Membrane Systems and Petri Net Synthesis
(Invited Paper)

A multiset over a finite set $X$ is a function $\theta:X\to{\mathbb{N}}=\{0,1,2,\ldots\}$. $\theta$ may be represented by listing its elements with repetitions, e.g., $\theta=\{y,y,z\}$ is such that $\theta(y)=2$, $\theta(z)=1$, and $\theta(x)=0$ otherwise. $\theta$ is said to be empty (and denoted by $\varnothing$) if there are no $x$ such that $x\in\theta$ by which we mean that $x\in X$ and $\theta(x)\geq 1$.

For two multisets $\theta$ and $\theta^{\prime}$ over $X$, the sum $\theta+\theta^{\prime}$ is the multiset given by $(\theta+\theta^{\prime})(x)=\theta(x)+\theta^{\prime}(x)$ for all $x\in X$, and for $k\in{\mathbb{N}}$ the multiset $k\cdot\theta$ is given by $(k\cdot\theta)(x)=k\cdot\theta(x)$ for all $x\in X$. The difference $\theta-\theta^{\prime}$ is given by $(\theta-\theta^{\prime})(x)=\max\{\theta(x)-\theta^{\prime}(x),0\}$ for all $x\in X$. We denote $\theta\leq\theta^{\prime}$ whenever $\theta(x)\leq\theta^{\prime}(x)$ for all $x\in X$, and $\theta<\theta^{\prime}$ whenever $\theta\leq\theta^{\prime}$ and $\theta\neq\theta^{\prime}$. The restriction $\theta|_{Z}$ of $\theta$ to a subset $Z\subseteq X$ is given by $\theta|_{Z}(x)=\theta(x)$ for $x\in Z$, and $\theta|_{Z}(x)=0$ otherwise. The size $|\theta|$ of $\theta$ is given by $\sum_{x\in X}\theta(x)$. If $f:X\to Y$ is a function then $f(\theta)$ is the multiset over $Y$ such that $f(\theta)(y)=\sum_{x\in f^{-1}(y)}\theta(x)$, for every $y\in Y$.

A step transition system over a finite set (of actions) $A$ is a triple $\mathit{TS}=(Q,\mathcal{A},q_{0})$, where: $Q$ is a set of nodes called states; $\mathcal{A}$ is the set of arcs, each arc being a triple $(q,\alpha,q^{\prime})$ such that $q,q^{\prime}\in Q$ are states and $\alpha$ is a multiset over $A$; and $q_{0}\in Q$ is the initial state. We may write $q\xrightarrow{\alpha}q^{\prime}$ whenever $(q,\alpha,q^{\prime})$ is an arc, and denote by

$$\mathit{tsSteps}_{q}=\{\alpha\mid\alpha\neq\varnothing~\wedge~\exists q^{\prime}:~q\xrightarrow{\alpha}q^{\prime}\}$$

the set of nonempty steps enabled at a state $q$ in $\mathit{TS}$. We additionally assume that:

• •

  if $q\xrightarrow{\alpha}q^{\prime}$ and $q\xrightarrow{\alpha}q^{\prime\prime}$ then $q^{\prime}=q^{\prime\prime}$ (i.e., $\mathit{TS}$ is deterministic);

• •

  for every state $q\in Q$, there is a path from $q_{0}$ leading to $q$;

• •

  for every action $a\in A$, there is an arc $q\xrightarrow{\alpha}q^{\prime}$ in $\mathit{TS}$ such that $a\in\alpha$; and

• •

  for every state $q\in Q$, we have $q\xrightarrow{\varnothing}q^{\prime}$ iff $q=q^{\prime}$.

Let $\mathit{TS}=(Q,\mathcal{A},q_{0})$ be a step transition system over a set of actions $A$, and $\mathit{TS}^{\prime}=(Q^{\prime},\mathcal{A}^{\prime},q^{\prime}_{0})$ be a step transition system over a set of actions $A^{\prime}$. $\mathit{TS}$ and $\mathit{TS}^{\prime}$ are isomorphic if there are two bijections, $\phi:A\to A^{\prime}$ and $\nu:Q\to Q^{\prime}$, such that $\nu(q_{0})=q^{\prime}_{0}$ and, for all states $q,q^{\prime}\in Q$ and multisets $\alpha$ over $A$:

$$(q,\alpha,q^{\prime})\in\mathcal{A}~~~\Longleftrightarrow~~~(\nu(q),\phi(\alpha),\nu(q^{\prime}))\in\mathcal{A}^{\prime}\;.$$

We denote this by $\mathit{TS}\sim_{\phi,\nu}\mathit{TS}^{\prime}$ or $\mathit{TS}\sim\mathit{TS}^{\prime}$.

