 Today I'm going to introduce our work about tennismional results for probabilistic BDL. The structure of the talk will be as follows. First, we will recall some background knowledge about PBDL and tennismional properties. Also, we introduce some other related works and our motivation. Next, we will see the characterization results of PBDL. After that, we go on to introduce two notions of bisimilarity and see that two extensions of PBDL categorizes these two notions. Let's first have a look at the subject of our paper, namely probabilistic, probabilistic, propositional dynamic logic, also known as PBDL. It was first introduced by Dexter Cawson in 1985 to reason about the probabilistic programs at a propositional level. So instead of problems like the determination of a program, PBDL describes properties such as what is the expected running time for a program involving coin flipping. Such demands determines that the nature of the logic must be real valued rather than boolean valued. We will come to more details about PBDL later and before that we introduce some knowledge about tennismional properties. Tennismional properties are basically bridges between logical equivalence and structural similarity. The benchmark work is traced back to Tennismional's paper in 1985, in which they proposed a simple model logic characterizing bisimulation for image finite label transition systems. The importance of such properties is that once we have the bridge between logical equivalence and structural similarity, then in order to check that two structures are not behavioral equivalence, if equivalence, one can simply do it by finding a logical formula which distinguishes the two models, and this is usually much easier than the formal work. Our paper is based on many previous work on labeled Markov processes. Labeled Markov processes are continuous probabilistic transition systems, and as one will see later, they are basically a PBDL model without the function valuations. We applied both the results and some proof techniques in the first three papers listed here. There they proved that a simple logic L0 categorizes both states and event bisimulation between labeled Markov processes. Other related work also mostly considers labeled Markov processes. For example, this paper here gives a real-valued logic which gives the same Tennismional properties for labeled Markov processes. However, as far as we know, there's no Tennismional results for PBDL and behavioral equivalence for PBDL models. So the first question we ask is that what behavioral equivalence does PBDL actually describe? After this, we defined the state and event bisimulation for PBDL models, which we think are natural extensions of that for labeled Markov processes. We then tried to extend PBDL in a natural way so that the new logic characterizes these two notions of bisimulation. Finally, we take this paper at the first step towards further research on Tennismional properties about co-algebraic dynamic logic. Alright, we are now ready to introduce our first characterization result, namely, PBDL characterizes trace equivalence. Before that, we briefly talk about PBDL. The syntax for PBDL is fourfolded, and we only present the non-trivial parts here, namely the programs and the functions. The programs involve primitive programs, sequential composition, and Boolean tests. Besides, it has arbitrary linear combination and iteration. For the functions, the most important part is the modalities, namely the diamond Pf in green here. So this syntax is rich enough to express programs like wild loops and conditionals. As for the semantics, a PBDL model is a tuple where sigma x here is a measurable space. VP and Vf are respectively valuations for primitive programs and primitive functions. Programs are interpreted as Markov kernels here, which are basically probabilistic state transition functions. And function symbols are interpreted as real-valued functions. In particular, the value of diamond Pf is the expected value of function f after executing program P. Next, we introduce the notion of trace. The terminology trace comes from automata theory, and our definition of trace on PBDL is actually a natural generalization of that for probabilistic automata. For the alphabet set, we have not only the primitive programs, but also those Boolean tests. The reason, put it simple, is that Boolean tests also appear as primitives in the formal construction of PBDL programs. Then the trace row at state x is a function which takes two inputs, a word omega in sigma star and the primitive function capital F. And then it returns the expected value of function f after the transition omega starting from state x. We illustrate our definition through a simple example here. So the primitive programs are A and B, and we only have one function symbol F. The states with red circle are states whose value of F are 1, and the rest all have value 0 for F. One can simply calculate that the trace at x0 with inputs A, B and F have value 1 over 2 times 1 over 3 times the value of F at x2, which is 1 over 6. Similarly, one can calculate the trace of y0 at with the same inputs. And since these two values are different, we say that these two pointed PBDL models are not trace equivalent. In fact, one can notice that these two pointed models are also distinguished by some PBDL function. For example, diamond A then B, F. Notice that this PBDL function is actually looks very similar as the trace at the input of the trace we had before. So this somehow already implies one direction of our categorization result. Namely, PBDL equivalence implies trace