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Polynomial expansions of Boolean functions with respect to nondegenerate functions. (English. Russian original) Zbl 0773.06018
Algebra Logic 30, No. 6, 411-416 (1991); translation from Algebra Logika 30, No. 6, 631-637 (1991).
The derivative of a Boolean function $$f(x_ 1,\dots,x_ n)$$ with respect to variables $$x_{i_ 1},\dots,x_{i_ m}$$ $$(m\leq n)$$ is defined by $f^{(m)}_{x_{i_ 1}\cdots x_{i_ m}}(x_ 1,\dots,x_ n)=\oplus f(x_ 1,\dots,\sigma_{i_ 1},\dots,\sigma_{i_ m},\dots,x_ n),$ where the sum on the right hand side is taken over all the binary strings $$(\sigma_{i_ 1},\dots,\sigma_{i_ m})$$. A function $$g$$ is nondegenerate if $$f^{(n)}_{x_ 1\cdots x_ n}(x_ 1,\dots,x_ n)\neq 0$$. A Boolean function $$f(x_ 1,\dots,x_ n)$$ has a polynomial expansion with respect to function $$g(x_ 1,\dots,x_ m,y)$$, $$m\leq n$$, if $f(x_ 1,\dots,x_ n)=\oplus g\bigl(x^{\tau_ 1}_ 1,\dots,x^{\tau_ m}_ m,\;f^ \tau(\sigma_ 1,\dots,\sigma_ m,x_{m+1},\dots,x_ n)\bigr),$ where $$\tau=g^{(m)}_{x_ 1\cdots x_ m}(x_ 1,\dots,x_ m,1)$$ and the sum is taken over all the binary strings $$(\sigma_ 1,\dots,\sigma_ m)$$, $$(\tau_ 1,\dots,\tau_ m)$$ for which $$g_ y'(\sigma^{\tau_ 1}_ 1,\dots,\sigma^{\tau_ m}_ m,y)=1$$.
It is proved that a function $$f(x_ 1,\dots,x_ n)$$ has a polynomial expansion with respect to function $$g(x_ 1,\dots,x_ m,y)$$ iff the function $$g$$ is nondegenerate. A canonical form presentation over the set $$\{\neg,\land,\lor,\to\}$$ of nondegenerate binary functions follows.
Reviewer: J.Henno (Naasa)

##### MSC:
 06E30 Boolean functions 03G05 Logical aspects of Boolean algebras
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##### References:
 [1] S. V. Yablonskii, G. P. Gavrilov, and V. B. Kudryavtsev, Functions of Algebra of Logic and Post Classes [in Russian], Nauka, Moscow (1966). [2] É. K. Machikenas and V. P. Suprun, On the Polynomial Expansion of Boolean Functions [in Russian], Deposited in the All-Union Institute of Scientific and Technical Information at No. 1899-V88 on March 9, 1988.
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