A Place/Transition net (or pt-net) is specified as a tuple $\mathit{PT}=(P,T,W,M_{0})$, where: $P$ and $T$ are finite disjoint sets of respectively places and transitions; $W:(T\times P)\cup(P\times T)\rightarrow{\mathbb{N}}$ is the arc weight function; and $M_{0}:P\rightarrow{\mathbb{N}}$ is the initial marking (in general, any multiset of places is a marking). We assume that, for each transition $t$, there is at least one place $p$ such that $W(p,t)>0$. In diagrams, such as that in Figure 1, places are drawn as circles, and transitions as boxes. If $W(x,y)\geq 1$, then $(x,y)$ is an arc leading from $x$ to $y$. An arc is annotated with its weight if the latter is greater than one. A marking $M$ is represented by drawing in each place $p$ exactly $M(p)$ tokens (small black dots).

A step $U$ of $\mathit{PT}$ is a multiset of transitions. Its pre-multiset and post-multiset of places, ${\rm Phys.~Rev.~E}{U}$ and ${{U}}^{\bullet}$, are respectively given by

$${\rm Phys.~Rev.~E}{U}(p)=\sum_{t\in U}U(t)\cdot W(p,t)~~~\mbox{and}~~~{{U}}^{\bullet}(p)=\sum_{t\in U}U(t)\cdot W(t,p)\;,$$

for each place $p$. For the pt-net in Figure 1 we have:

$${\rm Phys.~Rev.~E}{\{\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{31}}_{3}\}}=\{\mathtt{p}^{b}_{1},\mathtt{p}^{b}_{1},\mathtt{p}^{a}_{3}\}~\mbox{and}~{{\{\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{31}}_{3}\}}}^{\bullet}=\{\mathtt{p}^{a}_{1},\mathtt{p}^{a}_{1},\mathtt{p}^{c}_{1},\mathtt{p}^{a}_{3},\mathtt{p}^{a}_{3},\mathtt{p}^{c}_{3}\}.$$

We distinguish two basic modes of execution of pt-nets. To start with, a step of transitions $U$ is free-enabled at a marking $M$ if ${\rm Phys.~Rev.~E}{U}\leq M$. We denote this by $M[U\rangle_{\mathit{free}}$, and then say that a free-enabled $U$ is max-enabled at $M$ if $U$ cannot be extended by a transition to yield a step which is free-enabled at $M$, i.e., there is no $t\in T$ such that $M[U+\{t\}\rangle_{\mathit{free}}$. We denote this by $M[U\rangle_{\mathit{max}}$. In other words, $U$ is free-enabled at $M$ if in each place there are sufficiently many tokens for the specified multiple occurrence of each of its transitions. Maximal concurrency (max-enabledness) means that extending $U$ would demand more tokens than $M$ supplies. For the pt-net in Figure 1 we have that, at the given marking $M_{0}$, the step $\{\mathtt{t}^{\mathtt{r}_{12}}_{1},\mathtt{t}^{\mathtt{r}_{21}}_{2}\}$ is free-enabled but not max-enabled, and $\{\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{12}}_{1},\mathtt{t}^{\mathtt{r}_{21}}_{2},\mathtt{t}^{\mathtt{r}_{22}}_{2}\}$ is max-enabled.

