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This tool analyzes the expected termination time of population protocols. It takes as input a population protocol and outputs a parametric upper bound on its expected termination time. The tool is a prototype of the algorithm described in:

Michael Blondin, Javier Esparza and Antonín Kučera. Automatic Analysis of Expected Termination Time for Population Protocols. Proc. 29th International Conference on Concurrency Theory (CONCUR), 2018.


The tool requires Python ≥3.6 and the following dependencies:

  • z3
  • graph_tool

z3 is available at and works on the major operating systems. It can, e.g., be installed as follows:

  • Ubuntu: sudo apt-get install z3
  • OS X: brew install z3

You may have to install the Python bindings:

graph_tool is available at and works on most Linux distributions and OS X. Follow these installation instructions:


The tool can be used by running

> python3 src/ protocols/ -ov

If a protocol requires some arguments, then they can be passed in the JSON format. For example,

> python3 src/ protocols/ "[[1, 2], 0, 3]" -ov

for modulo([1, 2], 0, 3).

The flag -o outputs statistics in the JSON format. The -v flag outputs the progress of the execution. The stage tree can also be generated as a PDF as follows:

> python3 src/ protocols/ -t tree.pdf

If the tree is too large, it is possible to only generate its structure:

> python3 src/ protocols/ -s -t tree.pdf

For the list of possible arguments, run:

> python3 src/ -h


The instructions regarding benchmarking can be found in ./benchmarks.


Protocols can be specified as simple Python scripts. Many examples are given in ./protocols and can serve as a basis to add more protocols.