2.1 Mathematical background

We assume the reader to be familiar with the basic notions of topology (topology, continuous function, compact, locally compact, compactification, $\sigma$ -compact, Hausdorff), Polish spaces, Banach spaces, measure theory ( $\sigma$ -algebra, measurable functions, measure, probability space). For a detailed introduction, we refer the reader to Dudley (Reference Dudley1989); Panangaden (Reference Panangaden2009); Billingsley (Reference Billingsley2008).

2.2 Discrete-time systems

Markov processes are stochastic processes where what “happens next” depends only on the current state and not the full history of how that state was reached. They are used in many fields as statistical models of real-world problems and have therefore been extensively studied.

In some cases, the environment or a user may influence how things will evolve, for instance, by pressing a key on a keyboard. This is modelled by using a countable set Act of actions that determine at each step which stochastic process is used.

We will also assume that an LMP is equipped with a measurable function $obs: S \to2^{AP}$ , where AP denotes a set of atomic propositions and $2^{AP}$ is equipped with the discrete $\sigma$ -algebra. The function obs is useful to isolate areas of the state space. For instance, if the state space is a set of temperatures, there could be atomic propositions “freezing” and “heat wave” isolating, respectively, the subsets of the state space $({-}273, 0)$ and $(30, + \infty)$ .

Two states are bisimilar if there exists a bisimulation that relates them.

There is a greatest bisimulation which corresponds to the equivalence “are bisimilar”. This greatest bisimulation is called “bisimilarity”.

Example 3. Random walk is a very standard example. The most basic version of it takes place on the set of signed integers $\mathbb{Z,}$ and the intuition is that at each step, the process goes either to the left or to the right in an unbiased manner. More formally, the corresponding Markov kernel is the following for all integers n, m:

\[ \tau (n,m) = \begin{cases}\frac{1}{2} \qquad \text{if } m = n-1 \text{ or } n+1,\\[3pt]0 \qquad \text{otherwise.}\end{cases} \]

The state space $\mathbb{Z}$ can be equipped with the function $obs(x) = 0$ if $x = 0$ and $obs(x) = 1$ if $x \neq 0$ . In this case, two states n and m are bisimilar if and only if $|n| = |m|$ .

In Desharnais et al. (Reference Desharnais, Edalat and Panangaden2002), a more categorical approach to bisimulation is also provided.

It is quite easy to see that for a state s and a zigzag f, the states s and f(s) are bisimilar. In Desharnais et al. (Reference Desharnais, Edalat and Panangaden2002), the authors compare bisimulation to spans of zigzags and showed that the two notions coincided if the state space is analytic.

It was shown that the notions induced respectively by spans of zigzags and by cospans of zigzags were equivalent on analytic Borel spaces Doberkat (Reference Doberkat2005) and on separable universally measurable spaces (isomorphic to a universally measurable subset of a separable completely metrisable space) Pachl and Sánchez Terraf (2020).

3.1 Definition of Feller-Dynkin processes

We will quickly review the theory of continuous-time processes on continuous state space; much of this material is adapted from Rogers andWilliams (2000), and we use their notations. Another useful source is Bobrowski (Reference Bobrowski2005).

This concept is used to capture the idea that at time t what is “known” or “observed” about the process is encoded in the sub- $\sigma$ -algebra $\mathcal{F}_t$ .

Define the filtration $(\mathcal{G}_t)_{t \geq 0}$ as follows: for every $t \geq 0$ , the $\sigma$ -algebra $\mathcal{G}_t$ is the one generated by all the random variables $\{Y_s| s\leq t\}$ , that is

\[ \mathcal{G}_t = \sigma\! \left( \{ Y_s ^{-1}(A) ~|~ s \leq t \text{ and } A \in \Sigma_X \} \right) \]

This filtration $(\mathcal{G}_t)_{t\geq 0}$ is also referred to as the natural filtration associated with the stochastic process $(Y_t)_{t \geq 0}$ . A stochastic process is always adapted to its natural filtration.

Let E be a locally compact, Hausdorff space with countable base. These standard topological hypotheses allow us to perform the one-point compactification of the set E by adding the absorbing state $\partial$ . We write $E_\partial$ for the corresponding set: $E_\partial = E \uplus \{ \partial\}$ . We also equip the set E with its Borel $\sigma$ -algebra $\mathcal{B}(E)$ that we will denote $\mathcal{E}$ . The previous topological hypotheses also imply that E is $\sigma$ -compact and Polish.

Definition 7. A semigroup of operators on any Banach space X is a family of linear continuous (bounded) operators $\mathcal{P}_t: X \to X$ indexed by $t\in\mathbb{R}_{\geq 0}$ such that

\[ \forall s,t \geq 0, \mathcal{P}_s \circ \mathcal{P}_t = \mathcal{P}_{s+t}\]

and

\[ \mathcal{P}_0 = I \qquad \text{(the identity)}. \]

The first equation above is called the semigroup property. The operators in a semigroup are continuous operators. Moreover, there is a useful continuity property with respect to time t of the semigroup as a whole.

Definition 8. For X a Banach space, we say that a semigroup $\mathcal{P}_t:X\to X$ is strongly continuous if

\[ \forall x\in X, \lim_{t\downarrow 0}\| \mathcal{P}_t x - x \| \to 0. \]

What the semigroup property expresses is that we do not need to understand the past (what happens before time t) in order to compute the future (what happens after some additional time s, so at time $t+s$ ) as long as we know the present (at time t). However, it is simple to have such a semigroup describe a free fall as long as we are working on $\mathbb{R}^2 \to \mathbb{R}$ , as physics teaches us. The first coordinate represents the position and the second the velocity. In this case, we have that

\[ (\mathcal{P}_t f) (x,v) = f(5 t^2 + vt + x, 10t + v) \]

We say that a continuous real-valued function f on E “vanishes at infinity” if for every $\varepsilon > 0$ there is a compact subset $K \subseteq E$ such that for every $x\in E\setminus K$ , we have $|f(x)| \leq \varepsilon$ . To give an intuition, if E is the real line, this means that $\lim_{x \to \pm \infty} f(x) = 0$ . The space $C_0(E)$ of continuous real-valued functions that vanish at infinity is a Banach space with the $\sup$ norm.

Definition 10. A Feller-Dynkin (FD) semigroup is a strongly continuous semigroup $(\hat{P}_t)_{t \geq 0}$ of linear operators on $C_0(E)$ satisfying the additional condition:

\[\forall t \geq 0 ~~~ \forall f \in C_0(E) \text{, if }~~ 0 \leq f \leq 1 \text{, then }~~ 0 \leq \hat{P}_t f \leq 1\]

Under these conditions, strong continuity is equivalent to the apparently weaker condition (see Lemma III. 6.7 in Rogers andWilliams (2000) for the proof):

\[ \forall f \in C_0(E)~~ \forall x\in E, ~~ \lim_{t\downarrow 0} (\hat{P}_t f) (x) = f(x)\]

The following important proposition relates these FD-semigroups with Markov kernels which allows one to see the connection with more familiar probabilistic transition systems. We provide a detailed proof of the following proposition in Appendix A.

Proposition 11. Given an FD-semigroup $(\hat{P}_t)_{t \geq 0}$ on $C_0(E)$ , it is possible to define a unique family of sub-Markov kernels $(P_t)_{t \geq 0} : E \times \mathcal{E} \to [0,1]$ such that for all $t \geq 0$ and $f \in C_0(E)$ ,

\[ \hat{P}_t f(x) = \int f(y) P_t(x, dy). \]

A very important ingredient in the theory is the space of trajectories of a FD-process (FD-semigroup) as a probability space. This space does not appear explicitly in the study of labelled Markov processes but one does see it in the study of continuous-time Markov chains and jump processes.

Definition 12. A function $\omega : [0,\infty)\to E_{\partial}$ is called cadlagFootnote ¹ if for every $t \geq0$ ,

\[ \lim_{s > t, s \to t} \omega(s) = \omega (t) ~~\text{ and } ~~ \omega(t{-}) := \lim_{s < t, s \to t} \omega(s) ~\text{ exists} \]

We define a trajectory $\omega$ on $E_{\partial}$ to be a cadlag function $[0,\infty)\to E_{\partial}$ such that if either $\omega(t{-}) := \lim_{s < t, s \to t} \omega(s) = \partial$ or $\omega(t)=\partial$ then $\forall u\geq t, \omega(u) = \partial$ . We can extend $\omega$ to a map from $[0,\infty]$ to $E_{\partial}$ by setting $\omega(\infty)=\partial$ .

As an intuition, a cadlag function $\omega$ is an “almost continuous” function with some jumping in a reasonable fashion.

The intuition behind the additional condition of a trajectory is that once a trajectory has reached the terminal state $\partial$ , it stays in that state, and similarly if a trajectory “should” have reached the terminal state $\partial$ , then there cannot be a jump to avoid $\partial$ and once it is in $\partial$ , it stays there.

It is possible to associate with such an FD-semigroup a canonical FD-process. Let $\Omega$ be the set of all trajectories $\omega : [0, \infty) \to E_\partial$ .

Definition 13. The canonical FD-process associated with the FD-semigroup $(\hat{P}_t)_{t \geq 0}$ is

\[(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \geq 0}, (X_t)_{0 \leq t \leq \infty}, (\mathbb{P}^x)_{x \in E_\partial})\]

where

• the random variable $X_t: \Omega \to E_\partial$ is defined as $X_t(\omega) = \omega (t)$

• $\mathcal{G} = \sigma (X_s ~|~ 0 \leq s < \infty)$ ,Footnote ² $\mathcal{G}_t = \sigma (X_s ~|~ 0 \leq s \leq t)$

• given any probability measure $\mu$ on $E_\partial$ , by the Daniell-Kolmogorov theorem, there exists a unique probability measure $\mathbb{P}^\mu$ on $(\Omega, \mathcal{G})$ such that for all $n \in \mathbb{N}, 0 \leq t_1 \leq t_2 \leq ... \leq t_n$ and $x_0, x_1, ..., x_n$ in $E_\partial$ ,

  \[ \mathbb{P}^\mu (X_0 \in dx_0, X_{t_1} \in dx_1, ..., X_{t_n} \in dx_n) = \mu (dx_0) P_{t_1}^{+\partial}(x_0, dx_1)...P_{t_n - t_{n-1}}^{+\partial}(x_{n-1}, dx_n)\]
  Footnote ³ where $P_t^{+\partial}$ is the Markov kernel obtained by extending $P_t$ to $E_\partial$ by $P_t^{+\partial} (x, \{ \partial \}) = 1 - P_t (x, E)$ and $P_t^{+ \partial} (\partial, \{ \partial \}) = 1$ . We set $\mathbb{P}^x = \mathbb{P}^{\delta_x}$ .

The distribution $\mathbb{P}^x$ is a measure on the space of trajectories for a system started at the point x.