For each mode of execution $\mathfrak{m}\in\{\mathit{free},\mathit{max}\}$, a step $U$ which is $\mathfrak{m}$-enabled at a marking $M$ can be $\mathfrak{m}$-executed leading to the marking $M^{\prime}$ given by $M^{\prime}=M-{\rm Phys.~Rev.~E}{U}+{{U}}^{\bullet}$. We denote this by $M[U\rangle_{\mathfrak{m}}M^{\prime}$. For the pt-net in Figure 1 we have

$$M_{0}[\{\mathtt{t}^{\mathtt{r}_{12}}_{1},\mathtt{t}^{\mathtt{r}_{21}}_{2}\}\rangle_{\mathit{free}}\{\mathtt{p}^{b}_{1},\mathtt{p}^{b}_{1},\mathtt{p}^{b}_{2},\mathtt{p}^{b}_{2},\mathtt{p}^{c}_{2},\mathtt{p}^{c}_{2},\mathtt{p}^{a}_{3}\}\;.$$

pt-nets are a general model of concurrent computation. To capture the compartmentisation of membrane systems, [13] adds explicit localities to transitions. Though not necessary from a modelling point of view, we associate in this paper — only for notational convenience — also each place with a locality.

A pt-net with localities (or ptl-net) is a tuple ${\mathit{PTL}}=(P,T,W,\ell,M_{0})$ such that $(P,T,W,M_{0})$ is a pt-net, and $\ell$ is a location mapping for the transitions and places. Whenever $\ell(x)=\ell(z)$, we call $x$ and $z$ co-located. In diagrams, nodes representing co-located transitions and/or places will be shaded in the same way, as shown in Figure 2.

Co-locating transitions leads to one more way of enabling for steps of transitions. We say that a step $U$ of ${\mathit{PTL}}$ is lmax-enabled at a marking $M$ if $M[U\rangle_{\mathit{free}}$ and $U$ cannot be extended by a transition co-located with a transition in $U$ to yield a step which is free-enabled at $M$; i.e., there is no $t\in T$ such that $\ell(t)\in\ell(U)$ and $M[U+\{t\}\rangle_{\mathit{free}}$. We denote this by $M[U\rangle_{\mathit{lmax}}$, and then denote the lmax-execution of $U$ by $M[U\rangle_{\mathit{lmax}}M^{\prime}$, where $M^{\prime}=M-{\rm Phys.~Rev.~E}{U}+{{U}}^{\bullet}$. Note that locally maximal (lmax) concurrency is similar to maximal concurrency, but now only active localities¹¹1 By active localities of a step $U$ we mean the localities of transitions present in $U$. cannot execute further transitions. For the ptl-net in Figure 2 we have that $\{\mathtt{t}^{\mathtt{r}_{11}}_{1},\mathtt{t}^{\mathtt{r}_{12}}_{1}\}$ is lmax-enabled at the given marking, but $\{\mathtt{t}^{\mathtt{r}_{11}}_{1}\}$ is not.

Let $\mathfrak{m}\in\{\mathit{free},\mathit{max},\mathit{lmax}\}$ be a mode of execution of a ptl-net ${\mathit{PTL}}$. Then an $\mathfrak{m}$-step sequence is a finite sequence of $\mathfrak{m}$-executions starting from the initial marking, and an $\mathfrak{m}$-reachable marking is any marking resulting from the execution of such a sequence. Moreover, the $\mathfrak{m}$-concurrent reachability graph of ${\mathit{PTL}}$ is the step transition system:

$$\mathit{CRG}_{\mathfrak{m}}({\mathit{PTL}})=\Big{(}~[M_{0}\rangle_{\mathfrak{m}}~,~\big{\{}(M,U,M^{\prime})\mid M\in[M_{0}\rangle_{\mathfrak{m}}\;\wedge\;M[U\rangle_{\mathfrak{m}}M^{\prime}\big{\}}~,~M_{0}~\Big{)}\;,$$

where $[M_{0}\rangle_{\mathfrak{m}}$ is the set of all $\mathfrak{m}$-reachable markings which are the nodes of the graph; $M_{0}$ is the initial node; and the arcs between the nodes are labelled by $\mathfrak{m}$-executed steps of transitions. Concurrent reachability graphs provide complete representations of the dynamic behaviour of ptl-nets evolving according to the chosen mode of execution.

A membrane structure $\mu$ (of degree $m\geq 1$) is given by a rooted tree with $m$ nodes identified with the integers $1,\ldots,m$. We will write $(i,j)\in\mu$ or $i=\mathit{parent}(j)$ to indicate that there is an edge from $i$ (parent) to $j$ (child) in the tree of $\mu$, and $i\in\mu$ means that $i$ is a node of $\mu$. The nodes of a membrane structure represent nested membranes which in turn determine compartments (compartment $i$ is enclosed by membrane $i$ and lies in-between $i$ and its children, if any), as shown in Figure 3.

