Reduced number of errors
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elvis
111
background.tex
111
background.tex
@ -88,7 +88,7 @@
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The complete RS \(\mathcal{B}_n\) is defined as follows: \(\mathcal{B}_n = \left(S_n, B_n\right)\) where\[S_n = \{p_0, p_1, \ldots, p_{n-1}\} \cup \{dec, inc\}\] and \[B_n = \{a_j, b_j, d_j | 0 \leq k < n\} \cup \{c_{j, k}, e_{j, k} | 0 \leq l < k < n\}\]
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To illustrate the system in action consider the sequence of contexts:
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\(C_0 = \{p1, p3\},\\C_1 = \emptyset, C_2 = \{inc\}, C_3 = \{inc\}, C_4 = \{dec\}, C_5 = \{dec, inc\}\). This gives the result sequence \(\delta = \emptyset.\{p1, p3\}.\{p1, p3\}.\{p0, p1, p3\}.\{p2, p3\}.\{p0, p1, p3\}.\emptyset\) and state sequence\\\(\tau = \{p1, p3\}.\{p1, p3\}.\{p1, p3, inc\}.\{p0, p1, p3, inc\}.\{p2, p3, dec\}.\{p0, p1, p3, dec, inc\}.\emptyset\)\\that in binary representation is \(\{1010_2\}.\{1010_2\}.\{1010_2\}.\{1011_2\}.\{1100_2\}.\{1011_2\}.\{0000_2\}\) by ignoring \(inc\) and \(dec\).
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\(C_0 = \{p1, p3\},\\C_1 = \emptyset, C_2 = \{inc\}, C_3 = \{inc\}, C_4 = \{dec\}, C_5 = \{dec, inc\}\). This gives the result sequence \(\delta = \emptyset.\{p1, p3\}.\{p1, p3\}.\{p0, p1, p3\}.\{p2, p3\}.\{p0, p1, p3\}.\emptyset\) and state sequence\\\(\tau = \{p1, p3\}.\{p1, p3\}.\{p1, p3, inc\}.\{p0, p1, p3, inc\}.\{p2, p3, dec\}.\{p0, p1, p3, dec, inc\}.\emptyset\),\\in binary representation is \(\{1010_2\}.\{1010_2\}.\{1010_2\}.\{1011_2\}.\{1100_2\}.\{1011_2\}.\{0000_2\}\) by ignoring \(inc\) and \(dec\).
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\end{subsection}
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\begin{subsection}{Simple loops}
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@ -226,6 +226,7 @@
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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.
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\end{definition}
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\begin{definition}[Positive RS\cite{Brodo_Bruni_Falaschi_Gori_Milazzo_2025}]\label{positive_rs}
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A Positive RS is a Reaction System \(\mathcal{A}^+ = (\textbf{S}, A)\) that satisfies the following conditions:
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\begin{enumerate}
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@ -254,7 +255,7 @@
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A^+_{neg} &\defeq \bigcup_{a \in S} \left\{ (\textbf{T}, \bar{a}) \vert \textbf{T} \in \mathit{Proh}_{\mathcal{A}}(a) \right\}
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\end{align*}
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The resulting \({\mathcal{A}}^+\) satisfies the two conditions from Definition\ \ref{positive_rs} and thus is a Positive RS.\@
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The resulting \({\mathcal{A}}^+\) satisfies the two conditions from Definition\ \ref{positive_rs} and thus is a Positive RS.%
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\end{definition}
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The states of the new Positive RS are in bijection with the states of the old system and can be proven that the two systems compute exactly the same states at each step.
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@ -517,7 +518,7 @@
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\item Refining \(\pi\) with respect to \(B\) splits a block \(D \in \pi\) into two blocks \(D_1 = D \cap pre(B)\) and \(D_2 = D \setminus pre(B)\) if, and only if, \(D\) is not stable with respect to \(B\).
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\item Refining further \(\texttt{split}(B, \pi)\) with respect to \(I \setminus B\) splits the block \(D_1\) into two blocks \(D_{11} = D_1 \cap pre(S \setminus B) \) and \(D_{12} = D_1 \setminus D_{11}\) if, and only if, \(D_1\) is not stable with respect to \(S \setminus B\).
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\end{itemize}
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\begin{figure}[!h]
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\begin{figure}[!ht]
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\centering
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\begin{tikzpicture}[
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place/.style={rectangle,draw=blue!50,fill=blue!20,thick,
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@ -560,8 +561,8 @@
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Finally for every process \(P\) in \(T\) we create in \(T'\) a newly added path of length \(l+1\) starting from \(P\).
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A small optimization can be added by sorting the frequency of labels and thus creating the lowest possible number of auxiliary nodes for each label.