Here, the Markov kernels correspond to the family of subkernels $(P_t)_{t \geq 0}$ and the step-indexed family of random variables corresponds to the time-indexed family $(X_t)_{t \geq 0}$ .

We have constructed here one process which we called the canonical FD-process. Note that there could be other tuples

\[(\Omega', \mathcal{G}', (\mathcal{G}'_t)_{t \geq 0}, (Y_t)_{0 \leq t \leq \infty}, (\mathbb{Q}^x)_{x \in E_\partial})\]

satisfying the same conditions except that the space $\Omega'$ would not be the space of trajectories.

In order to bring the FD-processes more in line with the kind of transition systems that have hitherto been studied in the computer science literature, we introduce a countable set of atomic propositions AP and such an FD-process is equipped with a measurable function $obs : E \to 2^{AP}$ . This function is extended to a function $obs : E_\partial \to 2^{AP} \uplus \{ \partial \}$ by setting $obs (\partial) = \partial$ .

We further request that the function obs satisfies the following hypothesis: for each atomic proposition A, the atomic proposition partitions the state space into two subsets: one where A is satisfied ( $obs^{-1} (S_A)$ where $S_A = \{ a: AP \to \{ 0,1\} ~|~ a(A) = 1 \}$ ) and another one where it is not satisfied ( $obs^{-1} (S_A)$ where $S_A = \{ a: AP \to \{ 0,1\} ~|~ a(A) = 0 \}$ ). We assume that one of these spaces is open (and the other is closed). Note that it can be that for one atomic proposition $A_0$ , the set where it is satisfied is open and for a different atomic proposition $A_1$ , the set where it is satisfied is closed. This hypothesis will be useful in Section 6.

3.2 A classic example of FD-process: Brownian motion

Brownian motion is a stochastic process first introduced to describe the irregular motion of a particle being buffeted by invisible molecules. Now its range of applicability extends far beyond its initial application Karatzas and Shreve (Reference Karatzas and Shreve2012): it is used in finance, in physics, in biology or as a way to model noise to mention only a few domains.

We refer the reader to Karatzas and Shreve (Reference Karatzas and Shreve2012) for the complete definition. A standard one-dimensional Brownian motion is a Markov process on the real line $\mathbb{R}$ that is invariant under reflexion around any real value and under translation by any real value.

The following fundamental formula is useful in computations: If the process is at x at time 0, then at time t the probability that it is in the (measurable) set D is given by

\[ P_t(x,D) = \int_{y\in D} \frac{1}{\sqrt{2\pi t}} \exp\left({-}\frac{(x-y)^2}{2t}\right)\mathrm{d}y. \]

The associated FD-semigroup is the following: for $f \in C_0(\mathbb{R})$ and $x \in \mathbb{R}$ ,

\[ \hat{P}_t (f) (x) = \int_{y} \frac{f(y)}{\sqrt{2\pi t}} \exp\left({-}\frac{(x-y)^2}{2t}\right)\mathrm{d}y. \]

There are many variants of Brownian motions that we will use and describe in detail in further examples. For instance, a drift may be added. Another classic variant is when “walls” are added: when the process hits a wall, it can either vanish (boundary with absorption) or bounce back (boundary with reflection). These correspond to the variants of random walk that we discussed in Example 3.

3.3 Hitting times and stopping times

We will use the notions of hitting times and stopping times later. We will follow the definitions of Karatzas and Shreve (Reference Karatzas and Shreve2012).

Throughout this section, we assume that $(X_t)_{t \geq 0}$ is a stochastic process on $(\Omega, \mathcal{F})$ such that $(X_t)_{t \geq 0}$ takes values in a state space $(E, \mathcal{B}(E))$ , has right-continuous paths (for every time t, $\omega(t) = \lim_{s \to t, s > t} \omega (s)$ ) and is adapted to a filtration $(\mathcal{F}_t)_{t \geq 0}$ .

Definition 16. Given a measurable set $C \in \mathcal{B}(E)$ , the hitting time is the random time

\[ T_C (\omega) = \inf \{ t \geq 0 ~|~ X_t(\omega) \in C \}. \]

Intuitively, $T_C(\omega)$ corresponds to the first time when the trajectory $\omega$ “touches” the set C. The verb “touches” in previous sentence is important: this time is obtained as an infimum, not a minimum.

Recall that the intuition behind the filtration $(\mathcal{F}_t)_{t \in \mathbb{R}_{\geq 0}}$ is that the information about the process up to time t is stored in the $\sigma$ -algebra $\mathcal{F}_t$ . So intuitively, the random time T is a stopping time (resp. an optional time) if and only if there is enough information gathered at time t to decide if the event described by the random T is less ( $\leq$ and $<,$ respectively) than t. We provide examples and counterexamples to this definition in the remaining of this section.

A stopping time is also an optional time but the converse may not be true. The next lemma may provide some additional intuition and some examples behind those notions.

Lemma 18. If the set C is open, $T_C$ is an optional time.

If the set C is closed and the paths of the process X are continuous, then $T_C$ is a stopping time.

Let us give an example of a random time that is neither a stopping time nor an optional time. Given a continuous trajectory $\omega : [0, + \infty) \to \mathbb{R}$ , we define

\[T_{\max} (\omega) = \inf \{ t ~|~ \omega (t) = \max \omega \}\]

So $T_{\max}$ is the time that $\omega$ reaches its maximum value. In order to know if $T_{\max}(\omega) < t$ , one would need to know the maximum value attained by the trajectory $\omega$ . More generally, any such random time that requires to know about the future behaviour of a trajectory is neither a stopping time nor an optional time.

4.1 Naive definition

A naive extension of the discrete-time definition of bisimulation where the induction condition would only require that steps of time t (for every such $t \geq0$ ) preserve the relation does not work in continuous time.

Indeed, consider Brownian motion on the real line with a single atomic proposition such that $obs(x) = 0$ if and only if $x = 0$ . The equivalence $x ~R~y$ if and only if $x = y = 0$ or $xy \neq 0$ would satisfy the hypothesis of this naive extension of bisimulation. This would mean that any two non-zero states, regardless of their distance to 0 would be considered equivalent. This does not extend the intuition that we should get from random walk (see example 3) nicely into continuous time.

As it turns out, the problem lies with considering single time-steps since for every state $z \neq 0$ and every time $t \geq 0$ , $P_t(z, \{ 0\}) = 0$ . Had we considered instead probabilities of reaching the state 0 over an interval of time ( $\mathbb{P}^z (T_0 < t)$ for instance), we would not have had the same equivalence.

This is why we deal with trajectories instead of steps.

4.2 Closedness for sets of trajectories

Throughout the remaining of Section 4, we fix an FD-process as in Section 3.1.

Since the conditions for being a behavioural equivalence have to be stated for trajectories, we have to extend the notion of R-closedness of a set of states to a set of trajectories. This is what is done in the following definition.

That last condition deals with the question “how often should someone be allowed to observe the system?” and answers that an outside observer should at least be allowed to probe the process once every unit of time.

4.3 Bisimulation

There are two conditions (initiation and induction conditions) that can be modified to account for trajectories. Depending on which one is adapted, we get either temporal equivalence or bisimulation. In Chen et al. (Reference Chen, Clerc and Panangaden2019), we introduced only the notion of “bisimulation” where the induction condition is adapted to trajectories.

.

4.4 Temporal equivalence

We later came up with other notions of behavioural equivalences, and in [Chen et al.(Reference Chen, Clerc and Panangaden2020)], we proposed three other notions of behavioural equivalences. It is quite likely that there are other interesting notions to be studied in future works.

(initiation) for all time-obs-closed sets B, $\mathbb{P}^x (B) = \mathbb{P}^y (B)$ , and

(induction) for all measurable R-closed sets C, for all times t, $P_t(x, C) = P_t(y, C)$ .

4.5 Trace equivalence

Another well-known concept is that of trace equivalence which corresponds to the initiation condition of temporal equivalence. Temporal equivalence can thus be viewed as trace equivalence which additionally accounts for step-like branching.

We will later see in details how it relates to the standard notion of trace equivalence for discrete-time processes (see Section 4.8).

4.6 Relation between these equivalences

We are going to show that the weakest equivalence is trace equivalence, and the strongest is bisimulation as hinted.

Consider a time-obs-closed set B. Then, it is also time-R-closed. Using the induction condition of bisimulation, we get that $\mathbb{P}^x (B) = \mathbb{P}^y (B)$ .

Consider a measurable R-closed set C and a time t. The set $X_t^{-1}(C) = \{ \omega ~|~ \omega (t) \in C \}$ is measurable and time-R-closed. We can then apply the induction condition, and we get

\[ P_t(x, C) = \mathbb{P}^x(X_t^{-1}(C)) = \mathbb{P}^y(X_t^{-1}(C)) = P_t(y, C) \]

This concludes the proof that R is a temporal equivalence.

The second part of the lemma follows directly from the initiation condition of a temporal equivalence: this is precisely trace equivalence.

We provide in Section 5.1.2 an example where the greatest temporal equivalence is strictly greater than trace equivalence. It is still an open question as to whether bisimulation and temporal equivalence are the same notions or if they coincide only for a class of processes. We refer the reader to Section 5 for examples of these equivalences.

4.7 Symmetries of systems

An interesting notion that appears when looking at examples is that of symmetries of a system. It was the third notion introduced in Chen et al.(Reference Chen, Clerc and Panangaden2020).

Given a function $h : E_\partial \to E_\partial$ , we define $h_*: \Omega \to \Omega$ by $h_*(\omega) = h \circ \omega$ .

One of the requirements for being a group of symmetries is to be closed under inverse and composition. This condition is useful for getting an equivalence on the state space; however, it is usually easier (if possible) to view a group of symmetries as generated by a set of homeomorphisms.

Then the set $\mathcal{H}$ is a group of symmetries.

The proof of Theorem 34 requires two additional lemmas. We will omit the proof of the first one since it is straightforward.

To prove the converse implication, consider $\omega \in B$ and define $\omega'$ as $\omega' = (h^{-1})_* (\omega)$ . Note that for all times t, $h(\omega' (t)) = \omega (t)$ , that is $\omega(t) ~R_\mathcal{H}~ \omega'(t)$ .

Since B is time- $R_\mathcal{H}$ -closed, $\omega' \in B$ . We have defined $\omega' = (h^{-1})_* (\omega) = h^{-1} \circ \omega$ . A direct consequence is that $\omega = h \circ \omega' = h_* (\omega')$ , and therefore, $\omega \in h_*(B)$ .

Proof of Theorem 34. Consider two equivalent states $x ~R_\mathcal{H}~y$ , that is there exists $h \in \mathcal{H}$ such that $h(x) = y$ .

First, $obs(x) = obs \circ h (x) = obs (y)$ .

