productofsums

Product of sums (POS) is a form of Boolean expression in which a function is written as an AND (product) of one or more OR terms (sums) of literals. It is the dual of the more common sum of products (SOP) form and is commonly associated with conjunctive normal form (CNF) representations in logic.

A canonical POS expression uses a maxterm for each input combination that yields the function value 0. Each maxterm is a disjunction of all input variables, with each variable negated if its value in that row is 1 and unnegated if its value is 0. The full canonical POS is the product of all such maxterms, so the function is false for any input that makes at least one maxterm false.

Example: for two variables a and b, a function that is true only for a = 1 and b = 1 can be written as f(a,b) = (a + b)(a + ¬b)(¬a + b). Each factor is a maxterm corresponding to the three input combinations (0,0), (0,1), and (1,0) that yield f = 0.

Canonical POS can be simplified by Boolean algebra techniques or Karnaugh maps to reduce the number of literals per clause or the number of clauses. The POS form is the dual of SOP, and both are used in digital circuit design and logic synthesis, depending on whether a CNF-oriented or SOP-oriented representation is more appropriate for a given problem.