with a binary symbol $G$, and define for all glue type $g \in \mathcal{G}$ the term $\mathrm{gl}(g)(x)=G(g(x), s^{\mathrm{str}(g)}(0))$. Temperature is dealt with by adding *ambiance stars* that are required to connect between tile encodings. Ambiance stars connect to a tile translation on several sides and force the sum of forces to be larger than the temperature. Ambiance stars $A^{a,b,c,d}$ are defined as $$\begin{array}{p{12cm}}
$[-v_{-}(G(x_a, s^{a}(y_a)), z, w),$\\
\hspace{1cm} $+v_{+}(G(x_b, s^{b}(y_b)), z, w),$\\
\hspace{2cm} $-h_{-}(G(x_c, s^{c}(y_c)), z, w),$\\
\hspace{3cm} $+h_{+}(G(x_d, s^{d}(y_d)), z, w),$\\
\hspace{4cm} $+v_{+}(G(x_a,s^{a}(y_a),z_1,w_1)),$\\
\hspace{5cm} $-v_{-}(G(x_b,s^{b}(y_b),z_2,w_2)),$\\
\hspace{6cm} $+h_{+}(G(x_c, s^{c}(y_c)), z_3, w_3),$\\
\hspace{7cm} $-h_{-}(G(x_d, s^{d}(y_d)), z_4, w_4)]$
\end{array}$$ for all $0\leqslant a,b,c,d\leqslant \mathtt{t}$ such that $a+b+c+d=\mathtt{t}$. We also require the addition of stars $[-v_{+}(x)]$, $[+v_{-}(x)]$, $[-h_{+}(x)]$, $[+v_{-}(x)]$ to *plug* connexions where no tiles can be added.

Let $t^i = (g^i_w, g^i_e, g^i_s, g^i_n)$, $i\in I$ (countable but possibly infinite) be a set of self-assembling tiles. The tile $t^i$ is encoded as the star $(t^i)^\bigstar$: $$\begin{array}{c}
[-h_{+}(\mathrm{gl}(g^i_w)(x), x, y),\\
\hspace{2cm} -v_{+}(\mathrm{gl}(g^i_s)(y), x, y),\\
\hspace{4cm} +h_{-}(\mathrm{gl}(g^i_e)(s(x)), x, y),\\
\hspace{6cm} +v_{-}(\mathrm{gl}(g^i_n)(s(y)), x, y)].\end{array}$$

The aTAM model with tiles $(t^i)_{i\in I}$ and temperature $\mathtt{t}$ is encoded as $$\sum_{a+b+c+d=\mathtt{t}}A^\mathtt{a,b,c,d}+\sum_{i\in I} (t^i)^\bigstar+[-v_{+}]+[+v_{-}]+[-h_{+}]+[+v_{-}].$$

Interpreting the computational content of MLL {#sec:computational}
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In this section, we restrict the definitions 