Second, let us consider a measurable, time- $R_\mathcal{H}$ -closed B. Using Lemma 36, we know that for every $f \in \mathcal{H}$ , $f_*(B) = B$ , and hence, $\mathbb{P}^x(B) = \mathbb{P}^{h(x)}(B) = \mathbb{P}^y(B)$ , which concludes the proof.

4.8 Comparison to discrete time

Our work aims at extending the notion of bisimulation from discrete time to continuous time. It is possible to view a discrete-time process as a continuous-time one as explained below. It is important to check that our notions encompass the pre-existing notion of bisimulation in discrete time.

It is common in discrete time to consider several actions. Everything that was exposed for continuous time can easily be adapted to accommodate several actions. However, we will not mention actions in this section for the sake of readability.

Consider an LMP $(S, \Sigma, \tau, obs)$ where $\Sigma $ is the Borel-algebra generated by a topology on S. We also assume that this LMP has at most finitely many atomic propositions. We can always view it as an FD-process where transitions happen at every unit of time. Since the corresponding continuous-time process has to satisfy the Markov property, it cannot keep track of how long it has spent in a state of the LMP. This is why we need to include a “timer” into states. A state in the corresponding FD-process is thus a pair of a state in S and a time explaining how long it has been since the last transition. That time is in [0,1), the right open bound is in order for trajectories to be cadlag.

Formally, the state space of the FD-process is $(E, \mathcal{E})$ where the space is defined as $ E = S \times [0, 1)$ and is equipped with the product topology and the corresponding Borel $\sigma$ -algebra $\mathcal{E} = \Sigma \otimes \mathcal{B}([0, 1))$ . The corresponding kernel is defined for all $x \in S$ and $C \in \mathcal{E}$ , $t \geq 0$ and $s \in [0,1)$ as $P_t ((x,s), C) = \tau_{\lfloor t + s\rfloor} (x, C')$ where $C' = \{ z ~|~ (z,t+s - \lfloor t+s \rfloor) \in C\}$ and for $k \geq 1$ ,

\[\tau_0 (x, C') = \unicode{x1d7d9} _{C'}(x), \qquad\tau_1(x,C') = \tau (x,C') \quad \text{and} \quad\tau_{k + 1}(x,C') = \int _{y \in X} \tau(x,dy) \tau_k(y ,C')\]

We also define $obs(x,s) = obs(x)$ .

This gives us a way to view an LMP as a continuous-time process and thus to compare the definition of bisimulation on the original LMP to the behavioural equivalences for the corresponding continuous-time process. In order to make clear when we talk about discrete-time bisimulation and in order to avoid confusion for the reader, we will write “DT-bisimulation” for bisimulation in discrete time as was defined in Section 2.2.

Lemma 38. If two states x and y of an LMP $(S, \Sigma, \tau, obs)$ are DT-bisimilar, then for all $n \geq 1$ , for all measurable R-closed (where R is the greatest DT-bisimulation) sets $A_1, ..., A_n$ ,

\begin{align*}& \int_{x_1 \in A_1} ... \int_{x_n \in A_n} \tau (x, dx_1) \tau (x_1, dx_2)... \tau(x_{n-1}, dx_n)\\& \qquad = \int_{x_1 \in A_1} ... \int_{x_n \in A_n} \tau (y, dx_1) \tau (x_1, dx_2)... \tau(x_{n-1}, dx_n)\end{align*}

Proof. Denote $\pi_R : S \to S/R$ the corresponding quotient. We equip the quotient space $S/R$ with the largest $\sigma$ -algebra which makes the map $\pi_R$ measurable. We can also define the function

\[ \overline{\tau} (\pi_R(x), A) = \tau (x, \pi_R^{-1}(A)) \]

Note that the choice of x (within an R-class) does not change the right term since R is a DT-bisimulation and $\pi_R^{-1}(A))$ is an R-closed set: you can replace x by x’ as long as $ x ~R~ x'$ . A sequence of changes of variables yields:

\begin{align*}& \int_{x_1 \in A_1} ... \int_{x_n \in A_n} \tau (x, dx_1) \tau (x_1, dx_2)... \tau(x_{n-1}, dx_n)\\& = \int_{x_1 \in A_1} ... \int_{x_{n-1} \in A_{n-1}} \int_{y_n \in A_n/R} \tau (x, dx_1) \tau (x_1, dx_2)... \tau(x_{n-2}, dx_{n-1}) \tau(x_{n-1}, \pi_R^{-1}(dy_n))\\& = \int_{x_1 \in A_1} ... \int_{x_{n-1} \in A_{n-1}} \int_{y_n \in A_n/R} \tau (x, dx_1) \tau (x_1, dx_2)... \tau(x_{n-2}, dx_{n-1}) \overline{\tau}(\pi_R(x_{n-1}), dy_n)\\& = \int_{x_1 \in A_1} ... \int_{x_{n-2} \in A_{n-2}} \int_{y_{n-1} \in A_{n-1}/R} \int_{y_n \in A_n/R} \tau (x, dx_1) \tau (x_1, dx_2)... \tau(x_{n-2}, \pi_R^{-1}(dy_{n-1})) \overline{\tau}(y_{n-1}, dy_n)\\& = \int_{x_1 \in A_1} ... \int_{x_{n-2} \in A_{n-2}} \int_{y_{n-1} \in A_{n-1}/R} \int_{y_n \in A_n/R} \tau (x, dx_1) \tau (x_1, dx_2)... \overline{\tau}(\pi_R(x_{n-2}), dy_{n-1}) \overline{\tau}(y_{n-1}, dy_n)\\& = \int_{y_1 \in A_1/R} ... \int_{y_n \in A_n/R} \tau (x, \pi_R^{-1}(dy_1)) \overline{\tau} (y_1, dy_2)... \overline{\tau} (y_{n-1}, dy_n)\end{align*}

And since the two measures $\tau (x, \pi_R^{-1}({\cdot}))$ and $\tau (y, \pi_R^{-1}({\cdot}))$ are equal as R is a DT-bisimulation, this concludes the proof.

It is possible to define the notion of trajectories in the LMP and that of trace equivalence just as we did in the case of FD-processes. A trajectory is a function $\omega: \mathbb{N} \to S \uplus \{ \partial\}$ such that if $\omega(n) = \partial$ , then for every $k \geq n$ , $\omega(k) = \partial$ . Similarly to what was done in Section 3.1, for a state x we can define the probability distribution $\mathbb{P}^x$ on the set of trajectories that extends the finite-time distributions using the Daniell-Kolmogorov theorem, and this allows us to define trace equivalence: two states x and y are trace equivalent if for every time-obs-closed sets B, $\mathbb{P}^x(B) = \mathbb{P}^y(B)$ .

An important remark has to be made here. Traditionally, trace equivalence for discrete time is much simpler than our definition of trace equivalence: two states x and y are DT-trace equivalent if for every $C_0, ..., C_n$ measurable sets such that for every i, for every z, z’ such that $obs(z) = obs(z')$ , $z \in C_i$ if and only if $z' \in C_i$ ,

\[\mathbb{P}^x (X_0 \in C_0, ..., X_n \in C_n) = \mathbb{P}^y (X_0 \in C_0, ..., X_n \in C_n)\]

Lemma 38 shows that any two DT-bisimilar states are DT-trace equivalent. Our definition of trace equivalence is stronger than this: it requires that $\mathbb{P}^x$ and $\mathbb{P}^y$ agree for all time-obs-closed sets. The sets $\{ X_0 \in C_0, ..., X_n \in C_n\}$ are only examples of such time-obs-closed sets.

Recall the definition of $\Pi$ from the definition of time-obs-closed (see definition 21): for a trajectory $\omega: \mathbb{N} \to S \uplus \{ \partial\}$ , $\Pi (\omega)(n) = obs(\omega(n))$ . Now note that $B = \Pi^{-1} (\Pi (B))$ . The first inclusion ( $B \subset \Pi^{-1} (\Pi (B))$ ) always holds. For the converse direction, consider $\omega \in \Pi^{-1} (\Pi (B))$ , that is there exists $\omega' \in B$ such that $\Pi(\omega) = \Pi(\omega')$ . This means that for every n, $obs(\omega(n)) = \Pi(\omega)(n) = \Pi(\omega')(n) = obs(\omega'(n))$ and using the fact that B is time-obs-closed, we obtain that $\omega \in B$ which proves the desired equality.

Consider a finite cylinder on $\mathbb{N} \to 2^{AP}$ , that is a set of the form:

where n is an integer and for every i, $A_i$ is a subset of $2^{AP}$ . Since there are only finitely many atomic propositions, the set $2^{AP}$ is finite and the sets $A_i$ are measurable. Lemma 38 shows that for any finite cylinder C on $\mathbb{N} \to 2^{AP}$ ,

\[ \mathbb{P}^x (\Pi^{-1}(C)) = \mathbb{P}^y (\Pi^{-1}(C)) \]

The finite cylinders form a generating $\pi$ -system of the $\sigma$ -algebra on $\mathbb{N} \to 2^{AP}$ which means that for every measurable subset A of $\mathbb{N} \to 2^{AP}$ ,

\[ \mathbb{P}^x (\Pi^{-1}(A)) = \mathbb{P}^y (\Pi^{-1}(A)) \]

We can now look at our set B.

\[ \mathbb{P}^x(B) = \mathbb{P}^x(\Pi^{-1} (\Pi (B))) = \mathbb{P}^y(\Pi^{-1} (\Pi (B))) = \mathbb{P}^y(B) \]

The first and third equations hold since $B = \Pi^{-1} (\Pi (B)),$ and the second equation holds since $\Pi (B)$ is measurable and for every measurable subset A of $\mathbb{N} \to 2^{AP}$ ,

\[ \mathbb{P}^x (\Pi^{-1}(A)) = \mathbb{P}^y (\Pi^{-1}(A)) \]

Proposition 41. If the equivalence R is a DT-bisimulation, then the relation R’ defined as

\[ R' = \{ \left( (x,s), (y,s) \right) ~|~ s \in [0,1), x~R~y \} \]

is a temporal equivalence.

The set C’ is R-closed. Indeed, consider two states $z \in C'$ and $z' \in S$ such that $z ~R~ z'$ . These conditions imply that $(z,s') \in C$ and $(z,s') ~R'~ (z', s')$ . Since the set C is R’-closed, $(z', s') \in C$ and hence by definition of the set C’, $z' \in C'$ .

Since $(x,s) ~R'~ (y,s)$ , we also have that $x~R~y$ . By lemma 38, we have that $\tau_{\lfloor t + s \rfloor} (x, C') = \tau_{\lfloor t + s \rfloor} (y, C')$ . This allows us to conclude that $P_t((x,s),C) = P_t(y,s), C)$ .

The initiation condition (trace equivalence) is a direct consequence of Lemma 39.

Proposition 42. If the equivalence R is a temporal equivalence, then the relation R’ defined as the transitive closure of the relation

\[ \{ (x,y) ~|~ \exists t, t' \in [0,1) \text{ such that } (x,t) ~R~(y,t') \} \]

is a DT-bisimulation.

