Minimal POS

Also called: minimal product of sums, minimum POS, minimum product of sums, POS simplification, minimized POS

A product-of-sums expression for a function that uses the fewest possible sum terms and, among those, the fewest literals.

A minimal POS is the cheapest product of sums for a function: fewest sum terms first, then fewest literals. It is the mirror image of a minimal sop, and everything works the same way on the 0s:

Start from the 0-rows, the ΠM list.
Combine adjacent maxterms with the dual rule = X. In cube notation, the differing bit becomes a dash.
Read each pattern back with maxterm polarity: a 0 bit gives the plain variable, a 1 bit gives the barred one. 10- is .
Drop redundant sums, including consensus terms.

A second route: find the minimal SOP of the complement F′ (group the 0s), then apply de morgans laws to it.

Why bother? The minimal SOP and minimal POS of a function can have different costs. Compare both and build the cheaper one: NAND-NAND for the SOP or NOR-NOR for the POS.

Worked example

Example

Minimal POS from the 0s

F(A, B, C) = ΠM(0, 4, 6). Find the minimal POS, and compare it with the minimal SOP.

1.
The 0-rows are 000, 100, 110.
2.
M0 + M4 → -00 → (both bits 0, so both literals plain).
3.
M4 + M6 → 1-0 → (1 → barred, 0 → plain).
4.
Minimal POS: , 4 literals.
5.
The 1s are rows 1, 2, 3, 5, 7, giving the minimal SOP , 3 literals. Here the SOP is cheaper. (By OR over AND, the two are the same function.)

Common mistakes

Reading a combined pattern with minterm polarity. For a sum, a 1 bit means the barred variable.
Grouping the 1s when you want a POS. A POS comes from the 0s.
Assuming the minimal POS has the same cost as the minimal SOP.

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Minimal POS is taught in Boolean Simplification and Karnaugh Maps.

Boolean Simplification →Karnaugh Maps →

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