The Complexity of Boolean State Separation (Technical Report)

Providing a powerful mechanism for the modeling of conflicts, dependencies and parallelism, Petri nets are widely used for studying and simulating concurrent and distributed systems. In system analysis, one aims to check behavioral properties of system models, and many of these properties are decidable [7] for Petri nets and their reachability graphs, which represent systems’ behaviors. The task of system synthesis is opposite: a (formal) specification of the system’s behavior is given, and the goal is then to decide whether this behavior can be implemented by a Petri net. In case of a positive decision, such a net should be constructed.

Boolean Petri nets form a simple yet rich and powerful family of Petri nets [3, 4, 8, 10, 12, 13, 16], applied in asynchronous circuits design [24, 26], concurrent constraint programs [9] and analysis of biological systems models [6]. In Boolean nets, each place contains at most one token, for any reachable marking. Hence, a place can be interpreted as a Boolean condition that is true if marked and false otherwise. A place $p$ and a transition $t$ of such a net are related by one of the Boolean interactions that define in which way $p$ and $t$ influence each other. The interaction inp (out) defines that $p$ must be true (false) before and false (true) after $t$’s firing; free (used) implies that $t$’s firing proves that $p$ is false (true); nop means that $p$ and $t$ do not affect each other at all; res (set) implies that $p$ may initially be both false or true but after $t$’s firing it is false (true); swap means that $t$ inverts $p$’s current Boolean value. Boolean Petri nets are classified by the sets of interactions that can be applied. A set $\tau$ of Boolean interactions is called a type of net, and a net $N$ is of type $\tau$ (a $\tau$-net) if it applies at most the interactions of $\tau$. For a type $\tau$, the $\tau$-synthesis problem consists in deciding whether a specification given in the form of a labeled transition system (TS) is isomorphic to the reachability graph of some $\tau$-net $N$, and in constructing $N$ if it exists. The complexity of $\tau$-synthesis has been studied in different settings [17, 20, 22], and varies substantially from polynomial [16] to NP-complete [2].

In order to perform synthesis, that is, to implement the behavior specified by the given TS with a $\tau$-net, two general problems have to be resolved: The $\tau$-net has to distinguish the global states of the TS, and the $\tau$-net has to prevent actions at states where they are not permitted by the TS. In the literature [3], the former requirement is usually referred to as $\tau$-state separation property ($\tau$-SSP), while the latter – $\tau$-event/state separation property ($\tau$-ESSP). Both $\tau$-SSP and $\tau$-ESSP define decision problems that ask whether a given TS fulfills the respective property. The present work focuses exclusively on the computational complexity of $\tau$-SSP. The interest to state separation is motivated in several ways. First, many synthesis approaches are very sensitive to the size of the input’s state space. This raises the question if some initial, so-called pre-synthesis procedures [5, 25] can be employed as a quick-fail mechanism, i.e., techniques with a small computational overhead that would gain some helpful information for the main synthesis, or reject the input if exact (up to isomorphism) synthesis is not possible. Since $\tau$-synthesis allows a positive decision if and only if $\tau$-SSP and $\tau$-ESSP do [3], an efficient decision procedure for $\tau$-SSP could serve as a quick-fail pre-process check. Second, if exact synthesis is not possible for the given TS, one may want to have a simulating model, i.e., a $\tau$-net that over-approximates [14, 15] the specified behavior with some possible supplement. Formally, the TS then has to be injectively embeddable into the reachability graph of a $\tau$-net. It is well known from the literature [3] that a TS can be embedded into the reachability graph of a $\tau$-net if and only if it has the $\tau$-SSP. Finally, in comparison to $\tau$-ESSP, so far the complexity of $\tau$-SSP is known to be as hard [1, 16, 22, 23] or actually less hard [18, 19]. On the contrary, in this paper, for some types of nets, deciding the $\tau$-SSP is proven harder (NP-complete) than deciding the $\tau$-ESSP (polynomial), e.g., for $\tau=\{\textsf{nop},\textsf{res},\textsf{set}\}$. From the contribution perspective, the $\tau$-SSP has been previously considered only in the broader context of $\tau$-synthesis, and only for selected types [16, 19, 21, 22]. In this paper, we completely characterize the complexity of $\tau$-SSP for all 256 Boolean types of nets and discover 150 new hard types, cf. §1 - §3, §6, §9 of  Figure 1, and 4 new tractable types, cf. Figure 1 §10, and comprise the known results for the other 102 types as well, cf. Figure 1 §4, §5, §7, §8. In particular, our characterization categorizes Boolean types with regard to their behavioral capabilities, resulting from the Boolean interactions involved. This reveals the internal organization of the entire class of Boolean nets, suggesting a general approach for reasoning about its subclasses.

