Zhegalkin (also Žegalkin, Gégalkine or Shegalkin[1]) polynomials (Russian: полиномы Жегалкина), also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927,[2] 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.
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Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials (Russian: полиномы Жегалкина), also known as algebraic normal form, are a representation...
interpolation, using multivariate polynomials with two or three variables Zhegalkinpolynomial, a multilinear polynomial over F 2 {\displaystyle \mathbb...
which uniquely identifies the function: Algebraic normal form or Zhegalkinpolynomial, as a XOR of ANDs of the arguments (no complements allowed) Full...
&r=p+q{\pmod {2}}\\\end{matrix}}} The description of a Boolean function as a polynomial in F 2 {\displaystyle \mathbb {F} _{2}} , using this basis, is called...
Conjunctive x ¯ ⋅ y ¯ {\displaystyle {\overline {x}}\cdot {\overline {y}}} Zhegalkinpolynomial 1 ⊕ x ⊕ y ⊕ x y {\displaystyle 1\oplus x\oplus y\oplus xy} Post's...
Disjunctive x y {\displaystyle xy} Conjunctive x y {\displaystyle xy} Zhegalkinpolynomial x y {\displaystyle xy} Post's lattices 0-preserving yes 1-preserving...
theory of the ring of integers mod 2, via what are now called Zhegalkinpolynomials. Zhegalkin was professor of mathematics at Moscow State University. He...
{\overline {x}}+y} Conjunctive x ¯ + y {\displaystyle {\overline {x}}+y} Zhegalkinpolynomial 1 ⊕ x ⊕ x y {\displaystyle 1\oplus x\oplus xy} Post's lattices 0-preserving...
\left(a\land b\land c\right)} Formulas written in ANF are also known as Zhegalkinpolynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM)...
decompression. Similar data structures include negation normal form (NNF), Zhegalkinpolynomials, and propositional directed acyclic graphs (PDAG). A Boolean function...
x + y {\displaystyle x+y} Conjunctive x + y {\displaystyle x+y} Zhegalkinpolynomial x ⊕ y ⊕ x y {\displaystyle x\oplus y\oplus xy} Post's lattices 0-preserving...
Quine–McCluskey algorithm Reed–Muller expansion Venn diagram (1880) Zhegalkinpolynomial This should not be confused with the negation of the result of the...
-- Wisdom of repugnance -- Witness (mathematics) -- Word sense -- Zhegalkinpolynomial -- Philosophy portal List of logicians List of rules of inference...
transition can cause a 1-0 output transition; affine, representable with Zhegalkinpolynomials that lack bilinear or higher terms, e.g. x⊕y⊕1 but not xy; self-dual...
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