K-Map POS Simplification

POS minimization from a K-map is used whenever a circuit's active-low enable lines dominate the design — NOR-based PLAs and CMOS transmission gate logic are classic examples. Grouping the 0-cells directly saves complement inversions compared to complementing the SOP result.

Core Concept

K-map POS minimization groups the 0-cells instead of the 1-cells. Each group of 0s produces one sum term (a maxterm or simplified maxterm). The final expression is the AND of all these sum terms.

The variable rule for 0-cell groups is the inverse of the SOP rule. If a variable is 0 across all cells in the group, write it uncomplemented in the sum term. If it is 1 across all cells, write it complemented. If it changes, omit it. This is because a sum term evaluates to 0 when all its literals are 0 — so an uncomplemented variable contributes 0 when the variable itself is 0.

The 74HC4002 dual 4-input NOR gate at 5 V (t_pd ≈ 8 ns, fan-out 10) implements a POS sum term in a single gate. A 2-level NOR-NOR structure realizes a complete POS function the same way AND-OR realizes SOP.

Boolean Expression

If the K-map 0-cells group into terms G1, G2, …, the minimal POS is F = G1 · G2 · … where each Gi is a sum (OR) of literals. This is equivalent to ΠM notation but gives simplified rather than canonical maxterms.

Given:
  F(A,B,C) defined by truth table:
  ABC=000→0, 001→1, 010→0, 011→1, 100→1, 101→1, 110→0, 111→1
  F=0 at rows 0(000), 2(010), 6(110)

Formula / Rule:
  Draw 3-variable K-map, group the 0-cells, apply POS variable rule

Step by step:
  K-map (rows AB=00,01,11,10 × cols C=0,1):
            C=0   C=1
  AB=00:     0     1   (m0=0, m1=1)
  AB=01:     0     1   (m2=0, m3=1)
  AB=11:     0     1   (m6=0, m7=1)
  AB=10:     1     1   (m4=1, m5=1)

  0-cells: m0, m2, m6 — all in column C=0

  Group: m0, m2, m6 — three cells... not a power of 2.
  Check: can we include m4 (which is 1)? No — we group only 0-cells.
  Try pair: m0+m2 (AB=00,01; C=0) — group of 2
    A: 0,0 → omit (changes? no, both 0 → constant 0 → write A uncomplemented)
    Wait: A=0 both → A is constant at 0 → include A (uncomplemented in sum term)
    B: 0,1 → changes → omit
    C: 0,0 → constant 0 → include C (uncomplemented)
    Sum term for m0,m2: (A+C) — both variables 0 in their cells, both uncomplemented

  Remaining 0-cell m6 (AB=11, C=0):
    A=1,B=1,C=0 → A'·B in the AND world → sum term: (A'+B'+C)
    But m6 alone is a group of 1 → term = (A'+B'+C)

  Check: can m6 pair with m2? AB=01 and AB=11 are adjacent. C=0 both. Group m2,m6:
    A: 0,1 → changes → omit; B: 1,1 → constant 1 → B'; C: 0,0 → constant 0 → C
    Sum term: (B'+C)
    Now m0 alone: sum term (A+B+C) [all zeros, all uncomplemented]

  Best cover: group {m0,m2} gives (A+C); group {m2,m6} gives (B'+C)
  m0 covered by first group; m6 covered by second; m2 covered by both.
  F = (A+C)·(B'+C)

Final Answer:
  F = (A+C)·(B'+C)

Exam Tip: In POS K-map grouping, the variable rule flips relative to SOP. A variable constant at 0 in a 0-cell group appears uncomplemented in the sum term; a variable constant at 1 appears complemented. This is the exact opposite of the SOP rule and is the single most common mistake. Also, grouping 0-cells with wrap-around follows the same power-of-2 rectangle rules as grouping 1-cells — the geometry does not change.

Key Properties

• POS K-map: group 0-cells; each group yields a sum (OR) term; AND all terms for final expression
• Variable rule: constant 0 across group → uncomplemented in sum term; constant 1 → complemented
• Groups must be powers-of-2 rectangles; wrap-around adjacency applies identically to SOP grouping
• Minimal POS covers all 0-cells with fewest and largest groups
• 74HC4002 dual 4-input NOR: Vcc 2–6 V, t_pd ≈ 8 ns — implements one POS sum term per gate
• Don't-care cells can be grouped with 0-cells to enlarge 0-groups — same flexibility as in SOP
• POS is more efficient than SOP when there are fewer 0-cells than 1-cells

Quick Revision

• POS from K-map: group 0s, not 1s
• Each 0-group → one sum term; AND all sum terms
• Constant-0 variable in group → uncomplemented; constant-1 → complemented (opposite of SOP rule)
• Group size and shape rules (power of 2, rectangle, wrap-around) are identical to SOP
• Essential 0-prime implicants: the only group covering a particular 0-cell — must include them
• POS and SOP minimal forms implement the same function — verify by checking truth tables match
• Exam trap: applying the SOP variable rule (constant-1 uncomplemented) to a POS group — the rule reverses for 0-cell grouping