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BooleanFunction—Wolfram Language Documentation

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• See Also
  • BooleanMinterms
  • BooleanMaxterms
  • BooleanCountingFunction
  • BooleanConsecutiveFunction
  • BooleanTable
  • BooleanConvert
  • BooleanVariables
  • BooleanMinimize
  • CellularAutomaton
  • And
  • Or
  • Not
  • Nand
• Related Guides
  • Boolean Computation
  • Logic & Boolean Algebra
  • Computational Systems
  • Discrete Mathematics

BooleanFunction[values]

represents the Boolean function corresponding to the specified vector of truth values.

BooleanFunction[{{i₁₁,i₁₂,…}o₁,…}]

represents the Boolean function defined by the specified mapping from inputs to outputs.

BooleanFunction[spec,{a₁,a₂,…}]

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

BooleanFunction[spec,{a₁,a₂,…},form]

gives the Boolean expression in the form specified by form.

• BooleanFunction[spec] gives a Boolean function object that works like Function.
• BooleanFunction[spec][a₁,a₂,…] gives an implicit representation equivalent to the explicit Boolean expression BooleanFunction[spec,{a₁,a₂,…}].
• BooleanConvert converts BooleanFunction[spec][vars] to an explicit Boolean expression.
• In BooleanFunction[values] etc., values can be specified either as True and False or as 1 and 0.
• The functions represented by BooleanFunction always return True or False.
• In BooleanFunction[values], the values are specified in binary order starting with 111, ….
• BooleanFunction[k,n] is equivalent to BooleanFunction[IntegerDigits[k,2,2^n]].
• In BooleanFunction[values], each value can be a list, representing a vector-valued Boolean function.
• In BooleanFunction[{{i₁₁,i₁₂,…}->o₁,…}] the o_i can be lists, representing vector-valued Boolean functions.
• Elements of both inputs and outputs can be specified either as True and False or as 1 and 0.
• Elements of inputs and outputs can also include any number of _, representing "don't cares".
• They can also include up to one __, representing a sequence of "don't cares".
• In BooleanFunction[spec,{a₁,a₂,…},form], the possible forms are as given for BooleanConvert.
• BooleanFunction[spec,{a₁,a₂,…}] by default gives an expression in DNF.
• BooleanFunction[k] gives the k Boolean function in n variables, where n has the smallest value for which .
• The numbering of Boolean functions in BooleanFunction[k,…] is consistent with CellularAutomaton.
• BooleanFunction[CellularAutomaton[n]] is equivalent to BooleanFunction[n,3].
• BooleanFunction[CellularAutomaton[{n,2,r}]] is equivalent to BooleanFunction[n,2r+1].
• Operations such as BooleanMinimize, BooleanTable, etc. can operate directly on BooleanFunction objects.
• BooleanFunction objects can be applied to variables just like other Boolean functions such as And, Or, etc.
• In StandardForm and related formats, BooleanFunction objects are printed in elided form, with only their number of variables displayed.
• BooleanVariables gives the number of variables of a BooleanFunction object.

Generate the 30 Boolean function of 3 variables:

Use f like any other Boolean operator:

Convert to a DNF expression:

Generate the formula directly:

Specify a Boolean function based on a table of truth rules:

Use an incompletely specified truth table:

Convert a Boolean expression to a BooleanFunction:

Test that they represent the same function:

Create a BooleanFunction with two arguments by indexing:

Compute its value for particular arguments:

The function remains unevaluated for symbolic arguments:

BooleanFunction can be used just like any other Boolean operator:

Any Boolean expression can be converted to a BooleanFunction expression:

Including combinations of BooleanFunction expressions:

A BooleanFunction expression that is equivalent to True or False automatically simplifies:

BooleanFunction is a canonical representation, and equivalence can be tested using SameQ:

Create a truth table in standard order:

Create an equivalent BooleanFunction expression:

Show that they are equivalent:

Alternatively, show that the resulting truth tables are the same:

Create a complete list of truth rules:

Create the corresponding BooleanFunction expression:

The ordering of truth rules for complete lists has no effect:

Use _ to indicate "don't cares" in a truth table:

Create a BooleanFunction:

The resulting truth table matches the original specification:

Use _ and __ to indicate "don't cares" in truth rules:

Create a BooleanFunction:

The original rules completely specify the function and the resulting truth tables are identical:

Use lists of lists to indicate a vector-valued truth table:

Create a BooleanFunction:

The resulting output matches the original specification:

Use vector-valued truth rules:

Create a BooleanFunction:

The resulting truth rules match the original specification:

Truth tables can also be given using 0 in place of False and 1 in place of True:

The resulting truth tables are identical:

Convert any Boolean expression to a BooleanFunction expression:

Show that they are equivalent:

Convert expressions involving any Boolean operators:

Show that they are equivalent:

Convert a BooleanFunction expression to other standard forms:

A number of different standard forms:

As well as a truth table:

Or truth rules:

Specify BooleanFunction using CellularAutomaton:

Enumerate all 2-variable Boolean functions:

All 3-variable Boolean functions:

Randomly sample 50 functions of 4 variables:

Compare the sizes of Boolean functions in their standard and minimized forms:

Three variables:

Four variables, the first 1000 functions:

Create new Boolean primitives corresponding to ≤, ≥, <, and >:

These are closely related:

Implies is equivalent to x<=y:

Define a relation for Boolean functions f≼g iff f[u]≤g[u]:

Here you have x≼x∨y as reflected in the truth tables:

Using standard inequalities:

For all Boolean functions f and g, f∧g≼f≼f∨g:

This proves that f≼g iff f∧g⇔f:

And similarly f≼g iff f∨g⇔g:

Or that f≼g and g≼h implies f≼h:

Generate the rule 30 elementary cellular automaton rule:

Simulate it:

Compare to the standard encoding:

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