Holographic Approximation: Difference between revisions
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\sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_{\ell+1} \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} | \sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_{\ell+1} \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} | ||
\end{align} | |||
</math> | |||
<math> | |||
\begin{align} | |||
R_T | |||
&= | |||
\frac{\#\{\sigma_T\mid\sigma_T(e)=0\}}{\#\{\sigma_T\mid\sigma_T(e)=1\}}\\ | |||
&= | |||
\frac{ | |||
\sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_\ell \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} | |||
\#\{\sigma_T\mid\sigma_T(e)=1\} | |||
}{ | |||
\sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_{\ell+1} \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} | |||
} | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
Revision as of 20:44, 17 April 2012
Holant Problem
Recursion on tree
[math]\displaystyle{ \begin{align} \#\{\sigma_T\mid\sigma_T(e)=0\} &= \sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_\ell \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\}\\ \#\{\sigma_T\mid\sigma_T(e)=1\} &= \sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_{\ell+1} \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} \end{align} }[/math]
[math]\displaystyle{ \begin{align} R_T &= \frac{\#\{\sigma_T\mid\sigma_T(e)=0\}}{\#\{\sigma_T\mid\sigma_T(e)=1\}}\\ &= \frac{ \sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_\ell \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} \#\{\sigma_T\mid\sigma_T(e)=1\} }{ \sum_{\ell=0}^k\sum_{S\in{[k]\choose \ell}}f_{\ell+1} \prod_{i\in S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=1\}\prod_{i\in [k]\setminus S}\#\{\sigma_{T_i}\mid \sigma_{T_i}(e_i)=0\} } \end{align} }[/math]