Let us consider $x ~R'~ y$ , that is there exists $(x_i)_{i = 1, ..., n}$ , $(t_i)_{i = 1, ..., n-1}$ and $(t'_i)_{i = 1, ..., n-1}$ such that $x = x_1$ , $y = x_n$ and for every $1 \leq i \leq n-1$ , $(x_i, t_i) ~R~ (x_{i+1}, t'_i)$ .

The fact that $obs(x) = obs(y)$ is a direct consequence of the initiation condition of a temporal equivalence.

Now, consider a measurable (in $\Sigma$ ) and R’-closed set C’. Define the set $C = \{ (z,s) ~|~ z \in C', s \in [0,1)\}$ . It is measurable (in $\mathcal{E}$ ) and R-closed. Since R is a temporal equivalence, for every i, $P_1((x_i, t_i), C) = P_1((x_{i+1}, t'_i), C)$ for every $i \leq n$ . Additionally, note that for every $z \in E$ and every $s \in [0,1)$ , $P_1((z,s), C) = \tau (z, C')$ . This proves that $\tau(x, C') = \tau (y, C')$ .

These results can be summed up in the following theorem relating temporal equivalence and DT-bisimulation.

5.1 Basic examples

5.1.1 Deterministic drift

Consider a deterministic drift on the real line $\mathbb{R}$ with constant speed $v \in \mathbb{R}_{>0}$ and a single atomic proposition. We study two cases: the first one with 0 as the only distinguished point and the second one with all the integers distinguished from the other points. In both cases, trace equivalence and greatest bisimulation are the same.

where we define for every $x \in \mathbb{R}$ and every $k \in \mathbb{Z}$ , $t_k (x) = x+k$ and for $x,y \in \mathbb{R}_{>0}$ the function $f_{x,y} : \mathbb{R} \to \mathbb{R}$ by

\[ f_{x,y}(z) = \begin{cases}z ~~ \text{ if } z \leq 0\\[3pt]\frac{y}{x}z ~~ \text{ if } 0 \leq z \leq x\\[3pt]z - x + y ~~ \text{ if } z>x\end{cases} \]

These functions are essentially identity functions with a rescaled portion in the middle to connect positive reals.

5.1.2 Fork

The following example is of particular interest because it shows that temporal equivalence and trace equivalence are not equivalent notions. It emphasises the importance of the induction conditions in the definitions of temporal equivalence and of bisimulation. It is an extension of a standard example in discrete time (see Section 4.1, p.86 of Milner (Reference Milner1989)) to our continuous-time setting. The process is a deterministic drift at constant speed with a single probabilistic fork (the process then goes to either branch with equal probability). We compare the case where the fork is at the start with the case where the fork happens later. There are two atomic propositions P and Q which enable the process to tell the difference between the ends of each fork.

One can find in Clerc (Reference Clerc2021) a detailed and tedious description of the process and of the proofs. We will focus here on intuition.

A state is a pair (x, b) where x represents the time to be reached from the “origins” $x_0$ and $y_0$ and b represents the different “branches” of the state space. The state space with its two atomic propositions P and Q is the following:

This process is a deterministic drift at constant speed towards the right. When the process encounters a fork (in $x_0$ or in $y_1$ ), it randomly picks between the upper and lower branch and then continues as a deterministic drift to the right.

Lemma 44. Given $s \neq t \in [0,100]$ and , the states (s,i) and (t, j) cannot be trace equivalent.

Proposition 45. Two states $x \neq y$ are trace equivalent if and only if

• $\{ x,y\} = \{ x_0, y_0\}$ , or

• $\{ x,y\} = \{ (t,2), (t,5)\}$ for some $t \in (95, 100]$ , or

• $\{ x,y\} = \{ (t,3), (t,6)\}$ for some $t \in (95, 100]$ .

Proof. First, let us show that $x_0$ and $y_0$ are trace equivalent. There are two possible trajectories from $x_0$ : $\omega_2$ and $\omega_3$ going up (branch 2) and down (branch 3), respectively. Similarly, there are two trajectories from $y_0$ : $\omega_5$ and $\omega_6$ going first though branch 4 (from $y_0$ to $y_1)$ and then going respectively up (branch 5) and down (branch 6). If a set B is time-obs-closed, $\omega_2$ (resp. $\omega_3$ ) is in the set B if and only if $\omega_5$ (resp. $\omega_6$ ) is in the set B. Furthermore,

\[ \mathbb{P}^{x_0} (B) = \frac{1}{2} \delta_B (\omega_2) + \frac{1}{2} \delta_B (\omega_3) ~\text{and}~ \mathbb{P}^{y_0} (B) = \frac{1}{2} \delta_B (\omega_5) + \frac{1}{2} \delta_B (\omega_6)\]

This shows that $x_0$ and $y_0$ are trace equivalent.

It is easy to see that two states satisfying condition 2 or 3 are trace equivalent.

In order to show that no other non-equal states can be trace equivalent, note that lemma 44 shows that it is enough to show that (t, 2), (t,3) and (t,4) are not trace equivalent. It is enough to consider the set of trajectories seeing the atomic proposition P after letting time go for 100:

\[ B_P = \{ \omega ~|~ obs(\omega(100)) = (1,0)\}. \]

This set is time-obs-closed, but we have that $\mathbb{P}^{(t,2)} (B_P) = 1$ , $\mathbb{P}^{(t,3)}(B_P) = 0$ and $\mathbb{P}^{(t,4)} (B_P) = 1/2$

Proposition 46. The states $x_0$ and $y_0$ are not temporally equivalent (and hence they are not bisimilar either).

Proof. First, the states $x_1, x_2$ and $y_1$ cannot be temporally equivalent. Indeed, consider the set of trajectories $B = \{ \omega ~|~ obs(\omega (5)) = (1,0)\}$ . This set is time-obs-closed but we have that $\mathbb{P}^{x_1} (B) = 1$ , $\mathbb{P}^{x_2}(B) = 0$ and $\mathbb{P}^{y_1} (B) = 1/2$ .

Using Lemma 44, this shows that the states $x_1$ , $x_2$ and $y_1$ can only be temporally equivalent to themselves.

Third, we can consider the set $\{ x_1\}$ . We have just shown that for any temporal equivalence R, the set $\{ x_1\}$ is R-closed. But then, $P_t(x_0, \{ x_1\}) = 1/2$ whereas $P_t(y_0, \{ x_1\}) = 0$ .

This bisimulation is generated by the group of symmetries $\{ f,g,id\}$ where f permutes branches 2 and 5 and g similarly permutes branches 3 and 6.

5.2 Examples based on Brownian motion

All these examples are variants of Brownian motion with either a single or no (first two cases of absorbing wall) atomic propositions.

5.2.1 Standard Brownian motion

where for every $x \in \mathbb{R}$ , $s(x) = -x$ , $s_k(x) = k-x$ and $t_k (x) = x+k$ for every $k \in \mathbb{Z}$ .

With all integers distinguished: Let us consider the case when there is a single atomic proposition and $obs (x) = 1$ if and only if $x \in \mathbb{Z}$ .

Proposition 47. The set $\{ id, t_k, s_k ~|~ k \in \mathbb{Z}\}$ is a group of symmetries.

Proof. For every $k \in \mathbb{Z}$ , the functions $s_k$ and $t_k$ are indeed homeomorphisms and $obs \circ t_k = obs$ for every $k \in \mathbb{Z}$ (and similarly for $s_k$ ).

Consider a state x and a measurable set B such that $(s_k)_*(B) = B$ and $(t_k)_*(B) = B$ for every $k \in \mathbb{Z}$ . Brownian motion is invariant under translation which means that

\[ \mathbb{P}^{x}(B) = \mathbb{P}^{t_k(x)} ((t_k)_*(B))\]

Using the additional constraint that $(t_k)_*(B) = B$ , we obtain the desired result. For $s_k$ , note that $s_k = t_k \circ s$ . Brownian motion is invariant under symmetry which means that

\[ \mathbb{P}^{x}(B) = \mathbb{P}^{s(x)} (s_*(B)) = \mathbb{P}^{s(x)}(B) \]

and therefore

\[ \mathbb{P}^{x}(B) = \mathbb{P}^{s_k(x)}(B) \]

Proposition 48. Two states x and y are trace equivalent if and only if $x - \lfloor x \rfloor = y - \lfloor y \rfloor$ or $\lceil y \rceil - y$ .

Proof. The equivalence generated by the group of symmetries $\{ id, t_k, s_k ~|~ k \in \mathbb{Z}\}$ is

\[ \{ (x,y ~|~ x - \lfloor x \rfloor = y - \lfloor y \rfloor \text{ or } \lceil y \rceil - y \} \]

And therefore any two states x and y such that $x - \lfloor x \rfloor = y - \lfloor y \rfloor$ or $\lceil y \rceil - y$ are trace equivalent.

Let us show that no other states are trace equivalent. First note that if $x \in \mathbb{Z}$ and $y \notin \mathbb{Z}$ , they cannot be trace equivalent. Let us now consider two trace equivalent states x and y that are not in $\mathbb{Z}$ . Consider the sets $B_t = \{ \omega ~|~ \exists s \in [0,t) ~ \omega (s) \in \mathbb{Z}\}$ . These sets are time-obs-closed:

• If $\omega \in B_t$ (i.e. there exists a time $s< t$ such that $\omega(s) \in \mathbb{Z}$ ) and $\omega'$ is such that for every time u, $obs(\omega(u)) = obs(\omega'(u))$ , then in particular $\omega'(s) \in \mathbb{Z}$ and $\omega' \in B_t$ .

• The sets $B_t$ can also be expressed as:

  \[B_t = \bigcup_{n \in \mathbb{Z}} \{ \omega ~|~ T_n(\omega) <t\} = \bigcup_{n \in \mathbb{Z}} T_n^{-1}([0,t)) \]
  where $T_n$ is the hitting time of $\{n\}$ . From Lemma 18, we get that $T_n^{-1}([0,t))$ is measurable and hence that the sets $B_t$ are measurable.

• We can now check the final condition: $\Pi (B_t) = \{0,1\} ^\mathbb{N}$ which is measurable.

