Algebraic normal form
algebraic normal form, closely related to Zhegalkin polynomial
Algebraic normal form (ANF) is a representation of functions in boolean algebra. Formulas written in ANF are also known as ring sum normal form (RSNF or RNF), Zhegalkin polynomials (Russian: полиномы Жегалкина), or Positive Polarity (or parity) Reed–Muller expansions (PPRM). These terms describe a way of writing propositional logic formulas in one of three sub-forms:
The entire formula is purely true or false:
1
{\displaystyle 1}
0
{\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:
a
⊕
b
⊕
(
a
∧
b
)
⊕
(
a
∧
b
∧
c
)
{\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:
1
⊕
a
⊕
b
⊕
(
a
∧
b
)
⊕
(
a
∧
b
∧
c
)
{\displaystyle 1\oplus a\oplus b\oplus \left(a\land b\right)\oplus \left(a\land b\land c\right)}
Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.
History
Prior to 1927, Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on. Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, representing the false and true values as 0 and 1, the integers mod 2. Logical conjunction is written as xy, and logical exclusive-or as arithmetic addition mod 2, (written here as x⊕y to avoid confusion with the common use of + as a synonym for inclusive-or ∨).
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