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\begin{figure}[!h]
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\begin{tblr}{width=\linewidth, colspec={X[1,c]X[4,c]}}
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\begin{figure}[!ht]
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\begin{tblr}{width=\linewidth, colspec={Q[m,c]X[4,c]}}
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\begin{minipage}{.3\textwidth}
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\begin{tikzpicture}[auto,
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place/.style={rectangle,draw=blue!50,fill=blue!20,thick,inner sep=0pt,
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@ -578,62 +579,60 @@
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\end{tikzpicture}
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\end{minipage}
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&
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\begin{minipage}{.48\textwidth}
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\begin{tikzpicture}[auto,
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place/.style={rectangle,draw=blue!50,fill=blue!20,thick,inner sep=0pt,
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minimum size=6mm},
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new/.style={circle,draw=blue!30,fill=blue!10,thick,inner sep=0pt,
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minimum size=6mm},
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pre/.style={<-,shorten <=1pt,>={Stealth[round]},semithick},
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post/.style={->,shorten >=1pt,>={Stealth[round]},semithick}
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]
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\node[place] (P1) {\(P_1\)};
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\node[new] (p1e1) [above=of P1] {}
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edge [pre] (P1);
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\node[new] (p1e2) [above=of p1e1] {}
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edge [pre] (p1e1);
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\node[new] (p1e3) [above=of p1e2] {}
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edge [pre] (p1e2);
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\begin{tikzpicture}[auto,
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place/.style={rectangle,draw=blue!50,fill=blue!20,thick,inner sep=0pt,
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minimum size=6mm},
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new/.style={circle,draw=blue!30,fill=blue!10,thick,inner sep=0pt,
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minimum size=6mm},
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pre/.style={<-,shorten <=1pt,>={Stealth[round]},semithick},
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post/.style={->,shorten >=1pt,>={Stealth[round]},semithick}
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]
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\node[place] (P1) {\(P_1\)};
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\node[new] (p1e1) [above=of P1] {}
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edge [pre] (P1);
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\node[new] (p1e2) [above=of p1e1] {}
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edge [pre] (p1e1);
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\node[new] (p1e3) [above=of p1e2] {}
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edge [pre] (p1e2);
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\node[new] (a1) [right=of P1] {}
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edge [pre] (P1);
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\node[new] (a1e1) [above=of a1] {}
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edge [pre] (a1);
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\node[new] (a1) [right=of P1] {}
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edge [pre] (P1);
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\node[new] (a1e1) [above=of a1] {}
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edge [pre] (a1);
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\node[place] (P2) [right=of a1] {\(P_2\)}
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edge [pre] (a1);
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\node[new] (p2e1) [above=of P2] {}
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edge [pre] (P2);
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\node[new] (p2e2) [above=of p2e1] {}
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edge [pre] (p2e1);
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\node[new] (p2e3) [above=of p2e2] {}
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edge [pre] (p2e2);
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\node[place] (P2) [right=of a1] {\(P_2\)}
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edge [pre] (a1);
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\node[new] (p2e1) [above=of P2] {}
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edge [pre] (P2);
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\node[new] (p2e2) [above=of p2e1] {}
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edge [pre] (p2e1);
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\node[new] (p2e3) [above=of p2e2] {}
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edge [pre] (p2e2);
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\node[new] (a2) [below=of P2] {}
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edge [pre] (P2);
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\node[new] (a2e1) [right=of a2] {}
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edge [pre] (a2);
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\node[new] (a2e2) [right=of a2e1] {}
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edge [pre] (a2e1);
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\node[new] (a2) [below=of P2] {}
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edge [pre] (P2);
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\node[new] (a2e1) [right=of a2] {}
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edge [pre] (a2);
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\node[new] (a2e2) [right=of a2e1] {}
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edge [pre] (a2e1);
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\node[new] (a3) [below=of a1] {}
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edge [pre] (P1);
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\node[new] (a3e1) [below=of a3] {}
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edge [pre] (a3);
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\node[new] (a3e2) [below=of a3e1] {}
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edge [pre] (a3e1);
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\node[new] (a3) [below=of a1] {}
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edge [pre] (P1);
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\node[new] (a3e1) [below=of a3] {}
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edge [pre] (a3);
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\node[new] (a3e2) [below=of a3e1] {}
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edge [pre] (a3e1);
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\node[place] (P3) [below=of a2] {\(P_3\)}
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edge [pre] (a2)
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edge [pre] (a3);
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\node[new] (p3e1) [right=of P3] {}
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edge [pre] (P3);
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\node[new] (p3e2) [right=of p3e1] {}
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edge [pre] (p3e1);
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\node[new] (p3e3) [right=of p3e2] {}
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edge [pre] (p3e2);
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\end{tikzpicture}
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\end{minipage} \\
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\node[place] (P3) [below=of a2] {\(P_3\)}
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edge [pre] (a2)
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edge [pre] (a3);
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\node[new] (p3e1) [right=of P3] {}
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edge [pre] (P3);
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\node[new] (p3e2) [right=of p3e1] {}
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edge [pre] (p3e1);
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\node[new] (p3e3) [right=of p3e2] {}
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edge [pre] (p3e2);
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\end{tikzpicture}
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\end{tblr}
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\caption{Example of reduction}
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\end{figure}
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