Let us compute $\mathbb{P}^z(B_t)$ for any $z \in \mathbb{R}$ . Since the distribution of Brownian motion is invariant under translation with respect to its starting point, we have that for every $z\in\mathbb{R}$ and $t>0$ ,

\[\mathbb{P}^{z}\left(B_{t}\right)=\mathbb{P}^{z}\left(T_{\left\lceil z\right\rceil }\wedge T_{\left\lfloor z\right\rfloor }<t\right)=\mathbb{P}^{z-\left\lfloor z\right\rfloor }\left(T_{0}\wedge T_{1}<t\right).\]

Therefore, it is sufficient to study the distribution of $T_{0}\wedge T_{1}$ under $\mathbb{P}^{c}$ for every $c\in[0,1).$ We claim that, for any pair $c,c^{\prime}\in[0,1)$ , $T_{0}\wedge T_{1}$ has identical distribution under $\mathbb{P}^{c}$ and $\mathbb{P}^{c^{\prime}}$ if and only if $\left|c-\frac{1}{2}\right|=\left|c^{\prime}-\frac{1}{2}\right|$ , that is, either $c=c^{\prime}$ or $c=1-c^{\prime}$ . To see this, we consider the Laplace transform of $T_{0}\wedge T_{1}$ under $\mathbb{P}^{c}$ and use the formula 1.3.0.1 of Borodin and Salminen (Reference Borodin and Salminen1997) to write

\[\mathbb{E}^{c}\left[e^{-\lambda\left(T_{0}\wedge T_{1}\right)}\right]=\frac{\cosh\left(\left(c-\frac{1}{2}\right)\sqrt{2\lambda}\right)}{\cosh\left(\frac{1}{2}\sqrt{2\lambda}\right)}\text{ for every }\lambda>0.\]

Then, $T_{0}\wedge T_{1}$ has identical distribution under $\mathbb{P}^{c}$ and $\mathbb{P}^{c^{\prime}}$ if and only if

\[\mathbb{E}^{c}\left[e^{-\lambda\left(T_{0}\wedge T_{1}\right)}\right]=\mathbb{E}^{c^{\prime}}\left[e^{-\lambda\left(T_{0}\wedge T_{1}\right)}\right]\text{ for every }\lambda>0,\]

which is equivalent to $\left|c-\frac{1}{2}\right|=\left|c^{\prime}-\frac{1}{2}\right|$ .

Thus, we have proven that for every pair $x,y\in\mathbb{R}$ ,

\[\mathbb{P}^{x}\left(B_{t}\right)=\mathbb{P}^{y}\left(B_{t}\right)\text{ for every }t>0\]

if and only $x-\left\lfloor x\right\rfloor =y-\left\lfloor y\right\rfloor $ or $x-\left\lfloor x\right\rfloor =\left\lceil y\right\rceil -y$ .

The equivalence generated by this group of symmetry is trace equivalence which shows that trace equivalence, the greatest bisimulation and the greatest temporal equivalence are identical.

5.2.2 Brownian motion with drift

Let us consider a Brownian process with an additional drift: $W_t^{(v)} = W_t + vt$ (where $W_t$ is the standard Brownian motion and $v > 0$ ).

where for every $x \in \mathbb{R}$ and every $k \in \mathbb{Z}$ , $t_k (x) = x+k$ .

With zero distinguished: Let us consider the case when there is a single atomic proposition and $obs (x) = 1$ if and only if $x = 0$ .

Proposition 49. Two states are trace equivalent if and only if they are equal.

Proof. The only thing to show is that two different states cannot be trace equivalent.

Define the set

\[B_t := \{ \omega ~|~ \exists s < t ~ \omega (s) = 0\} = \{ T_0 < t\} \]

Similarly to what was done in the proof of proposition 48, this set is time-obs-closed.

Therefore, it is sufficient to study the distribution of $T_{0}$ under $\mathbb{P}^{z}$ for every $z \in \mathbb{R}$ . We claim that, for any pair x,y, $T_{0}$ has identical distribution under $\mathbb{P}^{x}$ and $\mathbb{P}^{y}$ if and only if $x = y$ . To see this, we consider the Laplace transform of $T_{0}$ under $\mathbb{P}^{z}$ for an arbitrary state $z \in \mathbb{R}$ and use the formula 2.2.0.1 of Borodin and Salminen (Reference Borodin and Salminen1997) to write

\[\mathbb{E}^{z}\left[e^{-\lambda T_{0}}\right]=\exp \left( -zv - |z| \sqrt{2 \lambda + v^2} \right)\text{ for every }\lambda>0.\]

Then, $T_{0}$ has identical distribution under $\mathbb{P}^{x}$ and $\mathbb{P}^{y}$ if and only if

\[\mathbb{E}^{x}\left[e^{-\lambda T_0} \right]=\mathbb{E}^{y}\left[e^{-\lambda T_0} \right]\text{ for every }\lambda>0,\]

which is equivalent to

\[ xv + |x| \sqrt{2 \lambda + v^2} = yv + |y| \sqrt{2 \lambda + v^2} \text{ for every }\lambda>0, \]

which ultimately means that $x = y$ .

This shows that there is a single group of symmetries possible: {id} and that no two different states can be bisimilar or temporally equivalent.

5.2.3 Brownian motion with absorbing wall

Another common variation of Brownian motion is to add boundaries and to consider that the process does not move anymore or dies once it has hit a boundary. The boundary is called an “absorbing wall”.

The standard Brownian motion on the real line is denoted $W_t$ and $\Omega$ is its space of trajectories. Assume the boundaries are at 0 and z. We introduce two hitting times $T_0$ and $T_z$ (which are stopping times thanks to Lemma 18). We can now define the stopping time $T = T_0 \wedge T_z$ . Finally, we obtain the Brownian motion with absorbing walls at 0 and z: for any $\omega$ in $\Omega$

\[ A_t (\omega) =\begin{cases}W_t (\omega) \qquad \text{if } t < T(\omega)\\\partial \qquad \text{otherwise}\end{cases} \]

Note that here $\Omega$ is the space of trajectories for the original Brownian motion and serves as source of randomness for the new process.

Let us clarify why we have chosen this to be the state space; essentially the reason is to ensure that the trajectories are cadlag. The boundaries are not included in the state space, that is the process with absorbing walls at 0 and z has (0,z) as its state space. This is forced by the requirement that trajectories are cadlag. To see this, consider what would happen if the wall was included in the state space. We would have to modify the behaviour of the process at 0 and z. But what would these modified trajectories look like? They would look like a trajectory of a standard Brownian motion: continuous until it reaches either state 0 or z (assume it happens at time s) and then the trajectory would jump to the state $\partial$ , that is the trajectory would be continuous on [0,s] and then perform a jump and be continuous (constant even) on $(s,+\infty]$ . This is not a cadlag trajectory because it is continuous on the “wrong side”.

where $s_k(x) = k-x$ for every $x \in \mathbb{R}$ and $k = b$ or 2b.

5.3 Poisson process

This is an example that we considered in Chen et al.(Reference Chen, Clerc and Panangaden2020). A Poisson process models a continuous-time process in which a discrete variable is incremented. A standard example is the arrival of customers in a queue. Let us give some notations: a Poisson process is a non-decreasing process $(N_t)_{t \geq 0}$ onto the set of natural numbers $\mathbb{N}$ , a discrete space. We define the set $\Omega$ of trajectories as usual on the state space. The probability distribution on the set $\Omega$ is defined as

\[ \mathbb{P}^k (N_t = n) = \frac{(\lambda t)^{n-k}}{(n-k)!} e^{- \lambda t} \qquad \text{ for } n \geq k \]

We are going to study two cases. In the first case, we are able to test if there is an even or odd number of customers that have arrived. In the second case, we are able to test if there have been more customers in total than a critical value.

Testing evenness of number of customers: There is a single atomic proposition on the state space: $obs(k) = 1$ if and only if k is even.

First, it is indeed a bisimulation. Consider $y = x + 2n$ where $n \in \mathbb{N}$ (note that $obs(x) = obs(y)$ ) and B a measurable, time-R-closed set.

For a measurable set B’, $\mathbb{P}^{x}(B') = \mathbb{P}^{x+2n}(B'+2n)$ , where $B' + 2n = \{ t \mapsto \omega(t) + 2n\} ~|~ \omega \in B\}$ . In particular, $\mathbb{P}^{x}(B) = \mathbb{P}^{y}(B+2n)$ . Since B is time-R-closed, $B +2n \subset B$ , so $\mathbb{P}^{x}(B) \leq \mathbb{P}^{y}(B)$ .

For the reverse direction, let M be the set of non-decreasing trajectories. Note that M is measurable since its complement is measurable:

Write $B_0 = B \cap M \cap\{ \omega ~|~ \omega (0) = y\}$ . Note that the process can only realise non-decreasing trajectories, therefore $\mathbb{P}^{y} (B) = \mathbb{P}^{y} (B_0)$ . Define $B_1 = \{ t \mapsto \omega (t) - 2n ~|~ \omega \in B_0\}$ . Note that for every $\omega' \in B_1$ and $s \geq 0$ , $\omega'(s) \in \mathbb{N}$ since for every $\omega \in B_0$ and $t \geq 0$ , $\omega (t) \geq \omega (0) = x + 2n$ . Since $B_1 + 2n = B_0$ , we have that $\mathbb{P}^y (B_0) = \mathbb{P}^x(B_1)$ . Furthermore, $B_1 \subset B$ since B is time-R-closed. Putting all this together: $\mathbb{P}^y(B) = \mathbb{P}^y(B_0) = \mathbb{P}^x(B_1) \leq \mathbb{P}^x(B)$ .

This concludes the proof that R is a bisimulation. Now, notice that $x~R~y$ if and only if $obs(x) = obs(y)$ . Since this equivalence is weaker than trace equivalence which is itself weaker than bisimulation, we have that R is the trace equivalence and the greatest bisimulation (and similarly the greatest temporal equivalence).

Testing for a critical value: Fix $m \in \mathbb{N}_{\geq 0}$ , we define the function obs by $obs(x) = 1$ if and only if $x \geq m$ .

Proof. Denote

\[R = \{ (x,x) ~|~ x < m\} \cup \{ (x,y) ~|~ x,y \geq m\} \]

Let us show that it is a bisimulation. Consider $x~R~y$ and assume $x \neq y$ . This means that $x, y \geq m$ and hence $obs(x) = obs(y)$ .

Now, also consider a measurable time-R-closed set B. Define $B' = B \cap M \cap \{ \omega ~|~ \omega(0) \geq m \}$ where M is the set of non-decreasing functions. Similarly to previous example, M is measurable. Note that the process can only realise non-decreasing trajectories, therefore $\mathbb{P}^{y} (B) = \mathbb{P}^{y} (B')$ (and similarly for x).

There are now two cases to consider:

• If B’ is empty, $\mathbb{P}^{y} (B) = \mathbb{P}^{x} (B) = 0$ .

• Otherwise, there exists $\omega' \in B'$ . Note that for every time $t \geq 0$ , $\omega'(t) \geq \omega'(0) \geq m$ since $\omega'$ is non-decreasing.

We have that $B' = \{ \omega ~|~ \omega (0) \geq m \} \cap M$ . This means that for every $z \geq m$ , $\mathbb{P}^z (B') = \mathbb{P}^z (\{ \omega ~|~ \omega (0) \geq m \} \cap M) = 1$ . And in particular, this shows that $\mathbb{P}^x(B) = \mathbb{P}^y(B) = 1$ .

To prove that it is the greatest bisimulation, we show that it corresponds to trace equivalence. This proof can be adapted to show that $x,y \geq m$ are trace equivalent.

