When you have to design a standard cell library, you need to choose a number of combinatorial cells. You can simply ask a friend and start from an existing list or build your own from scratch. But how? More specifically, how many n-input Boolean functions are there?
This latter question connects to the mathematical concept of group action that is a pivotal part of Algebra. If you like Rubyk's cube, you should not feel too uncomfortable.
It can be reformulated in this context as follows.
What are the orbits in the action of the symmetric group
The name of this package is group_action.
The version of this package is 0.2.18.
It contains a library module named library and several applications : orbit, orbit_random, orbits, conjugacy_classes, burnside, powerset, and symmetric_functions.
orbit computes the orbit of a function specified through its number of inputs and signature generated by the natural action of Sn on Xn.
orbit_random computes the orbit of a random function specified through its number of inputs generated by the natural action of Sn on Xn.
orbits computes the orbits in the natural action of the symmetric group
The result is basically a list of signatures that is presented in 2 formats and 2 levels of details.
Here, a signature is a non negative integer representing a Boolean function.
When the problem becomes too big you might consider using the following commands to get the number of orbits.
conjugacy_classes computes the conjugacy classes of the symmetric group
burnside computes the Burnside's formula for a given number of inputs
powerset computes the orbits generated by the natural action of the symmetric group
symmetric_functions computes the signatures of the symmetric Boolean functions with n inputs and 1 output.
For instance, what 2-input Boolean function does 12 represent? I use Big Endian format for binary words. 12 = 0101 This gives the following truth table
x0 x1 f(x0, x1)
0 0 0
1 0 1
0 1 0
1 1 1
and the following expression f(x0, x1) = x0 & ~x1 | x0 & x1 = x0
Note: x1 does not figure in this expression.
So 12 represents a 1-input Boolean function. It gives a standard cell called a buffer.
To move forward, is 12 the only signature representing a buffer cell? Actually this answer is no! If you permut x0 and x1, you get the following truth table.
x0 x1 f(x0, x1)
0 0 0
1 0 0
0 1 1
1 1 1
It corresponds to f(x0, x1) = ~x0 & x1 | x0 & x1 = x1, another instance of the buffer cell.
This is what the group action concept is all about, and behyond :-)
Each permutation of
Each pair
From this analysis, one can get the number of orbits, a list of representatives, and the detailed contents of each orbit. Boolean functions are represented by their signatures as non negative integers.
The results are printed out to the screen or stored into a json file named data.json. Binary data is considered as Big Endian throughout the code.
Same as orbits but the symmetric group
Here the orbits are not populated like for both the previous commands, as the level of complexity grows and becomes untractable. burnside command builds the formula as a string upon two facts.
- Conjugated permutations give the same number of fixed points.
- The action of
$S_n$ on$B^n$ allows to derive the number of fixed functions for a given permutation. Then the formula is evaluated. Special attention to very long integers has to be payed.
Same as orbits and burside but with ultimate performances.
Here a single orbit is computed associated to a signature given as an input.
Same as orbit but a random signature is provided.
Compute the signatures of the symmetric functions for a given number of inputs.
Run pip install group_action.
The commands named orbit, orbit_random, orbits, conjugacy_classes, burnside, powerset, and symmetric_functions are automatically installed under $HOME/.local/bin under Ubuntu 22.04 when you install the package.
Make sure your path is updated with $HOME/.local/bin. Check the following link for more information.
You are ready to go :-)
usage: orbit [-h] [--version] [--s S] [--n N] [--c C]
Computation of the orbit of a n-input, 1-output Boolean function specified via its signature as a LE integer under the
action of the symmetric group Sn via its transpositions.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--s S Signature
--n N Number of inputs
--c C Number of cores
usage: orbit_random [-h] [--version] [--i I] [--n N] [--c C]
Iterate on computation of the orbit of a n-input, 1-output Boolean function specified via a random signature as a LE
integer under the action of the symmetric group Sn via its transpositions.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--i I iterations
--n N Number of inputs
--c C Number of cores
orbits [-h] [--version] [--n N] [--c C] [--r] [--v] [--j]
Brut force computation of orbits of n-input 1-output Boolean functions under the action of the symmetric group Sn.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--n N Number of inputs
--c C Number of cores
--v Output every element of each orbit
--j Output data.json file
usage: conjugacy_classes [-h] [--n N] [--c C] [--v] [--j]
Brut force computation of conjugacy classes of the symmetric group Sn.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--n N Number of elements in S
--c C Number of cores
--v Output every element of each orbit
--j Output data.json file
usage: burnside [-h] [--n N] [--c C]
Build Burnside's formula and compute the number of orbits generated by the action of the symmetric group Sn on the set of n-input Boolean
functions B^{B^n}
options:
-h, --help show this help message and exit
--version show program's version number and exit
--n N Number of inputs
--c C Number of cores
usage: powerset [-h] [--version] [--n N] [--c C] [--v] [--j]
Computation of the orbits generated by the action of the symmetric group Sn via its transpositions on n-input Boolean functions generated by actions on power sets.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--n N Number of inputs
--c C Number of cores
--v Output every element of each orbit
--j Output data.json file
usage: symmetric_functions [-h] [--version] [--n N]
Generate each symmetric n-input 1-output Boolean function.
options:
-h, --help show this help message and exit
--version show program's version number and exit
--n N Number of inputs
After the installation, run orbits --n 3 --c 12 in order to run on 12 cores and to get the number of orbits and a representative of each orbit as an integer signature for 3-input, 1-output Boolean functions.
Activate the verbose mode running orbits --n 3 --c 12 --v in order to get the orbits populated.
n=0 2 orbits
n=1 4 orbits
n=2 12 orbits
n=3 80 orbits
n=4 3 984 orbits
- The group action is concrete and set up to
$G=S_n$ and$X=B^{B^n}$ in this version. - The packaging is managed through PyPI, not yet synchronized to GitHub.
- The length of the integer strings used to represent the signatures is limited to 8 192 characters.
Any comment and/or improvement whether on optimization, packaging, documentation, or on any other appropriate topic is welcome :-)
You can reach me by email: antoine AT sirianni DOT ai. I'll do my best to provide you with support.
Check ORConf 2024 presentation video on Open Source Standard Cell Library Design to get to know me.
Check ORConf 2024 presentation slides on Open Source Standard Cell Library Design.
Type pydoc -w group_action.library to generate the HTML documentation of the library module.