We will say that a ptl-net ${\mathit{PTL}}=(P,T,W,\ell,M_{0})$ is spanned over the membrane structure $\mu$ if $\ell:P\cup T\to\mu$ and the following hold, for all $p\in P$ and $t\in T$:

• •

  if $W(p,t)>0$ then $\ell(p)=\ell(t)$; and

• •

  if $W(t,p)>0$ then $\ell(p)=\ell(t)$ or $(\ell(p),\ell(t))\in\mu$ or $(\ell(t),\ell(p))\in\mu$.

The ptl-net of Figure 2 is spanned over the membrane structure depicted in Figure 3.

Let $V$ be a finite alphabet of names of objects (or molecules) and let $\mu$ be a membrane structure of degree $m$. A basic membrane system (over $V$ and $\mu$) is a tuple

$${\mathit{BMS}}=(V,\mu,w_{1}^{0},\ldots,w_{m}^{0},R_{1},\ldots,R_{m})$$

such that, for every membrane $i$, $w_{i}^{0}$ is a multiset of objects from $V$, and $R_{i}$ is a finite set of evolution rules associated with membrane (compartment) $i$. Each evolution rule $r\in R_{i}$ is of the form $r:\mathit{lhs}^{r}\to\mathit{rhs}^{r}$, where $\mathit{lhs}^{r}$ (the left hand side of $r$) is a nonempty multiset over $V$, and $\mathit{rhs}^{r}$ (the right hand side of $r$) is a multiset over

$$V\cup\{a_{\mathit{out}}\mid a\in V\}\cup\{a_{{in}_{j}}\mid a\in V~and~(i,j)\in\mu\}\;.$$

Here a symbol $a_{{in}_{j}}$ represents an object $a$ that is sent to a child node (compartment) $j$ and $a_{\mathit{out}}$ means that $a$ is sent to the parent node. If $i$ is the root of $\mu$ then no indexed object of the form $a_{\mathit{out}}$ belongs to $\mathit{rhs}^{r}$. A configuration of ${\mathit{BMS}}$ is a tuple

$$C=(w_{1},\ldots,w_{m})$$

of multisets of objects, and $C_{0}=(w_{1}^{0},\ldots,w_{m}^{0})$ is the initial configuration. Figure 4 shows a basic membrane system over the membrane structure depicted in Figure 3.

A membrane system evolves from configuration to configuration as a consequence of the application of evolution rules. There are different execution modes ranging from fully synchronous — as many applications of rules as possible — to sequential — a single application of a rule at a time. Here, similarly as in the case of ptl-nets, we distinguish three modes, all based on the notion of a vector multi-rule.

A vector multi-rule of ${\mathit{BMS}}$ is a tuple $\mathbf{r}=\langle\mathbf{r}_{1},\ldots,\mathbf{r}_{m}\rangle$ where, for each membrane $i$ of $\mu$, $\mathbf{r}_{i}$ is a multiset of rules from $R_{i}$. For such a vector multi-rule, we denote by $\mathit{lhs}_{i}^{\mathbf{r}}$ the multiset

$$\sum_{r\in R_{i}}\mathbf{r}_{i}(r)\cdot\mathit{lhs}^{r}$$

in which all objects in the left hand sides of the rules in $\mathbf{r}_{i}$ are accumulated, and by $rhs_{i}^{\mathbf{r}}$ the multiset

$$\sum_{r\in R_{i}}\mathbf{r}_{i}(r)\cdot\mathit{rhs}^{r}$$

of all (indexed) objects in the right hand sides. The first multiset specifies how many objects are needed in each compartment for the simultaneous execution of all the instances of evolution rules in $\mathbf{r}$.

A vector multi-rule $\mathbf{r}$ of ${\mathit{BMS}}$ is

• •

  free-enabled at a configuration $C$ if $\mathit{lhs}_{i}^{\mathbf{r}}\leq w_{i}$, for each $i$.

Moreover, a free-enabled vector multi-rule $\mathbf{r}=\langle\mathbf{r}_{1},\ldots,\mathbf{r}_{m}\rangle$ is:

• •

  max-enabled if no $\mathbf{r}_{i}$ can be extended to a vector multi-rule which is free-enabled at $C$; and

• •

  lmax-enabled if no nonempty $\mathbf{r}_{i}$ can be extended to a vector multi-rule which is free-enabled at $C$.

