Zhegalkin polynomial
Zhegalkin (also Zegalkin or Gegalkine) polynomials form one of many possible representations of the operations of Boolean algebra. Introduced by the Russian mathematician I. I. Zhegalkin in 1927, they are the polynomials of ordinary high school algebra interpreted over the integers mod 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.
Prior to 1927 Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, etc. Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, thinking of the logical constants 0 and 1 as integers mod 2. The logical operation of conjunction is realized as the arithmetic operation of multiplication 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 ∨). Logical complement ¬x is then derived from 1 and ⊕ as x⊕1. Since ∧ and ¬ form a sufficient basis for the whole of Boolean algebra, meaning that all other logical operations are obtainable as composites of these basic operations, it follows that the polynomials of ordinary algebra can represent all Boolean operations, allowing Boolean reasoning to be performed reliably by appealing to the familiar laws of high school algebra without the distraction of the differences from high school algebra that arise with disjunction in place of addition mod 2.
An example application is the representation of the Boolean 2-out-of-3 threshold or median operation as the Zhegalkin polynomial xy⊕yz⊕zx, which is 1 when at least two of the variables are 1 and 0 otherwise.
Formally a Zhegalkin monomial is the product of a finite set of distinct variables (hence square-free), including the empty set whose product is denoted 1. There are 2n possible Zhegalkin monomials in n variables, since each monomial is fully specified by the presence or absence of each variable. A Zhegalkin polynomial is the sum (exclusive-or) of a set of Zhegalkin monomials, with the empty set denoted by 0. A given monomial's presence or absence in a polynomial corresponds to that monomial's coefficient being 1 or 0 respectively. The Zhegalkin monomials, being linearly independent, span a 2n-dimensional vector space over the Galois field GF(2) (NB: not GF(2n), whose multiplication is quite different). The 22n vectors of this space, i.e. the linear combinations of those monomials as unit vectors, constitute the Zhegalkin polynomials. The exact agreement with the number of Boolean operations on n variables, which exhaust the n-ary operations on {0,1}, furnishes a direct counting argument for completeness of the Zhegalkin polynomials as a Boolean basis.
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