Built-in Mathematica Symbol

BooleanFunction

BooleanFunction[k, n]
represents the k

Boolean function in n variables.

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.

MORE INFORMATION

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.

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.

EXAMPLES

CLOSE ALL

Basic Examples (3)

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:

Generate the 30

Boolean function of 3 variables:

In[1]:=

Out[1]=

Use f like any other Boolean operator:

In[2]:=

Out[2]=

Convert to a DNF expression:

In[3]:=

Out[3]=

Generate the formula directly:

In[4]:=

Out[4]=

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

In[1]:=

Out[1]=

Use an incompletely specified truth table:

In[2]:=

Out[2]=

Convert a Boolean expression to a BooleanFunction:

In[1]:=

Out[1]=

Test that they represent the same function:

In[2]:=

Out[2]=

Scope (13)

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:

Applications (4)

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

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

:

Here we have x

x

y as reflected in the truth tables:

Using standard inequalities:

We have f

g

f

g for all Boolean functions f and g:

We prove 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:

Properties & Relations (7)

BooleanFunction displays in an elided form indicating the number of arguments:

It is an atomic object:

The InputForm gives an encoding that can be used to reconstruct the object:

Use the encoding to construct a BooleanFunction:

The result is identical to the original:

The order of values for BooleanFunction is the same as BooleanTable:

The corresponding BooleanFunction has an identical truth table:

The ordering is consistent with Tuples:

Indexing of BooleanFunction is consistent with IntegerDigits:

Convert from a BooleanFunction to its index:

Convert from any Boolean expression to its index:

Show that it is equivalent with the indexed BooleanFunction expression:

The indexing of Boolean functions agrees with the indexing of cellular automata:

BooleanMinterms can also represent any BooleanFunction:

The mapping from minterms to index:

The mapping from index to minterms:

Using bit vectors:

Use BooleanConvert to convert to BooleanFunction from other forms:

Also use BooleanConvert to convert from BooleanFunction to other forms:

Show that they are all equivalent:

Use BooleanTable to convert BooleanFunction to truth tables:

Or convert to truth rules: