28 lines
2.4 KiB
TeX
28 lines
2.4 KiB
TeX
%root: main.tex
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In the following definitions and examples, we use the following polynomial as an example:
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\begin{equation}
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\label{eq:poly-eg}
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\poly(X, Y) = 2X^2 + 3XY - 2Y^2.
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\end{equation}
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\begin{Definition}[Pure Expansion]
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The pure expansion of a polynomial $\poly$ is formed by computing all product of sums occurring in $\poly$, without combining like monomials. The pure expansion of $\poly$ generalizes \Cref{def:smb} by allowing monomials $m_i = m_j$ for $i \neq j$.
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\end{Definition}
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Note that similar in spirit to \Cref{def:reduced-bi-poly}, $\expansion{\circuit}$ \Cref{def:expand-circuit} reduces all variable exponents $e > 1$ to $e = 1$. Further, it is true that $\expansion{\circuit}$ is the pure expansion of $\circuit$.
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\begin{Example}[Example of Pure Expansion]\label{example:expr-tree-T}
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Consider the factorized representation $(X+ 2Y)(2X - Y)$ of the polynomial in \Cref{eq:poly-eg}.
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Its circuit $\circuit$ is illustrated in \Cref{fig:circuit}.
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The pure expansion of the product is $2X^2 - XY + 4XY - 2Y^2$. As an additional example of \Cref{def:expand-circuit}, $\expansion{\circuit}=[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.
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\end{Example}
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$\expansion{\circuit}$ effectively\footnote{The minor difference here is that $\expansion{\circuit}$ encodes the \emph{reduced} form over the SOP pure expansion of the compressed representation, as opposed to the \abbrSMB representation} encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
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However, unlike the constraint on the input $\poly$ to compute $\rpoly$, the input circuit $\circuit$ does not need to be in \abbrSMB/SOP form.
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\begin{Example}[Example for \Cref{def:positive-circuit}]\label{ex:def-pos-circ}
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Using the same factorization from \Cref{example:expr-tree-T}, $\polyf(\abs{\circuit}) = (X + 2Y)(2X + Y) = 2X^2 +XY +4XY + 2Y^2 = 2X^2 + 5XY + 2Y^2$. Note that this \textit{is not} the same as the polynomial from \Cref{eq:poly-eg}. As an example of the slight abuse of notation we alluded to, $\polyf\inparen{\abs{\circuit}\inparen{1,\ldots, 1}} =2\inparen{1}^2 + 5\inparen{1}\inparen{1} + 2\inparen{1}^2 = 9$.
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\end{Example}
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\begin{Definition}[Subcircuit]
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A subcircuit of a circuit $\circuit$ is a circuit \subcircuit such that \subcircuit is a DAG \textit{subgraph} of the DAG representing \circuit. The sink of \subcircuit has exactly one gate \gate.
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\end{Definition}
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