Algebraic normal form

In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), Zhegalkin normal form, or Reed–Muller expansion is a way of writing propositional logic formulas in one of three subforms:

• The entire formula is purely true or false:

  ${\displaystyle 1}$

  ${\displaystyle 0}$

• One or more variables are combined into a term by AND (${\displaystyle \land }$), then one or more terms are combined by XOR (${\displaystyle \oplus }$) together into ANF. Negations are not permitted:

${\displaystyle a\oplus b\oplus \left(a\land b\right)\oplus \left(a\land b\land c\right)}$

• The previous subform with a purely true term:

${\displaystyle 1\oplus a\oplus b\oplus \left(a\land b\right)\oplus \left(a\land b\land c\right)}$

Formulas written in ANF are also known as Zhegalkin polynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM).