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BooleanConsecutiveFunction—Wolfram Language Documentation

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• See Also
  • BooleanCountingFunction
  • BooleanConvert
  • ReliabilityDistribution
  • FailureDistribution
  • UnateQ
• Related Guides
  • Boolean Computation
  • Reliability

BooleanConsecutiveFunction[{k₁,k₂,…,k_d},{n₁,n₂,…,n_d}]

represents a Boolean function of n₁ n₂ ⋯ n_d variables that gives True if all variables in a block of the variable array are True.

BooleanConsecutiveFunction[{{k₁,k₂,…,k_d},{c₁,c₂,…,c_d}},{n₁,n₂,…,n_d}]

treats the i level of the variable array as cyclic if c_i is True.

BooleanConsecutiveFunction[spec,{a₁,a₂,…}]

gives the Boolean expression in variables a_i corresponding to the Boolean consecutive function specified by spec.

• BooleanConsecutiveFunction[k,n] is also known as linear consecutive-k-out-of-n:F.
• BooleanConsecutiveFunction[{k,True},n] is also known as circular consecutive-k-out-of-n:F.
• BooleanConsecutiveFunction[{k,False},n] is equivalent to BooleanConsecutiveFunction[k,n].
• BooleanConsecutiveFunction[{{k₁,k₂,…,k_d},c},{n₁,n₂,…,n_d}] is equivalent to BooleanConsecutiveFunction[{{k₁,k₂,…,k_d},{c,c,…,c}},{n₁,n₂,…,n_d}].
• BooleanConsecutiveFunction[spec] gives a Boolean function object that works like Function.
• BooleanConsecutiveFunction[spec][a₁,a₂,…] gives an implicit representation equivalent to the explicit Boolean expression BooleanConsecutiveFunction[spec,{a₁,a₂,…}].
• In BooleanConsecutiveFunction[…,{n₁,n₂,…,n_d},…][vars], vars can be either an array of dimension-{n₁,n₂,…,n_d} variables, or a list of n₁ n₂ ⋯ n_d variables.
• In BooleanConsecutiveFunction[{k₁,k₂,…,k_d},vars], vars needs to be a depth-d array of variables.
• BooleanConvert converts BooleanConsecutiveFunction[spec][vars] to an explicit Boolean expression.
• In BooleanConsecutiveFunction[spec,vars,form], the possible forms are as given for BooleanConvert.
• BooleanConsecutiveFunction[spec,vars] by default gives an expression in disjunctive normal form (DNF).

BooleanConsecutiveFunction stays unevaluated:

Use BooleanConvert to expand it:

Define a two-dimensional function, in the shape of a torus:

Variables can also be given in a flat list:

A system fails if at least three out of four consecutive components in a circle fail:

BooleanConsecutiveFunction can be used in a structure:

A system works if at least three out of four consecutive components in a circle work:

BooleanConsecutiveFunction can be used in a structure:

Wrapping can be different in different dimensions. Define a cylinder:

Use BooleanConvert to expand it:

Connected cylinders stacked inside each other:

Use BooleanConvert to expand it:

A chain of 10 radio towers fails if two neighboring towers fail:

Each tower has an exponential lifetime distribution with an expected lifetime of 10 years:

The survival function for communication from one end of the tower chain to the other:

The probability of functioning communication after five years without maintenance:

Cameras are arranged in an overlapping grid. All areas are covered until a 2×2 grid fails:

Each camera has an expected lifetime of six years:

The lifetime distribution of a complete camera system is modeled with FailureDistribution:

To cover the same area with minimal overlap, four cameras are needed:

The overlapping structure has a higher reliability, but uses more cameras:

Use the same amount of cameras in standby mode instead:

Depending on mission time, a standby setup has higher reliability than an overlapping grid:

Simulate a linear consecutive-k-out-of-n system:

When requiring four out of 10 components to fail for the system to fail, with failure rate :

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