Clearly if $x< m$ and $y \geq m$ , then x and y cannot be trace equivalent since the set $\{\omega ~|~ obs(\omega(0)) = 0\}$ is time-obs-closed, but $\mathbb{P}^x(\{\omega ~|~ obs(\omega(0)) = 0\}) = 1$ and $\mathbb{P}^y(\{\omega ~|~ obs(\omega(0)) = 0\}) = 0$ .

Consider the case when $x \neq y$ are both less than m. Consider a time $t > 0$ and define $B_t = \{ \omega ~|~ \omega(t) \geq m\}$ . This set is time-obs-closed: the first two conditions are straightforward and $\Pi (B_t) = \{0,1\}^{\mathbb{N}}$ if $t \notin \mathbb{N}$ and $\Pi(B_t) = \{ \delta : \mathbb{N} \to \{0,1\} ~|~ \delta(t) =1 \}$ if $t \in \mathbb{N}$ . Both are measurable.

For $k < m$ ,

\[\mathbb{P}^k (B_t) = \sum_{n \geq m} \mathbb{P}^k (N_t = n) = e^{- \lambda t} \sum_{n \geq m-k} \frac{(\lambda t)^n}{n!}\]

This allows us to conclude that if $x \neq y$ , $\mathbb{P}^x(B_t) \neq \mathbb{P}^y(B_t)$ .

There are several observations that can be made. First, in all examples, greatest temporal equivalence and bisimulation matched. We do not know if this is the case for all FD-processes or a subfamily of FD-processes. In particular, all the processes studied here are quite well-behaved and simple: Brownian motion has continuous trajectories (and not cadlag) for instance. Another interesting observation is that group of symmetries corresponds to intuition when handling a problem. Finally, notice how hitting times played a huge role in these proofs.

6.1 Feller-Dynkin homomorphism

The definition of bisimulation can easily be adapted to states in different Markov processes by constructing the disjoint union of the Markov processes in the following manner. First, one constructs the disjoint union of the two state spaces as topological spaces. It is then possible to extend the semigroups on the two state spaces to a semigroup on the disjoint union of the state spaces. The construction of the time-indexed family of Markov kernels, of the space of trajectories and of and of the space-indexed family of probabilities on the space of trajectories is the same as the one described in Section 3.1.

In that context, a bisimulation is defined in the natural way: If $x ~R~ y$ where $x \in E_i$ and $y\in E_j$ (i and j can be either 1 or 2 depending on which state space x and y are on), then:

(initiation) $obs_i (x) = obs_j(y)$ , and

(induction) for all measurable time-R-closed sets B, $\mathbb{P}^x(B \cap \Omega_i) = \mathbb{P}^y (B \cap \Omega_j)$ .

This condition can also be stated as follows. For all sets $B_i \in \mathcal{G}_i$ and $B'_j \in \mathcal{G}_j$ , $\mathbb{P}_{i}^x (B_i) = \mathbb{P}_{j}^y (B'_j)$ if the sets $B_i$ and $B'_j$ satisfy the following conditions: for all trajectories $\omega$ in $B_i \cup B'_j$ ,

\[ \forall \omega' \in \Omega_i ~ (\forall t \geq 0 ~ \omega (t) ~ R~ \omega'(t)) \Rightarrow \omega' \in B_i \]

and

\[ \forall \omega' \in \Omega_j~ (\forall t \geq 0 ~ \omega (t) ~ R~ \omega'(t)) \Rightarrow \omega' \in B'_j \]

To go from the first statement of the induction condition to the other, write $B_i = B \cap \Omega_i$ and $B'_j = B \cap \Omega_j$ . To go from the second statement to the first statement is slightly trickier to write down since B may contain trajectories on both $E_1$ and $E_2$ (switching state space). However, those trajectories have probability zero.

Note that $R \cap (E_j \times E_j)$ is a bisimulation on $(E_j, \mathcal{E}_j, (P_{t}^j), (\mathbb{P}_{j}^x))$ . To proceed with our cospan idea, we need a functional version of bisimulation; we call these Feller-Dynkin homomorphisms.

Note that if f and g are FD-homomorphisms, then so is $g \circ f$ . This notion of FD-homomorphisms is designed to capture some aspects of bisimulation which is demonstrated in the following proposition.

Proposition 54. Given an FD-homomorphism f, the equivalence relation R defined on $E \uplus E'$ as

\[ R = \{ (x,y) \in E \times E ~|~ f(x) = f(y)\} \cup \{ (x,y), (y,x) ~|~ f(x) = y\} \cup \{ (y,y) ~|~ y \in E' \} \]

is a bisimulation on $E \uplus E'$ .

First note that $obs (x) = obs' \circ f(x)$ since $obs = obs' \circ f$ . Since $f(x) = f(y)$ , we have that $obs(x) = obs' \circ f(x) = obs' \circ f(y) = obs(y)$ . This gives us (initiation) for both cases.

Second, let us check the induction condition (initiation). Write $\hat{\Omega}$ and $\hat{\mathcal{G}}$ for the set of trajectories and its $\sigma$ -algebra on $E \uplus E'$ . Consider a measurable time-R-closed set $\hat{B} \in \hat{\mathcal{G}}$ .

Define the two sets

\begin{align*}B' & = \hat{B}\cap \Omega '\\B & = \{ \omega \in \Omega ~|~ f \circ \omega \in B'\}\end{align*}

As f is an FD-homomorphism, we have that $\mathbb{P}^{f(x)} (B') = \mathbb{P}^x (B)$ .

Let us show that $B = \hat{B} \cap \Omega$ .

• Consider $\omega \in B$ , that is $f \circ \omega \in \hat{B} \cap \Omega '$ . By definition of the set B, $\omega \in \Omega$ . Furthermore, $f \circ \omega \in \hat{B}$ . By definition of R, we have that for all $t \geq 0$ , $\omega(t) ~R~ (f \circ \omega (t))$ . Since the set $\hat{B}$ is time-R-closed and $f \circ \omega \in \hat{B}$ , we have that $\omega \in \hat{B}$ which proves the first inclusion.

• Consider $\omega \in \hat{B} \cap \Omega$ . The trajectory $f \circ \omega$ is well-defined and is in $\Omega'$ since f is continuous. Similarly to what was done for the first inclusion, we get that $f \circ \omega \in \hat{B} $ since $\omega \in \hat{B}$ and $\hat{B}$ is R-closed. This proves that $\omega \in B$

We get that $\mathbb{P}^{f(x)} (\hat{B} \cap \Omega ') = \mathbb{P}^x (\hat{B} \cap \Omega)$ .

Since $f(x) = f(y)$ , we also get that $\mathbb{P}^x (\hat{B} \cap \Omega) = \mathbb{P}^y (\hat{B} \cap \Omega)$ .

A nice consequence of this result is that the equivalence relation R defined on E as $R = \{ (x,y) \in E \times E ~|~ f(x) = f(y)\} = \ker(f)$ is a bisimulation on E.

Let us show that the map $\pi$ is open: for any open set U in E,

\begin{align*}\pi^{-1} \pi (U)& = \{ x ~|~ \exists y \in U ~ \pi(x) = \pi(y)\}\\& = \{ x ~|~ \exists y \in U ~ x ~R~ y\}\\& = \{ x ~|~ \exists y \in U ~ \exists h \in \mathcal{H} ~ x = h(y) \}\\& = \bigcup_{h \in \mathcal{H}} h(U)\end{align*}

Every symmetry $h \in \mathcal{H}$ is open since they are homeomorphisms; hence, h(U) are all open sets. This means that the set $\pi^{-1}\pi (U)$ is open and hence the map $\pi$ is open.

Let us show that $R \subset E \times E$ is closed. Consider a sequence $(x_n, y_n)$ in R that converges to (x,y). In particular, for every $n \in \mathbb{N}$ , $\pi(x_n) = \pi(y_n)$ . Furthermore, $\lim_{n} \pi(x_n) = \pi (x)$ and $\lim_{n} \pi(y_n) = \pi (y)$ since $\pi$ is continuous. By uniqueness of limit, $\pi(x) = \pi(y)$ and hence $(x,y) \in \approx$ .

Since the quotient map is open and the equivalence R is closed, the space $E/R$ is Hausdorff.

Since the map $\pi$ is open and the space E is locally compact, the space $E/R$ is also locally compact.

Let us now clarify what the FD-process is on that state space. First, we define for $x \in E/ R$

\[ obs'(x) = obs(y) \quad \text{where } \pi(y) = x \]

This is indeed well-defined as whenever $y ~R~ z$ for $y,z \in E$ , then there exists $h \in \mathcal{H}$ such that $h(y) = z$ and hence $obs(y) = obs(h(y)) = obs(z)$ . Note that the function $obs' : E' \to 2^{AP}$ is measurable (the proof is similar to the proof of Lemma 71). Furthermore, $obs = obs' \circ \pi$ by definition of $\pi$ .

Let us denote $\Omega'$ the set of trajectories on the one-point compactification of $E/R$ : $E'_\partial$ . This defines a time-indexed family of random variables $(X'_t)_{t \geq 0}$ by $X'_t(\omega) = \omega(t)$ for every trajectory $\omega \in \Omega'$ and every time $t \geq 0$ . The $\sigma$ -algebra $\mathcal{G}'$ on the set of trajectories $\Omega'$ is defined as usual for FD-processes:

\[ \mathcal{G}' = \sigma \{ (X'_s)^{-1}(C') ~|~ s \geq 0, C' \in \mathcal{E}' \} \]

We define the following family of probabilities on the set $\Omega'$ : for $x \in E/R$ , $B' \in \mathcal{G}'$ ,

\[ \mathbb{Q}^x(B') =\mathbb{P}^y(B) \qquad \text{if } x =\pi(y) \text{ with } B = \{\omega \in \Omega' ~|~ \pi \circ \omega \in B'\} \]

There are two things to show in order to check that $\mathbb{Q}^x$ is well-defined:

• The set B is measurable. The proof is done by induction on the structure of B’.

- First, if $B' = (X'_s)^{-1} (C') = \{ \omega \in \Omega' ~|~ \omega(s) \in C'\}$ for $C' \in \mathcal{E}'$ , then $B = (X_s)^{-1} (\pi^{-1}(C'))$ . And since $\pi$ is continuous, we know that $\pi^{-1}(C') \in \mathcal{E}$ and hence $B \in \mathcal{G}$ .

- If $B' = (A')^C$ with $A' \in \mathcal{G}'$ and $A = \{ \omega \in \Omega ~|~\pi \circ \omega \in A'\} \in \mathcal{G}$ , then $B = \Omega \setminus A$ . And since $A \in \mathcal{G}$ , we get that $B \in \mathcal{G}$ .

