NAME

simecek-tools - Decode Petr Šimeček's files on representability of CI structures on 4 random variables

SYNOPSIS

$ wget -qO- http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/pbin.dat | sim-ci
111111111111101110111101
111111111111101110111111
111111111111101010101111
111111111111100111111111
111111111111111111111111
111111111111101101111111
...

$ wget -qO- http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/pbin.rep | sim-dist --moments
0 0 0 0 -1/2 -1/2 -1/2 1/4 1/4 1/4 0 0 0 0 0
0 0 0 0 -1/2 -1/2 -1/2 1/4 1/4 1/2 0 0 -1/4 1/4 -1/4
0 0 0 0 -1/2 -1/2 -1/2 1/4 1/4 1/2 0 0 0 0 -1/4
0 0 0 0 -1/2 -1/2 -1/2 1/4 1/2 1/2 0 -1/4 1/4 0 -1/2
0 0 0 0 -1/2 -1/2 -1/2 1/2 1/2 1/2 -1/4 -1/4 -1/4 1/4 -1/4
0 0 0 0 -1/2 -1/2 -1/4 1/4 0 1/4 0 0 -1/4 1/4 0
...

$ wget -qO- http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/pbin.rep | sim-dist
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(1/64) +-+-(5/64) +--+(5/64) +---(13/64) -+++(13/64) -++-(5/64) -+-+(5/64) -+--(1/64) --++(5/64) --+-(1/64) ---+(1/64) ----(1/64)
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(1/64) +-+-(5/64) +--+(5/64) +---(13/64) -+++(17/64) -++-(1/64) -+-+(1/64) -+--(5/64) --++(5/64) --+-(1/64) ---+(1/64) ----(1/64)
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(3/64) +-+-(3/64) +--+(3/64) +---(15/64) -+++(15/64) -++-(3/64) -+-+(3/64) -+--(3/64) --++(5/64) --+-(1/64) ---+(1/64) ----(1/64)
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(5/64) +-+-(1/64) +--+(1/64) +---(17/64) -+++(17/64) -++-(1/64) -+-+(5/64) -+--(1/64) --++(1/64) --+-(5/64) ---+(1/64) ----(1/64)
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(1/64) +-+-(5/64) +--+(5/64) +---(13/64) -+++(21/64) -++-(1/64) -+-+(1/64) -+--(1/64) --++(1/64) --+-(1/64) ---+(1/64) ----(5/64)
++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(1/64) +-+-(5/64) +--+(9/64) +---(9/64) -+++(13/64) -++-(5/64) -+-+(1/64) -+--(5/64) --++(5/64) --+-(1/64) ---+(1/64) ----(1/64)
...

DESCRIPTION

This repository contains some small tools to decode the files created by Petr Šimeček's when he studied representability of CI structures on four random variables and which are published on his website:

• http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/

sim-ci

Takes one of the .dat files containing CI structures and prints its contents in a binary string format compatible with https://gaussoids.de. Please refer to Šimeček's and our web page for explanations of those two formats.

sim-dist

Takes one of the .rep files (there is actually only one, pbin.rep) and prints its contents. This file contains in each record the moments of a binary probability distribution as an array of fifteen IEEE 754 doubles (little endian). With the --moments option, these moments are printed, in order and separated by spaces. Otherwise the probability distribution is reconstructed and printed.

A sample --moments line looks like this (refer to the web page for the order of moments):

0 0 0 0 -1/2 -1/2 -1/2 1/4 1/4 1/4 0 0 0 0 0

whereas when printing the whole distribution, we prefix each probability with the 4-element vector which is the elementary event of the probability. The distributions are over the state space {+1, -1}, which are abbreviated to just + and - in the output:

++++(1/64) +++-(1/64) ++-+(1/64) ++--(5/64) +-++(1/64) +-+-(5/64) +--+(5/64) +---(13/64) -+++(13/64) -++-(5/64) -+-+(5/64) -+--(1/64) --++(5/64) --+-(1/64) ---+(1/64) ----(1/64)

This script uses the bigrat pragma in Perl to do all computations with arbitrary precision rational numbers. This makes some sense because the moments are mostly negative powers of two which have an exact representation in IEEE 754. Using bigrats prevents errors from occuring when these exact numbers are combined to compute the probabilities. The last three models could not be exactly stored, please refer to Šimeček's work for exact expressions.

dual4

This is an auxilliary script to compute the dual of a given semigraphoid in the binary string format, as output by sim-ci. It can be used to see that the discrete representable CI structures on four variables are not closed under duality, simply by comparing

$ wget http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/alldis.dat
$ diff -u0 <(sim-ci alldis.dat | dual4) <(sim-ci alldis.dat) | grep ^-

SEE ALSO

• The files are available on Petr Šimeček's website: http://atrey.karlin.mff.cuni.cz/~simecek/skola/models/

THANKS

Thanks go to Petr Šimeček who did the actual work.

AUTHOR

Tobias Boege <[email protected]>

COPYRIGHT AND LICENSE

This software is copyright (C) 2019 by Tobias Boege.

This is free software; you can redistribute it and/or modify it under the terms of the Artistic License 2.0.