====== Coherent systems ======


Below we give a complete list of all the coherent systems with n components and k minimal path sets for some n and k.

A system is a map s:{0,1}x...x{0,1}->{0,1}
where xi=1 (=0) if the ith component works (does not work) and s(x1,...,xn)=1  (=0) if the system works (does not work).

A system is coherent if s is increasing (in a wide sense) in all the variables and it is strictly increasing in each variable at least in a point.

They are given by their minimal path sets.

A path set is a set P such that if all the components in P work, then the system works.

A path set is minimal if its does not contain other path sets.

If the minimal path sets of the system are P1,...,Pk, then: 

s(x1,...,xn)=max_i (min_{j∈Pi} xj).


**List of coherent systems**

**n=5**

The 180 systems with 5 components were obtained in:


Jorge Navarro and Rafael Rubio (2010). Computations of coherent systems with five components.  Communications in Statistics Simulation and Computation 39 (1), 68-84. JCR2010 102/110. Citas JCR hasta 2014: 23.

**n=6**


The 16145 systems with 6 components are in: {{sys6.pdf|Systems}}


**n=7**

There is a unique coherent system with 7 components and 1 minimal path set:

1: s(x1,...,x7)=min(x1,...,x7), P1={1,...,7} (series).

There are 966 coherent systems with 7 components and 2 minimal path sets. They are given in the following file:

{{sistemasn7k2_copia.txt|Systems with 7 components and 2 minimal path sets}}

{{sistemasn7k2sig_copia.txt|Systems with 7 components and 2 minimal path sets and their minimal signatures}}



There are 76482 coherent systems with 7 components and 3 minimal path sets. They are given in the following file:

{{sistemasn7k3_copia.txt|Systems with 7 components and 3 minimal path sets}}

{{sistemasn7k3sig_copia.txt|Systems with 7 components and 3 minimal path sets and their minimal signatures}}


There are 1868650 coherent systems with 7 components and 4 minimal path sets. The file is too big. Please, contact with the author at: jorgenav@um.es.



**n=8**

There is a unique coherent system with 8 components and 1 minimal path set:

1: s(x1,...,x8)=min(x1,...,x8), P1={1,...,8} (series).

There are 3025 coherent systems with 8 components and 2 minimal path sets. They are given in the following file:

{{sistemasn8k2sig_copia_2.txt|Systems with 8 components and 2 minimal path sets and their minimal signatures}}


There are 638736 coherent systems with 8 components and 3 minimal path sets. They are given in the following file:

{{sistemasn8k3sigcopia2.txt|Systems with 8 components and 2 minimal path sets and their minimal signatures}}

***

This work was partially supported by Ministerio de Ciencia y Tecnología under grant MTM2009-08311.