- If $B' = \bigcup_{i \in \mathbb{N}}A'_i$ where for every $i \in \mathbb{N}$ , $A'_i \in \mathcal{G}'$ and $A_i = \{ \omega \in \Omega ~|~ \pi \circ \omega \in A'_i \} \in \mathcal{G}$ , then $B = \bigcup_{n \in \mathbb{N}} A_i$ . And since $A_i \in \mathcal{G}$ for every $i \in \mathbb{N}$ , we get that $B \in \mathcal{G}$ .

• If y, z are in E and such that $\pi(y) = \pi(z)$ , then $\mathbb{P}^y(B) = \mathbb{P}^z(B)$ . Indeed, having $\pi(y) = \pi(z)$ means that $y ~R~ z$ where R is the bisimulation generated by the group of symmetries $\mathcal{H}$ . It is therefore enough to show that the set B is time-R-closed. Consider two trajectories $\omega$ and $\omega'$ such that at every time $t \geq 0$ , $\omega(t) ~R~ \omega'(t)$ , that is $\pi(\omega(t)) = \pi(\omega'(t))$ . This means that $\pi \circ \omega = \pi \circ \omega'$ and hence by definition of B, $\omega \in B$ if and only if $\omega' \in B$ .

By definition of $\mathbb{Q}$ . we know that for all $x \in E$ and for all measurable sets $B' \subset \Omega '$ , $\mathbb{Q}^{\pi(x)}(B') = \mathbb{P}^x (B)$ where $B := \{ \omega \in \Omega ~|~\pi \circ \omega \in B'\}$ . We have already shown that the map $\pi$ was surjective, continuous and open and that $obs = obs' \circ \pi$ . Which shows that $\pi$ is an FD-homomorphism.

However, we still have to check that this family $(\mathbb{Q}^y)_{y \in E/R}$ corresponds to an FD-process. That family of probabilities $(\mathbb{Q}^x)_{x \in E/R}$ on $\Omega'$ yields a Markov kernel on $E/R$ :

\[ \text{for } x \in E/R, ~ t \geq 0 \text{ and } C \in \mathcal{E}' \quad P'_t (x, C) = \mathbb{Q}^x ((X'_t)^{-1} (C')) \]

which in turns yields an FD-semigroup:

\[ \text{for}\ f \in C_0(E') \text{ and } x \in E/R \quad (\hat{P'}_t f)(x) = \int f ~ dP'_t(x, {-}) \]

In order to see that $(\hat{P'}_t)_{t \geq 0}$ forms an FD-semigroup, it is convenient to note that for any z such that $x = \pi(z)$

\[ \hat{P'}_t f (x) = \int_{E/R} f ~ dP'_t(x, {-}) = \int_E f \circ \pi ~ dP_t(z, {-}) = [\hat{P}_t (f \circ \pi)] (z) \]

This concludes the proof.

Let us detail how the Markov kernel is obtained for $\mathcal{M}_{refl, 1}$ . Write $(P_t)_{t \geq 0}$ for the Markov kernel for the standard Brownian motion on $\mathbb{R}$ . We define the Markov kernel $(Q_t)_{t \geq 0}$ for $\mathcal{M}_{refl, 1}$ for every $x \in [0,1]$ and C measurable subset of [0,1]:

\[ Q_t(x, C) = P_t(x, C') \]

where

\[ C' = \bigcup_{k \in \mathbb{Z}} (C + 2k) \uplus \bigcup_{k \in \mathbb{Z}} (\overline{C} + 1 + 2k) \]

with $C + k = \{ x + k ~|~ x \in C\}$ is the translation by k of the original set C and $\overline{C} = \{ 1-x ~|~ x \in C\}$ is the symmetry of the original set C around $1/2$ . The intuition for that variation is that we “fold” the real line at each integer. The atomic proposition is therefore held at both extremities since they correspond to the integers of the real line once folded.

We will only provide intuition for the remaining two variations. First, the reflexive variation on $[0, 1/2]$ is obtained from the standard Brownian motion by folding the real line at each integer and at each half-integer (i.e. $k + 1/2$ where $k \in \mathbb{Z}$ ). The atomic proposition therefore only holds at 0. Indeed, all integers of the original real line get folded into 0.

Second $\mathcal{M}_{circ}$ is obtained from the standard Brownian motion by “wrapping” the real line around a circle of perimeter 1. All the integers of the initial real line are thus mapped to a single point on the circle (which we decide to be 0).

We can define some natural mappings between these processes:

where the upper two morphisms correspond to the intuition provided before:

\begin{align*} \phi_{wrap} : \mathbb{R} & \to [ 0 , 2 \pi ) \\ x & \mapsto 2\pi (x - \lfloor x \rfloor )\\ \text{and} \quad \phi_{fold, int} : \mathbb{R} & \to \left[ 0, 1 \right]\\ x & \mapsto x - 2n ~\text{with } n \in \mathbb{Z} \text{ such that } 2n \leq x < 2n+1\\ x & \mapsto 2n+2 - x ~\text{with } n \in \mathbb{Z} \text{ such that } 2n+1 \leq x < 2n+2\end{align*}

Now, the remaining two morphisms intuitively make sense. How do we obtain the reflexive Brownian motion on $[0,1/2]$ from a reflexive Brownian motion on [0,1]? We “fold” the interval [0,1] at its middle point $1/2$ :

\begin{align*}\phi_{middle} : [0, 1] & \to \left[ 0, 1/2 \right] \\x & \mapsto x ~\text{if } x \leq 1/2 \\x & \mapsto 1-x ~\text{otherwise}\end{align*}

In order to obtain the reflexive variation on $[0,1/2]$ from the circle, intuitively we “flatten” the circle by identifying the two halves that meet at 0 (the point with the atomic proposition):

\begin{align*}\phi_{flat} : [0, 2\pi) & \to \left[ 0, 1/2 \right] \\\theta & \mapsto \theta/2 \pi ~\text{if } \theta \leq \pi \\\theta & \mapsto (2 \pi - \theta )/2 \pi ~\text{if } \theta \geq \pi \\\end{align*}

All these morphisms correspond to the construction of one variation of Brownian motion from another one. It is therefore easy to deduce that all these morphisms are FD-homomorphisms.

6.2 Definition of cospans

It is time to retrieve the relational aspect of bisimulation in this functorial framework. For discrete time, the initial definition used spans of zigzags Desharnais et al. (Reference Desharnais, Edalat and Panangaden2002). This is equivalent to considering cospans of zigzags in analytic spaces, but this is not necessarily the case for non-analytic spaces Sánchez Terraf (Reference Terraf2011).

In order to show that FD-cospans behave like an equivalence, it is a necessity to show that they are somehow transitive. Namely, if we have two cospans $S \overset{f}{\rightarrow} E_1 \overset{h}{\leftarrow} E_2$ and $E_2 \overset{g}{\rightarrow} E_3 \overset{k'}{\leftarrow} S'$ , can we construct an FD-cospan relating S and S’? The following theorem shows that we can.

There are two inclusions $i_1 : E_1 \to E_1 \uplus E_3$ and $i_3 : E_3 \to E_1\uplus E_3$ . Define the equivalence relation $\sim$ on $E_1 \uplus E_3$ as the smallest equivalence such that for all $x_1 \in E_1$ and $x_3 \in E_3$ , $i_1(x_1) \sim i_3(x_3)$ if there exists $z \in E_2$ , such that $x_1 = h(z)$ , and $x_3 = g(z)$ . Define $E_4 = E_1 \uplus E_3 / \sim$ with its corresponding quotient $\pi_\sim : E_1 \uplus E_3 \to E_4$ and the two maps $\phi_3 = \pi_\sim \circ i_3$ and $\phi_1 = \pi_\sim \circ i_1$ . This corresponds to the pushout in Set.

We equip this set with the largest topology that makes $\pi_\sim$ continuous (where the topology on $E_1 \uplus E_3$ is the topology inherited from the inclusions). This means that a set $O \subset E_4$ is open if and only if $\pi_\sim ^{-1}(O) \subset (E_1 \uplus E_3)$ is open. This corresponds to the pushout in Top.

It is worth noting that this topological space $(E_1 \uplus E_3) / \sim$ is bijective to the space $E_2 / \approx$ where $\approx$ is defined on $E_2$ as the smallest equivalence such that if $g(z) = g(z')$ or $h(z) = h(z')$ , then $z \approx z'$ (see Lemma 68). We will write $\pi_\approx : E_2 \to E_4 = (E_1 \uplus E_3)/ \sim$ for the map $E_2 \overset{h}{\to} E_1 \overset{i_1}{\to} E_1 \uplus E_3 \overset{h}{\to} E_4$ . This is equal to the map $E_2 \overset{g}{\to} E_3 \overset{i_3}{\to} E_1 \uplus E_3 \overset{h}{\to} E_4$ . It is a quotient map and thus surjective.

Lemma 69 shows that the space $E_4$ equipped with its topology and Borel-algebra is locally compact and Hausdorff.

Note that both maps $\phi_1$ and $\phi_3$ are surjective, continuous and open (see Lemma 70)

We define $obs_4$ as such: $obs_4 (x_4) = obs_2 (x_2)$ where $x_4 = \pi_\approx (x_2)$ . This is well-defined since if $z \approx z'$ , then $obs_2(z) = obs_2(z')$ . Indeed, that would mean that there is a sequence $z = z_0, z_1, ..., z_n = z'$ where $z_i$ and $z_{i+1}$ have same image by either g or h. Since g and h are FD-homomorphisms, that means that $obs_2(z_i) = obs_2(z_{i+1})$ . It is equivalent to defining $obs_4$ as:

\[ obs_4 (x_4) = \begin{cases}obs_1 (x_1) \qquad \text{if } x_4 = \phi_1 (x_1)\\obs_3 (x_3) \qquad \text{if } x_4 = \phi_3 (x_3)\end{cases} \]

Lemma 71 shows that this map is indeed measurable.

Finally, let us define the FD-process on $E_4$ . Let $\Omega_4$ be the set of trajectories on $E_4$ . The random variable $X^4$ is straightforward to define: $X^4_t (\omega) = \omega(t)$ for every trajectory $\omega \in \Omega_4$ and time $t \geq~0$ . The set of trajectories $\Omega_4$ is equipped with a $\sigma$ -algebra $\mathcal{G}_4$ defined in the standard way for FD-processes:

\[\mathcal{G}_4 = \sigma (X^4_s ~|~ 0 \leq s < \infty) = \sigma(\{ (X^4_s) ^{-1}(C) ~|~ 0 \leq s < \infty, ~ C \in \mathcal{E}_4 \})\]

We are now equipped to define the FD-process on $E_4$ . Consider $x_4 \in E_4$ and $B_4 \in\mathcal{G}_4$ . We define

\[ \mathbb{P}_4^{x_4} (B_4) =\mathbb{P}_2^{x_2} (B_2)\]

where $x_2 \in E_2$ and $B_2 \subset \Omega_2$ are such that $x_4 = \pi_\approx (x_2)$ and $B_2 = \{ \omega \in \Omega_2 ~|~ \pi_\approx \circ \omega \in B_4\}$ . We show in Lemma 72 that this is indeed well-defined and that

\[ \mathbb{P}_4^{x_4} (B_4) =\mathbb{P}_2^{x_2} (B_2) = \mathbb{P}_3^{x_3} (B_3) =\mathbb{P}_1^{x_1} (B_1) \]

where $x_1 \in E_1$ is such that $\phi_1(x_1) = x_4$ , $x_3 \in E_3$ is such that $\phi_3(x_3) = x_4$ and $B_1 = \{ \omega \in \Omega_1 ~|~ \phi_1 \circ \omega \in B_4\}$ and $B_3= \{ \omega \in \Omega_3 ~|~ \phi_3 \circ \omega \in B_4\}$ .