For example, in Figure 4,

• •

  $\langle\varnothing,\varnothing,\{\mathtt{r}_{31}\}\rangle$ is not free-enabled;

• •

  $\langle\{\mathtt{r}_{11},\mathtt{r}_{12}\},\varnothing,\varnothing\rangle$ is lmax-enabled but not max-enabled; and

• •

  $\langle\{\mathtt{r}_{11},\mathtt{r}_{12}\},\{\mathtt{r}_{21},\mathtt{r}_{22}\},\varnothing\rangle$ is max-enabled.

If $\mathbf{r}$ is free-enabled ($\mathit{free}$) at a configuration $C$, then $C$ has in each membrane $i$ enough copies of objects for the application of the multiset of evolution rules $\mathbf{r}_{i}$. Maximal concurrency ($\mathit{max}$) requires that adding any extra rule makes $\mathbf{r}$ demand more objects than $C$ can provide. Locally maximal concurrency ($\mathit{lmax}$) is similar but in this case only those compartments which have rules in $\mathbf{r}$ cannot enable any more rules; in other words, each compartment either uses no rule, or uses a maximal multiset of rules.

The effect of the rules is independent of the mode of execution $\mathfrak{m}\in\{\mathit{free},\mathit{max},\mathit{lmax}\}$. A vector multi-rule $\mathbf{r}$ which is $\mathfrak{m}$-enabled at $C$ can $\mathfrak{m}$-evolve to a configuration $C^{\prime}=(w^{\prime}_{1},\ldots w^{\prime}_{m})$ such that, for each $i$ and object $a$:

$$w^{\prime}_{i}(a)=w_{i}(a)-\mathit{lhs}_{i}^{\mathbf{r}}(a)+\mathit{rhs}_{i}^{\mathbf{r}}(a)+\mathit{rhs}_{\mathit{parent}(i)}^{\mathbf{r}}(a_{{in}_{i}})+\!\!\!\!\!\!\sum_{i=\mathit{parent}(j)}\!\!\!\!\!\!\mathit{rhs}_{j}^{\mathbf{r}}(a_{\mathit{out}})$$

where $\mathit{rhs}_{\mathit{parent}(i)}^{\mathbf{r}}=\varnothing$ if $i$ is the root of $\mu$. We denote this by $C\stackrel{{\scriptstyle\mathbf{r}}}{{\longrightarrow}}_{\mathfrak{m}}C^{\prime}$. Moreover, an $\mathfrak{m}$-computation is a finite sequence of $\mathfrak{m}$-evolutions starting from the initial configuration; any configuration which can be obtained through such a computation is called $\mathfrak{m}$-reachable. For the basic membrane system depicted in Figure 4 we have, for example:

$$C_{0}\xrightarrow{\langle\{\mathtt{r}_{11},\mathtt{r}_{12}\},\varnothing,\varnothing\rangle}_{\mathit{lmax}}(\{a,b\},\{a,b,c,c,c\},\{a\})\xrightarrow{\langle\varnothing,\{\mathtt{r}_{21},\mathtt{r}_{22}\},\varnothing\rangle}_{\mathit{lmax}}(\{a,b\},\{a,b,c,c\},\{a\})\;.$$

Let $\mathfrak{m}\in\{\mathit{free},\mathit{max},\mathit{lmax}\}$ be a mode of execution of a basic membrane system ${\mathit{BMS}}$. Then the $\mathfrak{m}$-concurrent reachability graph of ${\mathit{BMS}}$ is given by:

$$\mathit{CRG}_{\mathfrak{m}}({\mathit{BMS}})=\Big{(}~[C_{0}\rangle_{\mathfrak{m}}~,~\big{\{}(C,\mathbf{r}_{1}+\ldots+\mathbf{r}_{m},C^{\prime})\mid C\in[C_{0}\rangle_{\mathfrak{m}}\;\wedge\;C\xrightarrow{\langle\mathbf{r}_{1},\ldots,\mathbf{r}_{m}\rangle}_{\mathfrak{m}}C^{\prime}\big{\}}~,~C_{0}~\Big{)}\;,$$

