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: 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:

Systems with 7 components and 2 minimal path sets

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:

Systems with 7 components and 3 minimal path sets

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:

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:

Systems with 8 components and 2 minimal path sets and their minimal signatures

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