This paper is organized as follows. In Section 2, all the necessary notions and definitions will be given. Section 3 presents NP-completeness results for the types with nop-interaction. $\tau$-SSP for types without nop is investigated in Section 4. Concluding remarks are given in Section 5. One proof is shifted to the appendix.

In this section, we introduce necessary notions and definitions, supported by illustrations and examples, and some basic results that are used throughout the paper.

Transition Systems. A (finite, deterministic) transition system (TS, for short) $A=(S,E,\delta)$ is a directed labeled graph with the set of nodes $S$ (called states), the set of labels $E$ (called events) and partial transition function $\delta:S\times E\longrightarrow S$. If $\delta(s,e)$ is defined, we say that $e$ occurs at state $s$, denoted by $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope\pgfsys@setlinewidth{0.24pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{} \pgfsys@moveto{-1.10999pt}{1.47998pt}\pgfsys@curveto{-1.01749pt}{0.92499pt}{0.0pt}{0.0925pt}{0.2775pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.0925pt}{-1.01749pt}{-0.92499pt}{-1.10999pt}{-1.47998pt}\pgfsys@stroke\pgfsys@endscope}} }{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}$. By $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s^{\prime}\in A$, we denote $\delta(s,e)=s^{\prime}$. This notation extends to paths, i.e., $q_{0}\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.87895pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{1}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\dots\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.49608pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{n}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}q_{n}\in A$ denotes $q_{i-1}\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.34169pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{i}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}q_{i}\in A$ for all $i\in\{1,\dots,n\}$. A TS $A$ is loop-free, if $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s^{\prime}\in A$ implies $s\not=s^{\prime}$. A loop-free TS $A$ is bi-directed if $s^{\prime}\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s\in A$ implies $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s^{\prime}\in A$. We say $s_{0}\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.3975pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.3975pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.87895pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{1}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\dots\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.3975pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.3975pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.49608pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{n}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s_{n}\in A$ is a simple bi-directed path if $s_{i}\not=s_{j}$ for all $i\not=j$ with $i,j\in\{0,\dots,n\}$. An initialized TS $A=(S,E,\delta,\iota)$ is a TS with a distinct initial state $\iota\in S$, where every state $s\in S$ is reachable from $\iota$ by a directed labeled path.

Boolean Types of Nets [3]. The following notion of Boolean types of nets allows to capture all Boolean Petri nets in a uniform way. A Boolean type of net $\tau=(\{0,1\},E_{\tau},\delta_{\tau})$ is a TS such that $E_{\tau}$ is a subset of the Boolean interactions: $E_{\tau}\subseteq I=\{\textsf{nop},\textsf{inp},\textsf{out},\textsf{res},\textsf{set},\textsf{swap},\textsf{used},\textsf{free}\}$. Each interaction $i\in I$ is a binary partial function $i:\{0,1\}\rightarrow\{0,1\}$ as defined in Figure 4. For all $x\in\{0,1\}$ and all $i\in E_{\tau}$, the transition function of $\tau$ is defined by $\delta_{\tau}(x,i)=i(x)$. Notice that a type $\tau$ is completely determined by $E_{\tau}$. Hence we often identify $\tau$ with $E_{\tau}$, cf. Figure 1. Moreover, since $I$ captures all meaningful Boolean interactions [22, p. 617] and $\tau$ is defined by $E_{\tau}\subseteq I$, there are 256 Boolean types of nets at all. For a Boolean type of net $\tau$, we say that its state $1$ is inside and $0$ is outside. An interaction $i\in E_{\tau}$ exits if $1\leavevmode\hbox to17.37pt{\vbox to11.94pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.15785pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$i$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}0$, enters if $0\leavevmode\hbox to17.37pt{\vbox to11.94pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.15785pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$i$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}1$, saves 1 if $1\leavevmode\hbox to17.37pt{\vbox to11.94pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.15785pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$i$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}1$ and saves 0 if $0\leavevmode\hbox to17.37pt{\vbox to11.94pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.15785pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$i$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}0$. Accordingly, we group interactions together by $\mathfrak{exit}=\{\textsf{inp},\textsf{res},\textsf{swap}\}$, $\mathfrak{enter}=\{\textsf{out},\textsf{set},\textsf{swap}\}$, $\mathfrak{save}_{1}=\{\textsf{nop},\textsf{set},\textsf{used}\}$, $\mathfrak{save}_{0}=\{\textsf{nop},\textsf{res},\textsf{free}\}$ and $\mathfrak{save}=\mathfrak{save}_{1}\cup\mathfrak{save}_{0}$.