Let us now check that this is indeed an FD-process. We define first the corresponding kernel for $t \geq 0$ , $x \in E_4$ and $C \in \mathcal{E}_4$ ,

\[ P_t^4 (x, C) = \mathbb{P}^x_4 (\{ \omega ~|~ \omega (t) \in C\}) \]

For $x_2 \in E_2$ such that $\pi_\approx(x_2) = x$ , we have that $P_t^4 (x, C) =P_t^2 (x_2, \pi_\approx^{-1}(C))$ We can now define the operator

\[ \hat{P}_t^4 f (x) = \int_{y \in E_4} f(y) P_t^4 (x, dy) \]

Using the corresponding equality for the kernel and a change of measure, we get that if $x = \pi_\approx(x_2)$ , then $\hat{P}_t^4 f (x) = \hat{P}_t^2 (f \circ \pi_\approx)(x_2)$ . The fact that $(\hat{P}_t^4)_{t \geq 0}$ is a semigroup follows from $(\hat{P}_t^2)_{t \geq 0}$ being itself a semigroup.

By construction, $\phi_1$ and $\phi_3$ are indeed FD-homomorphisms.

Let us now check the universal property. Assume there is an FD-process $(E_5, \mathcal{E}_5, (P_t^5)_t)$ and two FD-homomorphisms $\psi_1 : E_1 \to E_5$ and $\psi_3 : E_3 \to E_5$ such that $\psi_1 \circ h = \psi_3 \circ g$ . Since $E_4$ is the corresponding pushout in Top, there exists a continuous morphism $\gamma : E_4 \to E_5$ such that $\psi_j = \gamma \circ \phi_j$ (for $j = 1,2$ ). Let us show that it is an FD-homomorphism.

First, we want to show that $obs_4 = obs_5 \circ \gamma$ . Let $x_4 \in E_4$ , there are two cases: $x_4 = \phi_1 (x_1)$ or $x_4 = \phi_3 (x_3)$ . Consider the first case $x_4 = \phi_1 (x_1)$ (the other case is similar).

\begin{align*}obs_5 \circ \gamma (x_4)& = obs_5 \circ \gamma \circ \phi_1 (x_1)\\& = obs_5 \circ \psi_1 (x_1)\\& = obs_1 (x_1) \qquad \text{ since $\psi_1$ is an FD-homomorphism}\\& = obs_4 (\phi_1 (x_1)) \qquad \text{ since $\phi_1$ is an FD-homomorphism}\\& = obs_4 (x_4)\end{align*}

Second, we show that for $x_4 \in E_4$ and $B_5 \in \mathcal{G}_5$ , $\mathbb{P}_5^{\gamma (x_4)} (B_5) = \mathbb{P}_4^{x_4} (B_4)$ with $B_4 = \{ \omega \in \Omega_4 ~|~ \gamma \circ \omega \in B_5 \}$ . Consider the first case $x_4 = \phi_1 (x_1)$ (the other case $x_4 = \phi_3 (x_3)$ is similar). First, as $\psi_1$ is an FD-homomorphism,

\begin{align*}\mathbb{P}_5^{\gamma (x_4)} (B_5)& = \mathbb{P}_5^{\gamma (\phi_1 (x_1))} (B_5) = \mathbb{P}_5^{\psi_1 (x_1)} (B_5)\\& = \mathbb{P}_1^{x_1} (B_1) \quad \text{ where } B_1 = \{ \omega \in \Omega_1 ~|~ \psi_1 \circ \omega \in B_5\}\end{align*}

Second, as $\phi_1$ is an FD-homomorphism,

\[ \mathbb{P}_4^{x_4} (B_4) = \mathbb{P}_4^{\phi_1(x_1)} (B_4) = P_1^{x_1}(B'_1) \]

where $B'_1 = \{ \omega \in \Omega_1 ~|~ \phi_1 \circ \omega \in B_4\}$ . Finally, we have that

\begin{align*}B'_1 & = \{ \omega \in \Omega_1 ~|~ \phi_1 \circ \omega \in B_4\}\\& = \{ \omega \in \Omega_1 ~|~ \gamma \circ \phi_1 \circ \omega \in B_5\} = B_1\end{align*}

which concludes the proof.

What this theorem shows is that FD-cospans behave like equivalence relations. Indeed, this means that if we have an FD-cospan $(f_1, g_1)$ between FD-processes $\mathcal{M}_1$ and $\mathcal{M}_2$ and another FD-cospan $(f_2, g_2)$ between FD-processes $\mathcal{M}_2$ and $\mathcal{M}_3$ , then there exists an FD-cospan $(f_3, g_3)$ between the FD-processes $\mathcal{M}_1$ and $\mathcal{M}_3$ .

We have already showed how FD-homomorphisms yield bisimulations in Proposition 54. Now, consider an FD-cospan (f, g) with $f : E_1 \to E_3$ and $g: E_2 \to E_3$ inducing the bisimulations $R_f$ on $E_1 \uplus E_3$ and $R_g$ on $E_2 \uplus E_3$ respectively. We can define a bisimulation R on $E_1 \uplus E_2$ as

\[ R = \{ (x, y), (y,x) ~|~ \exists z \in E_3 \text{ such that } (x, z) \in R_f \text{ and } (y,z) \in R_g \} \]

Conversely, given a group of symmetries, it is also possible to define an FD-cospan corresponding to its corresponding bisimulation as is stated next:

Theorem 59. Consider two FD-processes on the state spaces $E_1$ and $E_2$ respectively. Let $\mathcal{H}$ be a group of symmetries of the FD-process on $E_1 \uplus E_2$ such that

\[ \text{for all } x_1 \in E_1 \text{ there exists } x_2 \in E_2 \text{ such that } i_1(x_1) ~R~ i_2(x_2) \]

and

\[ \text{for all } x_2 \in E_2 \text{ there exists } x_1 \in E_1 \text{ such that } i_1(x_1) ~R~ i_2(x_2) \]

where R is the bisimulation generated by the group of symmetries $\mathcal{H}$ on $E_1 \uplus E_2$ . Then, there exists (f,g) an FD-cospan such that

• for all $x_1 \in i_1(E_1), x_2 \in i_2(E_2)$ , $x_1~R~x_2$ if and only if $f(x_1) = g(x_2)$ ,

• or all $x_1,y_1 \in i_1(E_1)$ , $x_1 ~R~ y_1$ if and only if $f(x_1) = f(y_1)$ , and

• for all $x_2, y_2 \in i_2(E_2)$ , $x_2 ~R~ y_2$ if and only if $g(x_2) = g(y_2)$ .

Define the set $E = (E_1 \uplus E_2) / R$ . There are two inclusions $i_1 : E_1 \to E_1 \uplus E_2$ , $i_2 : E_2 \to E_1 \uplus E_2$ and a quotient $\pi : E_1 \uplus E_2 \to E$ . Using Proposition 55, we know that $\pi$ is an FD-homomorphism and we know what the FD-process $(E, \mathcal{E}, \Omega, \mathcal{G}, obs, (\mathcal{P}^y)_{y \in E})$ on E is.

We define $f = \pi \circ i_1 : E_1 \to E$ and $g = \pi \circ i_2 : E_2 \to E$ .

We only have to prove that f is indeed an FD-homomorphism.

First note that f is continuous as both $i_1$ and $\pi$ are. Similarly, f is relatively open (open on its image set) since $i_1$ is relatively open and $\pi$ is open. It is also surjective: consider $y \in E$ , then there exists $x \in E_1 \uplus E_2$ such that $\pi(x) = y$ . Then there are two possibilities:

• either $x = i_1 (x_1)$ where $x_1 \in E_1$ , in which case $f(x_1) = y$

• or $x = i_2 (x_2)$ where $x_2 \in E_2$ . But then, we know that there exists $x_1 \in E_1$ such that $i_1(x_1) ~R~ i_2(x_2)$ , that is such that $\pi \circ i_1 (x_1) = \pi \circ i_2(x_2) = y$

There are two remaining conditions to check for f to be indeed an FD-homomorphism: that $obs_1 \circ f = obs$ and for $x_1 \in E_1$ and $B \in \mathcal{G}$ , $\mathbb{P}^{f(x)} (B') = \mathbb{P}_1^x(B)$ where $B = \{ \omega \in \Omega_1 ~|~ f \circ \omega \in B'\}$ . These two conditions directly follow from the fact that $\pi$ itself satisfies these conditions.

We thus have the following correspondence between bisimulation, groups of symmetries and FD-cospans:

If there exists an FD-cospan (f,g) such that $f(x) = g(y)$ , then the two states x,y are bisimilar.

7.1 Game interpretation of bisimulation

We define the following game. Duplicator’s plays are pairs of trajectories that he claims are time-bisimilar. Spoiler is trying to prove him wrong.

• Given two trajectories $\omega$ and $\omega'$ , Spoiler chooses $t \geq 0$ and $B \neq \emptyset \in \mathcal{G}$ such that $\mathbb{P}^{\omega (t)} (B) \neq \mathbb{P}^{\omega' (t)} (B)$

• Duplicator answers by choosing $\omega_0 \in B$ and $\omega_1 \notin B$ such that $obs \circ \omega_0 = obs \circ \omega_1$ and the game continues from $(\omega_0, \omega_1)$

A player who cannot make a move at any point loses. Duplicator wins if the game goes on forever. The only way for Spoiler to win is to choose a time-obs-closed set.

7.2 Game interpretation of temporal equivalence

We define the following game. Duplicator’s plays are pairs of states that he claims are temporally equivalent. Spoiler is trying to prove him wrong.

• Given two states x and y, Spoiler chooses $t \geq 0$ and $C \neq \emptyset \in \mathcal{E}$ such that $P_t(x, C) \neq P_t(y, C)$ .

• Duplicator answers by choosing $x_1 \in C$ and $y_1 \notin C$ that are trace equivalent and the game continues from $(x_1, y_1)$

A player who cannot make a move at any point loses. Duplicator wins if the game goes on forever. The only way for Spoiler to win is to choose a set that is closed under trace equivalence. Duplicator’s only valid moves are pairs of trace equivalent states.