From bbdb5c72720c92cdc2a02efc1e23fb863e9a419d Mon Sep 17 00:00:00 2001 From: partowp Date: Thu, 20 Aug 2026 15:51:41 +0100 Subject: [PATCH] the counter-example added but it's sloppy --- draft/draft.tex | 17 +++++++++++++---- 1 file changed, 13 insertions(+), 4 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 43fc677..9ecfd4a 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2531,11 +2531,11 @@ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined \end{gather*} $\sigma$ is defined as below: \begin{gather*} - \sigma(w)=\{(1,2),(2,1)\} + \forall w\in R,\quad\sigma(w)=\{(1,2),(2,1)\} \end{gather*} -$R$ is symmetric, and $\sigma$ is a witness for $R$ to be an AM-simulation, but $R$ is not a bisimulation in the traditional sense because $(2,1)\in R$, and $2\to 3$, but $(3,1)$ or $(3,2)$ are not in $R$. It is easy to see that it is not an AM-bisimulation as well because we can not define a function that can serve as an evidence for it as $3$ does not appear in any pair in $R$, while it exists in $\alpha(2)$. +$R$ is symmetric, and $\sigma$ is a witness for $R$ to be an AM-simulation, but $R$ is not a bisimulation in the traditional sense because $(2,1)\in R$, and $2\to 3$, but $(3,1)$ or $(3,2)$ are not in $R$. It is easy to see that it is not an AM-bisimulation because we can not define a function that can serve as an evidence for it as $3$ does not appear in any pair in $R$, while it exists in $\alpha(2)$. -This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr\comp(FR)^\dagger\comp\appr=\powf X\times \powf X$, so $R\subseteq( \appr\comp(FR)^\dagger\comp\appr)$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. So, unlike their definition, in the context of the relator-based definitions that only work in $\Set$, this counter-example does not live. +This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr\comp(FR)^\dagger\comp\appr=\powf X\times \powf X$, so $R\subseteq( \alpha\comp\appr\comp(FR)^\dagger\comp\appr\comp\alpha)$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. Worth noting that Hughes and Jacobs give their definition in $\Set$, and they claim that it is easy to generalize to other categories. The mentioned ordering is not liftable. Assuming $h\in\Hom(\nats,\powfi \nats)$, $g\c \nats\to \nats$, and $k\in\Hom(\nats,\powfi \nats)$, and they are defined for every $n$ in $\nats$ as $h(n)=\{2\times n\}$, $g(n)=3\times n$, and $k(n)=\{n\}$, then $|h(n)|=|\powfi g(k(n))|=1$ that means $h\appr\powfi g\comp k$ is satisfied, but there is no $k'$ that $h=\powfi g\comp k$ because we can never have $h(1)=\powfi g\comp k'(1)$, as assuming $k'(1)=\{n\}$, and $n$ must be a natural number, then we should have $2=3\times n$ that is impossible. @@ -2549,8 +2549,17 @@ Indeed, $\sigma^\dagger$ is a witness for $R$ to be an HJ-simulation, but it is \begin{example} In~\autoref{ex:sym-sim-bisim-count} we gave a counter-example with an ordering that gives preorders. Now, we slightly change the ordering so that it gives posets. We define $\appr$ as follows: \begin{gather*} - A\appr B \iff + A\appr B \qquad\iff\qquad A=B \quad or\quad |A|<|B| \end{gather*} + For the defined $\alpha$ and $R$ in~\autoref{ex:sym-sim-bisim-count}, we define $\sigma\c R\to \powfi R$ as follows as a witness for $R$ to be an AM-simulation: + \begin{gather*} + \forall w\in R,\quad\sigma(w)=\{(1,1),(2,1),(3,1)\} + \end{gather*} + Additionally we define $\sigma^\dagger$ as follows as a witness for $R$ to be an HJ-simulation: + \begin{gather*} + \sigma^\dagger(w)=\{(\{1,2,3\},\{1\})\} + \end{gather*} + Although $R$ is not an AM-bisimulation not an HJ-bisimulation. \end{example} \subsection{From Symmetric Simulation To Bisimulation (Aczel-Mendler)}