For a net of type $\tau$ ($\tau$-net), the interactions of $\tau$ determine relations between places and transitions of the net. For instance, if a place $p$ and a transition $t$ are related via inp, then $p$ has to be marked (true) to allow $t$ to fire, and becomes unmarked (false) after the firing (cf. Figure 4). Since we are only concerned with state separation, we omit the formal definition of $\tau$-nets and rather refer to, e.g., [3] for a comprehensive introduction to the topic.

$\tau$-Regions. The following notion of $\tau$-regions is the key concept for state separation. A $\tau$-region $R=(sup,sig)$ of TS $A=(S,E,\delta,\iota)$ consists of the mappings support $sup:S\rightarrow\{0,1\}$ and signature $sig:E\rightarrow E_{\tau}$, such that for every edge $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s^{\prime}$ of $A$, the edge $sup(s)\leavevmode\hbox to27.22pt{\vbox to14.67pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{23.7875pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.7875pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.81444pt}{4.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$sig(e)$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}sup(s^{\prime})$ belongs to the type $\tau$; we also say $sup$ allows ($sig$ and thus the region) $R$. If $P=q_{0}\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.87895pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{1}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\dots\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.49608pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{n}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}q_{n}$ is a path in $A$, then $P^{R}=sup(q_{0})\leavevmode\hbox to30.81pt{\vbox to14.67pt{\pgfpicture\makeatletter\hbox{\;\lower-1.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{23.7875pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.7875pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.01999pt}{4.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$sig(e_{1})$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\dots\leavevmode\hbox to31.58pt{\vbox to14.67pt{\pgfpicture\makeatletter\hbox{\enspace\lower-1.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{23.7875pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.7875pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.36288pt}{4.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$sig(e_{n})$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}sup(q_{n})$ is a path in $\tau$. We say $P^{R}$ is the image of $P$ (under $R$). For region $R$ and path $P$, event $e_{i}$ with $1\leq i\leq n$ is called state changing on the path $P^{R}$, if $sup(q_{i-1})\neq sup(q_{i})$ for $q_{i-1}\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.34169pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{i}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}q_{i}$ in $P$. Notice that $R$ is implicitly defined by $sup(\iota)$ and $sig$: Since $A$ is reachable, for every state $s\in S$, there is a path $\iota\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.87895pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{1}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}\dots\leavevmode\hbox to17.37pt{\vbox to11.31pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-1.53299pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.49608pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e_{n}$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s_{n}$ such that $s=s_{n}$. Thus, since $\tau$ is deterministic, we inductively obtain $sup(s_{i+1})$ by $sup(s_{i})\leavevmode\hbox to37.99pt{\vbox to14.67pt{\pgfpicture\makeatletter\hbox{\enspace\lower-1.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{28.05525pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.05525pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.43341pt}{4.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$sig(e_{i+1})$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}sup(s_{i+1})$ for all $i\in\{0,\dots,n-1\}$ and $s_{0}=\iota$. Hence, we can compute $sup$ and thus $R$ purely from $sup(\iota)$ and $sig$, cf. Figure 4. If $\textsf{nop}\in\tau$, then a $\tau$-region $R=(sup,sig)$ of a TS $A$ is called normalized if $sig(e)=\textsf{nop}$ for as many events $e$ as $sup$ allows: for all $e\in E$, $sig(e)\not\in\{\textsf{used},\textsf{free}\}$ and if $sig(e)\in\mathfrak{exit}\cup\mathfrak{enter}$ then there is $s\leavevmode\hbox to17.37pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\thinspace\lower-0.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{16.67432pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.67432pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@invoke{ }\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6734pt}{3.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{\scalebox{0.8}{$e$}}$}} }}\pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope}}} \pgfsys@invoke{ }\pgfsys@endscope} \pgfsys@invoke{ }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{ }\pgfsys@endscope\hss}}\endpgfpicture}}s^{\prime}\in A$ such that $sup(s)\not=sup(s^{\prime})$.

