used by an algorithm that solves a packing-certifiable problem $\Pi$ by doing DP over a tree decomposition of the input graph. The encodings in ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I)$ correspond to the entries of the DP-tables of graphs with boundary labeled by the set of integers $I$. The language Łidentifies certificates which are partial solutions satisfying the boundary conditions imposed by an encoding.

The following definition differs from [@KviaDP Definition 3.3] as now the function [$f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}$]{}is incorporated in the definition of an encoder [[${\mathcal}{E}_{}$]{}]{}.

\[defi:pi-encod\] Let $\Pi$ be a packing-certifiable problem. An encoder [[${\mathcal}{E}_{}$]{}]{}is a *$\Pi$-encoder* if ${\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(\emptyset)$ is a singleton, denoted by $\{R_\emptyset\}$, such that for any $0$-boundaried graph $G$, ${\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R_\emptyset) = f^\Pi(G)$.

The following definition allows to control the number of possible distinct values assigned to encodings and plays a similar role to FII or *monotonicity* in previous work [@BFL+09; @KLP+12; @FLST10].

\[def:confined\] An encoder ${{\ensuremath{{\mathcal}{E}_{}}\xspace}}$ is *$g$-confined* if there exists a function $g : \mathbb{N} \to \mathbb{N}$ such that for any $t$-boundaried graph $G$ with $\Lambda(G) = I$ it holds that either $\{R \in {\ensuremath{{\mathcal}{C}^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(I) : {\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R) \neq - \infty \}= \emptyset\ $ or $\ \max_{R} \{{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R)\neq - \infty \}\  -\   \min_{R} \{{\ensuremath{f^{{{\ensuremath{{\mathcal}{E}_{}}\xspace}}}}\xspace}(G,R) \neq - \infty\}  \  {\leqslant}\ g(t)$.

For an 