Holographic Approximation

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]