BooleanConsecutiveFunction
BooleanConsecutiveFunction[k,n]
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.
Details
- 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,…,kd},c},{n1,n2,…,nd}] is equivalent to BooleanConsecutiveFunction[{{k1,k2,…,kd},{c,c,…,c}},{n1,n2,…,nd}].
- 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,…,nd},…][vars], vars can be either an array of dimension-{n1,n2,…,nd} variables, or a list of n1 n2 ⋯ nd variables.
- In BooleanConsecutiveFunction[{k1,k2,…,kd},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).
Examples
open allclose allBasic Examples (3)
Create a consecutive-2-out-of-3 Boolean function:
Use BooleanConsecutiveFunction in ReliabilityDistribution:
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:
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