BooleanMinimize
✖
BooleanMinimize
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
open allclose allBasic Examples (2)Summary of the most common use cases
Find the minimal disjunctive normal form:
https://wolfram.com/xid/0n4uf2pv9lm-dfe0db
A Boolean counting function in disjunctive normal form:
https://wolfram.com/xid/0n4uf2pv9lm-k9rnoy
Find a minimal disjunctive normal form:
https://wolfram.com/xid/0n4uf2pv9lm-c1dwp6
Scope (2)Survey of the scope of standard use cases
A Boolean function of five variables represented in DNF:
https://wolfram.com/xid/0n4uf2pv9lm-qtk6f4
https://wolfram.com/xid/0n4uf2pv9lm-uem3e
https://wolfram.com/xid/0n4uf2pv9lm-b2uhck
https://wolfram.com/xid/0n4uf2pv9lm-kaaiam
https://wolfram.com/xid/0n4uf2pv9lm-bmu9m
https://wolfram.com/xid/0n4uf2pv9lm-fhbshf
https://wolfram.com/xid/0n4uf2pv9lm-cqtm3d
Show that all the forms are equivalent:
https://wolfram.com/xid/0n4uf2pv9lm-jgvn56
Minimize a Boolean function using a "care set" or condition:
https://wolfram.com/xid/0n4uf2pv9lm-gf665x
https://wolfram.com/xid/0n4uf2pv9lm-sl2ga
https://wolfram.com/xid/0n4uf2pv9lm-wcxnn
https://wolfram.com/xid/0n4uf2pv9lm-fi8llh
The resulting forms are equivalent when cond is true:
https://wolfram.com/xid/0n4uf2pv9lm-eclid5
They are not equivalent without the condition:
https://wolfram.com/xid/0n4uf2pv9lm-hpcajv
Typically the forms are longer without conditions:
https://wolfram.com/xid/0n4uf2pv9lm-hzerfn
https://wolfram.com/xid/0n4uf2pv9lm-dgi52e
Applications (1)Sample problems that can be solved with this function
Distribution of Minimal Size (1)
Compute the minimal DNF representation:
https://wolfram.com/xid/0n4uf2pv9lm-dvc206
https://wolfram.com/xid/0n4uf2pv9lm-b5casr
Plot the size as a function of index:
https://wolfram.com/xid/0n4uf2pv9lm-e8exij
https://wolfram.com/xid/0n4uf2pv9lm-fy7gg2
Compute the size for the first 1000 four-variable functions:
https://wolfram.com/xid/0n4uf2pv9lm-ferrtc
https://wolfram.com/xid/0n4uf2pv9lm-ed3hut
https://wolfram.com/xid/0n4uf2pv9lm-ct3953
Properties & Relations (4)Properties of the function, and connections to other functions
The output from BooleanMinimize is equivalent to its input:
https://wolfram.com/xid/0n4uf2pv9lm-bky3nb
https://wolfram.com/xid/0n4uf2pv9lm-qitwn
https://wolfram.com/xid/0n4uf2pv9lm-bjs0y1
The output from BooleanMinimize with condition is conditionally equivalent to its input:
https://wolfram.com/xid/0n4uf2pv9lm-gmbo7y
https://wolfram.com/xid/0n4uf2pv9lm-g7x2t
https://wolfram.com/xid/0n4uf2pv9lm-6w4ro
The forms f and g are equivalent when cond is true:
https://wolfram.com/xid/0n4uf2pv9lm-m7i4l
They are not equivalent on their own:
https://wolfram.com/xid/0n4uf2pv9lm-kydu5
The minimal lengths "DNF", "CNF", "NAND", or "NOR" are not unique:
https://wolfram.com/xid/0n4uf2pv9lm-20ws
BooleanMinimize will produce an expression of length 3:
https://wolfram.com/xid/0n4uf2pv9lm-kae824
Another equivalent expression of length 3 is given by exchanging b and c:
https://wolfram.com/xid/0n4uf2pv9lm-dbkter
https://wolfram.com/xid/0n4uf2pv9lm-teehh
Similar examples for "CNF", "NAND", and "NOR":
https://wolfram.com/xid/0n4uf2pv9lm-dau3m9
https://wolfram.com/xid/0n4uf2pv9lm-j1xxhm
https://wolfram.com/xid/0n4uf2pv9lm-ejjvk
Use BooleanConvert when the minimal length form is not required:
https://wolfram.com/xid/0n4uf2pv9lm-fr8ncb
https://wolfram.com/xid/0n4uf2pv9lm-geoabe
BooleanConvert can also convert to additional forms:
https://wolfram.com/xid/0n4uf2pv9lm-f2nes
Wolfram Research (2008), BooleanMinimize, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanMinimize.html.
Text
Wolfram Research (2008), BooleanMinimize, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanMinimize.html.
Wolfram Research (2008), BooleanMinimize, Wolfram Language function, https://reference.wolfram.com/language/ref/BooleanMinimize.html.
CMS
Wolfram Language. 2008. "BooleanMinimize." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BooleanMinimize.html.
Wolfram Language. 2008. "BooleanMinimize." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BooleanMinimize.html.
APA
Wolfram Language. (2008). BooleanMinimize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BooleanMinimize.html
Wolfram Language. (2008). BooleanMinimize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BooleanMinimize.html
BibTeX
@misc{reference.wolfram_2024_booleanminimize, author="Wolfram Research", title="{BooleanMinimize}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BooleanMinimize.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_booleanminimize, organization={Wolfram Research}, title={BooleanMinimize}, year={2008}, url={https://reference.wolfram.com/language/ref/BooleanMinimize.html}, note=[Accessed: 10-January-2025
]}