where $[C_{0}\rangle_{\mathfrak{m}}$ is the set of all $\mathfrak{m}$-reachable configurations which are the nodes of the graph; $C_{0}$ is the initial node; and the arcs between the nodes are labelled by multisets of evolution rules involved in the $\mathfrak{m}$-executed vector multi-rules.²²2 Though it may be that rules from different membranes are the same in terms of the multisets defining their left hand and right hand sides, we assume here that evolution rules associated with different membranes can be distinguished, e.g., by giving them each their own name (an injective label). Similarly as in the case of ptl-nets, concurrent reachability graphs capture completely the dynamic behaviour of basic membrane system evolving according to the chosen mode of execution.

Let $T$, $\mu$, $\ell:T\to\mu$ and $\mathit{TS}=(Q,\mathcal{A},q_{0})$ be as in Problem 1. Assume that $Q=\{q_{0},\ldots,q_{h}\}$ and $T=\{t_{1},\ldots,t_{n}\}$. We use three vectors of non-negative variables:

$\displaystyle\mathbf{x}$

$\displaystyle=x_{0}\ldots x_{h}$

$\displaystyle\mathbf{y}$

$\displaystyle=y_{1}\ldots y_{n}$

$\displaystyle\mathbf{z}$

$\displaystyle=z_{1}\ldots z_{n}\;.$

We also denote $\mathbf{p}=\mathbf{x}\mathbf{y}\mathbf{z}$ and define a homogeneous linear system

$$\mathcal{P}~:~\left\{\begin{array}[]{ll}x_{i}\geq\alpha\cdot\mathbf{z}\\ x_{j}=x_{i}+\alpha\cdot(\mathbf{y}-\mathbf{z})\end{array}\mbox{for all $q_{i}\xrightarrow{\alpha}q_{j}$ in $\mathit{TS}$}\right.$$

where $\alpha\cdot\mathbf{z}$ denotes $\alpha(t_{1})\cdot z_{1}+\cdots+\alpha(t_{n})\cdot z_{n}$ and similarly for $\alpha\cdot(\mathbf{y}-\mathbf{z})$.

The regions (2) of $\mathit{TS}$ are then determined by the integer solutions $\mathbf{p}$ of the system $\mathcal{P}$ assuming that, for $0\leq i\leq h$ and $1\leq j\leq n$,

$\displaystyle\sigma(q_{i})$

$\displaystyle=x_{i}$

$\displaystyle\imath(t_{j})$

$\displaystyle=y_{j}$

$\displaystyle\omega(t_{j})$

$\displaystyle=z_{j}$

The set of rational solutions of $\mathcal{P}$ forms a polyhedral cone in $\mathbb{Q}^{h+2n+1}$. As described in [4], one can effectively compute finitely many integer generating rays $\mathbf{p}^{1},\ldots,\mathbf{p}^{k}$ of this cone such that any integer solution $\mathbf{p}$ of $\mathcal{P}$ can be expressed as a linear combination of the rays with non-negative rational coefficients:

$$\mathbf{p}=\sum_{l=1}^{k}c_{l}\cdot\mathbf{p}^{l}\;.$$

Such rays $\mathbf{p}^{l}$ are fixed and (some of them) turned into net places if Problem 1 has a solution. More precisely, if $\mathbf{p}^{l}$ is included in the constructed net, then

$\displaystyle M_{0}(\mathbf{p}^{l})$

$\displaystyle=x^{l}_{0}$

$\displaystyle W(\mathbf{p}^{l},t_{i})$

$\displaystyle=z^{l}_{i}$

$\displaystyle W(t_{i},\mathbf{p}^{l})$

$\displaystyle=y^{l}_{i}\;,$

(4)

where $M_{0}$ is the initial marking of the target net, and $t_{i}\in T$.

Clearly, not all such rays can be considered for the inclusion in the net being constructed, as the corresponding regions have to be compatible with $\mu$. We therefore ensure through a simple check that the generating rays $\mathbf{p}^{1},\ldots,\mathbf{p}^{k}$ are compatible with $\mu$, deleting in the process those which are not. Note that any $\mathbf{p}\in\mathbf{P}_{\mu}$ is a non-negative linear combination of rays compatible with $\mu$.

Having found the generating rays compatible with $\mu$, we proceed to check whether Problem 1 has any solutions at all.