$\tau$-State Separation Property. A pair $(s,s^{\prime})$ of distinct states of $A$ defines a state separation atom (SSP atom). A $\tau$-region $R=(sup,sig)$ solves $(s,s^{\prime})$ if $sup(s)\not=sup(s^{\prime})$. If $R$ exists, then $(s,s^{\prime})$ is called $\tau$-solvable. If $s\in S_{A}$ and, for all $s^{\prime}\in S_{A}\setminus\{s\}$, the atom $(s,s^{\prime})$ is $\tau$-solvable, then $s$ is called $\tau$-solvable. A TS has the $\tau$-state separation property ($\tau$-SSP) if all of its SSP atoms are $\tau$-solvable. A set $\mathcal{R}$ of $\tau$-regions of $A$ is called $\tau$-separative if for each SSP atom of $A$ there is a $\tau$-region $R$ in $\mathcal{R}$ that solves it. By the next lemma, if $\textsf{nop}\in\tau$, then $A$ has the $\tau$-SSP if and only if it has a $\tau$-separative set of normalized $\tau$-regions:

By the following lemma, $\tau$-SSP and $\tilde{\tau}$-SSP are equivalent if $\tau$ and $\tilde{\tau}$ are isomorphic:

In this paper, we consider the $\tau$-SSP also as decision problem that asks whether a given TS $A$ has the $\tau$-SSP. The decision problem $\tau$-SSP is in NP: By definition, $A$ has at most $\lvert S\rvert^{2}$ SSP atoms. Hence, a Turing-machine can (non-deterministically) guess a $\tau$-separative set $\mathcal{R}$ such that $|\mathcal{R}|\leq|S|^{2}$ and (deterministically) check in polynomial time its validity if it exists.

In what follows, some of our NP-completeness results base on polynomial-time reductions of the following decision problem, which is known to be NP-complete [11]:

Cubic Monotone 1-in-3 3Sat. (CM 1-in-3 3Sat) The input is a Boolean formula $\varphi=\{\zeta_{0},\dots,\zeta_{m-1}\}$ of negation-free three-clauses $\zeta_{i}=\{X_{i_{0}},X_{i_{1}},X_{i_{2}}\}$, where $i\in\{0,\dots,m-1\}$, with set of variables $X=\bigcup_{i=0}^{m-1}\zeta_{i}$; every variable $v\in X$ occurs in exactly three clauses, implying $|X|=m$. The question to decide is whether there is a (one-in-three model) $M\subseteq X$ satisfying $|M\cap\zeta_{i}|=1$ for all $i\in\{0,\dots,m-1\}$.

In this section, we investigate the computational complexity of nop-equipped Boolean types of nets. For technical reasons, we separately consider the types that include neither res nor set (§5 - §7 in Figure 1) and the ones that have at least one of them (§1 - §4 in Figure 1).

First of all, the fact that $\tau$-SSP is polynomial for the types of §7 in Figure 1 is implied by the results of [16][22, p. 619]. Moreover, for the types of Figure 1 §5 the NP-completeness of $\tau$-SSP has been shown in [23] ($\tau=\{\textsf{nop},\textsf{inp},\textsf{out}\}$), and in [19] ($\tau=\{\textsf{nop},\textsf{inp},\textsf{out},\textsf{used}\}$, there referred to as $1$-bounded P/T-nets). Thus, in order to complete the complexity characterization for the nop-equipped types that neither contain res nor set, it only remains to ascertain the complexity of $\tau$-SSP for the types Figure 1 §6. The following Subsection 3.1 proves that $\tau$-SSP is NP-complete for these types.

Then, we proceed with the types of §1 - §4 in Figure 1. The fact that $\tau$-SSP is polynomial for the types of §4 follows from [22, p. 619]. The NP-completeness of $\tau$-SSP for the remaining types (§1 - §3) will be demonstrated in Subsection 3.2.