Python Symbolic Information Theoretic Inequality Prover

## Project description

Python Symbolic Information Theoretic Inequality Prover

PSITIP is a computer algebra system for information theory written in Python. Random variables, expressions and regions are objects in Python that can be manipulated easily. Moreover, it implements a versatile deduction system for automated theorem proving. PSITIP supports features such as:

• Proving linear information inequalities via the linear programming method by Yeung and Zhang. The linear programming method was first implemented in the ITIP software developed by Yeung and Yan ( http://user-www.ie.cuhk.edu.hk/~ITIP/ ).

• Automated inner and outer bounds for multiuser settings in network information theory (see the Jupyter Notebook examples ).

• Numerical optimization over distributions, and evaluation of rate regions involving auxiliary random variables.

• Interactive mode and Parsing LaTeX code.

• Finding examples of distributions where a set of constraints is satisfied.

• Fourier-Motzkin elimination.

• Discover inequalities via the convex hull method for polyhedron projection [Lassez-Lassez 1991].

• Non-Shannon-type inequalities.

• Integration with Jupyter Notebook and LaTeX output.

• Drawing information diagrams.

• User-defined information quantities.

Documentation: https://github.com/cheuktingli/psitip

Jupyter Notebook examples: https://nbviewer.jupyter.org/github/cheuktingli/psitip/tree/master/examples/

Author: Cheuk Ting Li ( https://www.ie.cuhk.edu.hk/people/ctli.shtml ). The source code of PSITIP is released under the GNU General Public License v3.0 ( https://www.gnu.org/licenses/gpl-3.0.html ). The author would like to thank Raymond W. Yeung, Chandra Nair and Pascal O. Vontobel for their invaluable comments.

The working principle of PSITIP (existential information inequalities) is described in the following article:

If you find PSITIP useful in your research, please consider citing the above article.

## WARNING

This program comes with ABSOLUTELY NO WARRANTY. This program is a work in progress, and bugs are likely to exist. The deduction system is incomplete, meaning that it may fail to prove true statements (as expected in most automated deduction programs). On the other hand, declaring false statements to be true should be less common. If you encounter a false accept in PSITIP, please let the author know.

## Installation

To install PSITIP with its dependencies, use one of the following three options:

### A. Default installation

Run (you might need to use python -m pip or py -m pip instead of pip):

pip install psitip

If you encounter an error when building pycddlib on Linux, refer to https://pycddlib.readthedocs.io/en/latest/quickstart.html#installation .

This will install PSITIP with default dependencies. The default solver is ortools.GLOP. If you want to choose which dependencies to install, or if you encounter an error, use one of the following two options instead.

### C. Installation with pip

2. Run (you might need to use python -m pip or py -m pip instead of pip):

pip install numpy
pip install scipy
pip install matplotlib
pip install ortools
pip install pulp
pip install pyomo
pip install lark-parser
pip install pycddlib
pip install --no-deps psitip
3. If you encounter an error when building pycddlib on Linux, refer to https://pycddlib.readthedocs.io/en/latest/quickstart.html#installation .

4. (Optional) The GLPK LP solver can be installed on https://www.gnu.org/software/glpk/ or via conda.

5. (Optional) Graphviz (https://graphviz.org/) is required for drawing Bayesian networks and communication network model. A Python binding can be installed via pip install graphviz

6. (Optional) If numerical optimization is needed, also install PyTorch (https://pytorch.org/).

## References

The general method of using linear programming for solving information theoretic inequality is based on the following work:

• R. W. Yeung, “A new outlook on Shannon’s information measures,” IEEE Trans. Inform. Theory, vol. 37, pp. 466-474, May 1991.

• R. W. Yeung, “A framework for linear information inequalities,” IEEE Trans. Inform. Theory, vol. 43, pp. 1924-1934, Nov 1997.

• Z. Zhang and R. W. Yeung, “On characterization of entropy function via information inequalities,” IEEE Trans. Inform. Theory, vol. 44, pp. 1440-1452, Jul 1998.

Convex hull method for polyhedron projection:

• C. Lassez and J.-L. Lassez, Quantifier elimination for conjunctions of linear constraints via a convex hull algorithm, IBM Research Report, T.J. Watson Research Center, RC 16779 (1991)

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