1add D(n)  |       N-2add D(n)
--+-----------+---------------------+------------------
0 |         2 |         0 (00.00 %) |       0 (00.00 %)
1 |         3 |         0 (00.00 %) |       0 (00.00 %)
2 |         6 |         0 (00.00 %) |       0 (00.00 %)
3 |        20 |         0 (00.00 %) |       0 (00.00 %)
4 |       168 |        18 (10.71 %) |       0 (00.00 %)
5 |     7 581 |     4 294 (56.64 %) |       0 (00.00 %)
6 | 7 828 354 | 7 584 196 (96.88 %) | 145 502 (01.86 %)


The second table contains the number of different Inequivalent MBFs (IMBFs)
that are, R(n), and the number of IMBFs that are not 1-additive, N-1add R(n),
or 2-additive, N-2add R(n).

n  |   R(n) |      N-1add R(n) |   N-2add R(n)
---+--------+------------------+--------------
0  |      2 |      0 (00.00 %) |   0 (00.00 %)
1  |      3 |      0 (00.00 %) |   0 (00.00 %)
2  |      5 |      0 (00.00 %) |   0 (00.00 %)
3  |     10 |      0 (00.00 %) |   0 (00.00 %)
4  |     30 |      4 (13.33 %) |   0 (00.00 %)
5  |    210 |     91 (43.33 %) |   0 (00.00 %)
6  | 16 353 | 15 240 (93.19 %) | 338 (02.07 %)

References
----------

* E. Ersek Uyanic, O. Sobrie, V. Mousseau, and M. Pirlot. Listing the
  families of sufficient coalitions of criteria involved in sorting
  procedures. In DA2PL 2014 Workshop From Multiple Criteria Decision Aid to
  Preference Learning, pages 60-70, 2014. Paris, France

* Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean
  functions, Discrete Applied Mathematics, 167 (2014), 15-24.