reduction from the iterated monotone Boolean circuit problem [@GMST2016].

A depth-1 monotone Boolean circuit is a function $F:\{0,1\}^n \to \{0,1\}^n$ represented by $n$ constraints: For every $0\le i < n$ there is a constraint of the form $i'=j \otimes k,$ where $0\le j,k< n$ and $\otimes \in \{\wedge, \vee \}$, which expresses how the next value of bit $i$ depends on the current values of bits $j$ and $k$. For every bitvector ${\boldsymbol{v}}\in\{0,1\}^n$, the function $F$ then satisfies $F({\boldsymbol{v}})[i]{{\overset{\text{\tinydef}}{=}}}{\boldsymbol{v}}[j]\otimes {\boldsymbol{v}}[k]$. It is [$\mathsf{PSPACE}$]{}-complete to check whether for a given vector ${\boldsymbol{v}}\in\{0,1\}^n$ there exists a number $m\in{\mathbb{N}}$ such that $F^m({\boldsymbol{v}})[0]=1$.

Towards a lower bound for [$\exists$COVER ]{} (\[thm:LB\]) we construct a non-consuming TPN as follows, for a given circuit. The main idea is to simulate circuit constraints by transitions that reset tokens of age $1$ (encoding ${\boldsymbol{v}}$) to fresh ones of age $0$ (encoding $F({\boldsymbol{v}})$), and let time pass by one unit to enter the next round.

(Tj) \[place, label=right:[$\mathit{True}_j$]{}\] ; (Ti) \[place, below=of Tj, label=right:[$\mathit{True}_{i}$]{}\] ; (Tk) \[place, below=of Ti, label=right:[$\mathit{True}_{k}$]{}\] ; (Fj) \[place, right=of Tj, label=left:[$\mathit{False}_j$]{}\] ; (Fi) \[place, right=of Ti, label=left:[$\mathit{False}_{i}$]{}\] ; (Fk) \[place, right=of Tk, label=left:[$\mathit{False}_{k}$]{}\] ;

(iB) \[transition, left=of Ti, label=left:[$i.B$]{}\] [$x=y=1$]{}; (iL) \[transition, right=of Fj, 