Missing conclusion

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elvis
2025-11-18 15:37:25 +01:00
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A set \(\textbf{W} \subseteq \textbf{S}\) is non-contradictory if for all entities \(a \in S\) it holds that \(\{a, \bar{a}\} \nsubseteq \textbf{W}\). A non-contradictory state \(\textbf{W} \subseteq \textbf{S}\) is consistent if, for any entity \(a \in S\), either \(a \in \textbf{W}\) or \(\bar{a} \in \textbf{W}\) holds.
\end{definition}
\begin{definition}[Positive RS]\label{positive_rs}
\begin{definition}[Positive RS\cite{Brodo_Bruni_Falaschi_Gori_Milazzo_2025}]\label{positive_rs}
A Positive RS is a Reaction System \(\mathcal{A}^+ = (\textbf{S}, A)\) that satisfies the following conditions:
\begin{enumerate}
\item Each reaction \(r\) in \(A\) is positive, i.e., \(r = (\textbf{R}, \emptyset, \textbf{P})\) for some non-contradictory sets \(\textbf{R}\) and \(\textbf{P}\).
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Starting from the pair \(\frac{D_{\sigma}}{C_m}\) denoting the user's marking and proceeding backwards, apply iteratively a slicing step that deletes from the partial computation all information not related to \(D_{\sigma}\). The sliced trace will contain only the subsets of entities and reactions which are necessary for deriving the marked entities.
Since the algorithm\ \ref{slicing_algorithm} can only capture dependencies related to reactants, but ignores the ones related to inhibitors, converting the RS into a Positive RS makes possible the tracking of the absence of entities via negative entities. Minimizing the Positive RS reduces the noise in the output and is thus desirable.
Since the algorithm\ \ref{slicing_algorithm}\cite{Brodo_Bruni_Falaschi_Gori_Milazzo_2025} can only capture dependencies related to reactants, but ignores the ones related to inhibitors, converting the RS into a Positive RS makes possible the tracking of the absence of entities via negative entities. Minimizing the Positive RS reduces the noise in the output and is thus desirable.
\end{subsection}
\end{section}