Zhegalkin algebra

In mathematics, Zhegalkin algebra is a set of Boolean functions defined by the nullary operation taking the value $1$, use of the binary operation of conjunction $\land$, and use of the binary sum operation for modulo 2 $\oplus$. The constant $0$ is introduced as $1\oplus 1=0$. The negation operation is introduced by the relation $\neg x=x\oplus 1$. The disjunction operation follows from the identity $x\lor y=x\land y\oplus x\oplus y$.

Using Zhegalkin Algebra, any perfect disjunctive normal form can be uniquely converted into a Zhegalkin polynomial (via the Zhegalkin Theorem).

Basic identities

• $x\land (y\land z)=(x\land y)\land z$, $x\land y=y\land x$
• $x\oplus (y\oplus z)=(x\oplus y)\oplus z$, $x\oplus y=y\oplus x$
• $x\oplus x=0$
• $x\oplus 0=x$
• $x\land (y\oplus z)=x\land y\oplus x\land z$

Thus, the basis of Boolean functions ${\bigl \langle }\wedge ,\oplus ,1{\bigr \rangle }$ is functionally complete.

Its inverse logical basis ${\bigl \langle }\lor ,\odot ,0{\bigr \rangle }$ is also functionally complete, where $\odot$ is the inverse of the XOR operation (via equivalence). For the inverse basis, the identities are inverse as well: $0\odot 0=1$ is the output of a constant, $\neg x=x\odot 0$ is the output of the negation operation, and $x\land y=x\lor y\odot x\odot y$ is the conjunction operation.

The functional completeness of the these two bases follows from completeness of the basis $\{\neg ,\land ,\lor \}$.

See also

• Zhegalkin polynomial

References

• Zhegalkin, Ivan Ivanovich (1927). "On the technique of calculating propositions in symbolic logic" (PDF). Matematicheskii Sbornik. 34 (1): 9–28. Retrieved 12 January 2024.

Further reading

• https://encyclopediaofmath.org/wiki/Zhegalkin_algebra