Let $\mathit{TS}=(Q,\mathcal{A},q_{0})$ be as in Problem 1. We take in turn each pair of distinct states, $q_{i}$ and $q_{j}$, of $Q$ and decide whether there exists $\mathbf{p}=(\sigma,\imath,\omega)\in\mathbf{P}_{\mu}$ with coefficients $c_{1},\ldots,c_{k}$ such that $\sigma(q_{i})=x_{i}\neq x_{j}=\sigma(q_{j})$. Since the latter is equivalent to

$$\sum_{l=1}^{k}c_{l}\cdot x_{i}^{l}\neq\sum_{l=1}^{k}c_{l}\cdot x_{j}^{l}\;,$$

one can simply check whether there exists at least one $\mathbf{p}^{l}$ (called a witness [7]) such that $x_{i}^{l}\neq x_{j}^{l}$.

Again, let $\mathit{TS}=(Q,\mathcal{A},q_{0})$ be as in Problem 1, and $\mathfrak{m}\in\{\mathit{free},\mathit{max},\mathit{lmax}\}$. First, we take in turn each state $q_{i}$ of $Q$, and calculate the set of region enabled steps, denoted by $\mathit{regSteps}_{q_{i}}$. Intuitively, region enabled steps are those that cannot be disabled (or blocked) by compatible regions.

To build $\mathit{regSteps}_{q_{i}}$ one only needs to consider nonempty steps $\alpha$ with $|\alpha|\leq m\cdot\mathit{Max}$, where $\mathit{Max}$ is the maximum size of steps labelling arcs in $\mathit{TS}$, and $m$ is the number of membranes of $\mu$. The reason is that, for each membrane $i\in\mu$ there exists a compatible region $(\sigma,\imath,\omega)\in\mathbf{P}_{\mu}$ (called a witness) such that $\sigma(Q)=\{\mathit{Max}\}$ and, for every $t\in T$,

$$\omega(t)=\iota(t)=\left\{\begin{array}[]{ll}1&\mbox{if }\ell(t)=i\\ 0&\mbox{otherwise}\;.\end{array}\right.$$

Taken together, all such regions block any step $\alpha$ with $|\alpha|>m\cdot\mathit{Max}$.

For each nonempty step $\alpha$ with $|\alpha|\leq m\cdot\mathit{Max}$ it is the case that $\alpha\notin\mathit{regSteps}_{q_{i}}$ iff for some $\mathbf{p}\in\mathbf{P}_{\mu}$ with coefficients $c_{1},\ldots,c_{k}$ we have $x_{i}<\alpha\cdot\mathbf{z}$. Since the latter is equivalent to

$$\sum_{l=1}^{k}c_{l}\cdot(x_{i}^{l}-\alpha\cdot\mathbf{z}^{l})<0\;,$$

one simply checks whether there exists at least one $\mathbf{p}^{l}$ (again called a witness) such that $x_{i}^{l}-\alpha\cdot\mathbf{z}^{l}<0$.

Having determined the region enabled steps, in order to establish forward closure we need to verify that, for every state $q\in Q$,

$$\mathit{tsSteps}_{q}=\left\{\begin{array}[]{ll}\mathit{regSteps}_{q}&\mbox{if }\mathfrak{m}=\mathit{free}\\[2.84526pt] \{\alpha\in\mathit{regSteps}_{q}\mid\neg\exists t\in T:~\alpha+\{t\}\in\mathit{regSteps}_{q}\}&\mbox{if }\mathfrak{m}=\mathit{max}\\[2.84526pt] \{\alpha\in\mathit{regSteps}_{q}\mid\neg\exists t\in T:~\alpha+\{t\}\in\mathit{regSteps}_{q}~\wedge~\ell(t)\in\ell(\alpha)\}&\mbox{if }\mathfrak{m}=\mathit{lmax}\;.\end{array}\right.$$

If the above checks for the feasibility of Problem 1 are successful, one can construct a solution ptl-net spanned over $\mu$ by taking all the witness rays and regions, and treating them as places in the way indicated in (4). The resulting net ${\mathit{PTL}}$ satisfies

$$\mathit{CRG}_{\mathfrak{m}}({\mathit{PTL}})\sim\mathit{TS}\;.$$