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 logical formulas in one of three subforms:

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

ANF is a normal form, which means that two equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal forms, it can be represented as a simple list of lists of variable names. Conjunctive and disjunctive normal forms also require recording whether each variable is negated or not. Negation normal form is unsuitable for that purpose, since it doesn't use equality as its equivalence relation: a ∨ ¬a isn't reduced to the same thing as 1, even though they're equal.

Putting a formula into ANF also makes it easy to identify linear functions (used, for example, in linear feedback shift registers): a linear function is one that is a sum of single literals. Properties of nonlinear feedback shift registers can also be deduced from certain properties of the feedback function in ANF.

There are straightforward ways to perform the standard boolean operations on ANF inputs in order to get ANF results.

XOR (logical exclusive disjunction) is performed directly:

NOT (logical negation) is XORing 1:

AND (logical conjunction) is distributed algebraically

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