# BooleanConsecutiveFunction

represents a Boolean function of n variables that gives True if k consecutive variables are True.

BooleanConsecutiveFunction[{k,True},n]

treats the variable list as cyclic.

BooleanConsecutiveFunction[{k1,k2,,kd},{n1,n2,,nd}]

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

BooleanConsecutiveFunction[{{k1,k2,,kd},{c1,c2,,cd}},{n1,n2,,nd}]

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

BooleanConsecutiveFunction[spec,{a1,a2,}]

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

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

gives the Boolean expression in the form specified by form.

# Examples

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## Basic Examples(3)

Create a consecutive-2-out-of-3 Boolean function:

Convert to explicit form:

Compute the survival function:

Compute the survival function:

## Scope(10)

### Linear Model(4)

BooleanConsecutiveFunction stays unevaluated:

Use BooleanConvert to expand it:

Define a two-dimensional function, shaped as a grid:

Variables can also be given in a flat list:

A system works if at least three out of four consecutive components on a line work:

BooleanConsecutiveFunction can be used in a structure:

A system fails if at least three out of four consecutive components on a line fail:

BooleanConsecutiveFunction can be used in a structure:

### Circular Model(4)

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:

### Mixed Model(2)

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:

## Applications(2)

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:

## Properties & Relations(2)

Lower dimensions are special cases of higher dimensions:

A ReliabilityDistribution with a BooleanConsecutiveFunction is equal to the negated corresponding Boolean expression in a FailureDistribution:

## Neat Examples(1)

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 :

Wolfram Research (2012), BooleanConsecutiveFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html.

#### Text

Wolfram Research (2012), BooleanConsecutiveFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html.

#### CMS

Wolfram Language. 2012. "BooleanConsecutiveFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html.

#### APA

Wolfram Language. (2012). BooleanConsecutiveFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html

#### BibTeX

@misc{reference.wolfram_2024_booleanconsecutivefunction, author="Wolfram Research", title="{BooleanConsecutiveFunction}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html}", note=[Accessed: 23-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_booleanconsecutivefunction, organization={Wolfram Research}, title={BooleanConsecutiveFunction}, year={2012}, url={https://reference.wolfram.com/language/ref/BooleanConsecutiveFunction.html}, note=[Accessed: 23-June-2024 ]}