sumofproducts

Sum of products (SOP) is a standard form for expressing a Boolean function in digital logic. In an SOP expression, the function is the OR (sum) of one or more product terms. Each product term is the AND of literals, where a literal is a variable or its negation. Thus, an SOP expression is a disjunction of conjunctions of variables or their complements.

Canonical SOP, also called the sum of minterms, is obtained by listing every minterm corresponding to an input combination that yields a true output. A minterm is a product that contains every input variable exactly once, either in true or complemented form, representing a single row of the truth table. The canonical SOP is the OR of all such minterms for which the function outputs 1. Non-canonical (minimized) SOP allows for fewer literals or terms by combining adjacent minterms to simplify the expression.

To obtain an SOP form, you can start from a truth table or a Boolean expression and apply distributive and absorption laws to factor into a sum of products. Common methods for minimization include Karnaugh maps and the Quine–McCluskey algorithm. Software tools may perform Boolean simplification to produce a minimal SOP expression.

Use in circuits: an SOP representation maps directly to a network of AND gates feeding into an OR gate, where each product term is implemented by an AND gate and the terms are ORed. This structure is convenient for certain hardware implementations but may require many gates if not minimized. SOP is one of two main normal forms in Boolean algebra; its dual form is the product of sums (POS).