# BooleanMinimize

BooleanMinimize[expr]

finds a minimal-length disjunctive normal form representation of expr.

BooleanMinimize[expr,form]

finds a minimal-length representation for expr in the specified form.

BooleanMinimize[expr,form,cond]

finds a minimal-length expression in the specified form that is equivalent to expr when cond is true.

# Details and Options • BooleanMinimize[expr,form] always produces an expression equivalent to expr.
• Available forms are:
•  "DNF","SOP" disjunctive normal form, sum of products "CNF","POS" conjunctive normal form, product of sums "ANF" algebraic normal form "NOR" two-level Nor and Not "NAND" two-level Nand and Not "AND" two-level And and Not "OR" two-level Or and Not
• In general, there may be several minimal-length representations for a particular expression in a certain form. BooleanMinimize gives one of them.
• BooleanMinimize supports a Method option that specifies the detailed method to use.

# Examples

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## Basic Examples(2)

Find the minimal disjunctive normal form:

A Boolean counting function in disjunctive normal form:

Find a minimal disjunctive normal form:

## Scope(2)

A Boolean function of five variables represented in DNF:

Minimal-length DNF:

Minimal-length CNF:

Minimal-length NAND form:

Minimal-length NOR form:

Minimal-length ANF:

Show that all the forms are equivalent:

Minimize a Boolean function using a "care set" or condition:

The resulting forms are equivalent when cond is true:

They are not equivalent without the condition:

Typically the forms are longer without conditions:

## Applications(1)

### Distribution of Minimal Size(1)

Compute the minimal DNF representation:

Plot the size as a function of index:

Get its distribution:

Compute the size for the first 1000 four-variable functions:

## Properties & Relations(4)

The output from BooleanMinimize is equivalent to its input:

The output from BooleanMinimize with condition is conditionally equivalent to its input:

The forms f and g are equivalent when cond is true:

They are not equivalent on their own:

The minimal lengths "DNF", "CNF", "NAND", or "NOR" are not unique:

BooleanMinimize will produce an expression of length 3:

Another equivalent expression of length 3 is given by exchanging b and c:

Similar examples for "CNF", "NAND", and "NOR":

Use BooleanConvert when the minimal length form is not required:

BooleanConvert can also convert to additional forms: