Canonical POS Form

Programmable logic devices store logic functions as lists of maxterms in hardware registers. The canonical product of sums (POS) form is the standard representation used when specifying active-low outputs in PALs and GALs.

Core Concept

A maxterm is a sum (OR) of all variables in either complemented or uncomplemented form such that the maxterm evaluates to 0 for exactly one input combination. For row i, a variable appears uncomplemented if its bit in i is 0, and complemented if its bit is 1. This is the opposite rule compared to minterms.

The canonical POS expression is the product (AND) of all maxterms for which F=0. It is written as F=ΠM(list of row numbers). A GAL16V8 programmable logic device stores functions in AND-OR-INVERT form, which is structurally identical to POS. Supply voltage is 5 V, with a maximum propagation delay of 10 ns.

Every canonical SOP and canonical POS for the same function are complements of each other in their index lists. If the SOP uses minterms {1,3,4,6,7}, the POS uses maxterms {0,2,5}. The two sets always partition all 2^n row numbers.

Boolean Expression

For n variables, maxterm Mi evaluates to 0 when inputs equal the binary value i. The canonical POS is F = ΠM(i₁, i₂, …) = M_i₁ · M_i₂ · … where each M is an OR of all n literals. Maxterms and minterms are related: Mi = (mi)', where mi is the corresponding minterm.

Given:
  3-variable function F with F=0 at rows 0, 2, 5

Formula / Rule:
  For row i: variable X appears as X if bit=0, as X' if bit=1

Step by step:
  Row 0 → A=0,B=0,C=0 → M0 = (A+B+C)           [all uncomplemented, bit=0]
  Row 2 → A=0,B=1,C=0 → M2 = (A+B'+C)          [B bit=1 → B complemented]
  Row 5 → A=1,B=0,C=1 → M5 = (A'+B+C')        [A,C bits=1 → complemented]

  F = M0 · M2 · M5

Verify row 0: A=0,B=0,C=0
  M0 = 0+0+0 = 0 → F=0 ✓
Verify row 2: A=0,B=1,C=0
  M2 = 0+0+0 = 0 → F=0 ✓
Verify row 1: A=0,B=0,C=1
  M0 = 0+0+1 = 1, M2 = 0+1+1 = 1, M5 = 1+0+0 = 1 → F=1 ✓

Final Answer:
  F = (A+B+C)·(A+B'+C)·(A'+B+C')

Exam Tip: The maxterm variable rule is the inverse of the minterm rule. In a minterm, a variable is complemented when its bit is 0. In a maxterm, a variable is complemented when its bit is 1. This reversal catches almost every student at least once. Also remember: ΠM notation lists the rows where F=0, not where F=1. Mixing up these two is the most frequent POS error in GATE papers.

Key Properties

• Maxterm Mi = 0 only when inputs equal binary i; it equals 1 for all other input combinations
• Canonical POS: product of maxterms for all rows where F=0
• Minterm index set and maxterm index set are complementary — they together cover 0 to 2^n-1
• Relation: Mi = (mi)' where mi is the minterm at the same row number
• GAL16V8: 5 V supply, 10 ns max t_pd, stores POS in AND-OR-INVERT programmable array
• POS is preferred when the number of 0-rows is smaller than 1-rows — fewer maxterms to write
• K-map POS simplification groups 0-cells instead of 1-cells to find minimal POS

Quick Revision

• Maxterm = OR of all n literals; one maxterm per row where F=0
• Variable rule: uncomplemented if row bit=0, complemented if row bit=1
• ΠM(0,2,5) means F=0 at rows 0, 2, 5
• SOP minterm list + POS maxterm list = complete set {0 … 2^n-1}
• Mi and mi are complements of each other
• Canonical POS has exactly n literals per maxterm — no simplification has occurred yet
• Exam trap: students write maxterm variables complemented when bit=0, which is the minterm rule — for maxterms the complement goes with bit=1