equivalent. The non-trivial part here is that the relatively weak notion, seemingly weak notion of trace equivalence actually implies PBDL logical equivalence. And this is based on the following observation that diamonds somehow distribute the linear combination, somehow distribute over the modality here. And this will give us our first theorem, namely, PBDL characterizes trace equivalence. Next, we go on to states by simulation and event by simulation. We first have a look at state by simulation, which is more intuitive. So let's start from state by simulation between labeled Markov processes. So labeled Markov processes, as shown here, is simply a measurable space together with some transition functions on it. And labeled Markov process state by simulation is a binary relation on the state set, such that for any r-related pair x and x' and for any r-closed subset A of x, the probability of x-transit to A and the probability of x'-transit to A by program P are always the same for any primitive program P. This is illustrated by the figure here. So this y, r, y' and z, r, z' illustrates the meaning of A, B, r-closed. Namely, for any r-related pair y and y' either both of them are in A or both of them are out of A. This notion of state by simulation is actually very intuitive in the sense that they are very like the notion of by simulation in the classical case between labeled transition systems. The only difference here is that the reachability condition now is replaced with the probability of the transition. For the state by simulation between PDR models, since PDR models are basically labeled Markov processes with this red extra valuation on primitive functions, we can define state by simulation between PDR models naturally by taking into consideration this extra weight structure. So we say that besides the above conditions, we require that for R to be a state by simulation, any r-related pair should have the same value on any primitive function F, which is illustrated by the red part here. We also illustrate this by a simple example. So here we also have two discrete PDR models. The states with red circles have value 1 for F while the red states have value 0. We claim that the states xx0 and yy0 are not state by similar. This is basically because if we do want to have a by simulation relation r which relates x0 and y0, then we indeed want to relate x1 with y1 and x2 with y1. However, this is not possible because for example, y1 can transit to state y3 via B where y3 has value 1 for F. But for the state x2 via program B, it can only transit to state x4 whose value of F is not 1. This simple argument tells us that there is no by simulation which relates x0 and y0. However, one can calculate that actually these two pointed models are trace equivalents. This implies that state by simulation is actually a strictly stronger notion than trace equivalents. Furthermore, it implies that in order to characterize trace equivalents and in order to characterize state by simulation, we do need to strengthen our logic namely PBDL. The notion of state by simulation has some shortcomings namely it requires an electricity even just between labeled Markov processes. Our idea comes from the simple logic L0 which can write state by simulation for labeled Markov processes. This simple logic L0 is defined here whose key part is the yellow part. The yellow part basically does a threshold check. It says that the probability to be transferred to a state which satisfies phi via program P is bigger than the value r. r is the number between 0 and 1 here. Since it characterizes state by simulation for labeled Markov processes, and our notion of state by simulation between PBDL models are kind of natural extension of that for labeled Markov processes, we imagine that if we could add L0 somehow to our PBDL, then our new logic should be able to characterize this new notion of state by simulation for PBDL models. The difficulty here is that L0 is a boolean value logic while our PBDL is essentially real valued. So what we did is that we simply add a new constructor bigger than r for our PBDL boolean functions. And then we can express the yellow threshold check via the new PBDL plus function in green here. Essentially, it says that after executing p, the probability, the expected value of function f is bigger than r. And since f basically denotes the formula phi, so the green part basically does the threshold check as we said before for the yellow part. And here we have this new logic PBDL plus and one direction is kind of easy, namely state by simulation implies logical equivalence. The non-trivial direction here is PBDL plus equivalence implies state by similarity. We solved this by using some proof techniques which they used to prove that this L0 categorizes state by simulation. And this gives us our second theorem, namely PBDL plus categorizes state by similarity for our PBDL models. Finally, we come to the event by simulation. So the notion of event by simulation is a bit more evolved than the notion of state by simulation because it is no longer following the idea of state to state transition. Spelling out the definition and we still start from labeled marker processes. So a labeled marker process as state event by simulation is a sub-sigma algebra of the measurable space on x such that if we replace lambda with, replace sigma x with lambda, the new type of we have is still a labeled marker process. This definition comes from the observation that for a relation to be R to be a state by simulation is equivalent to having this typo to be a labeled marker process while this sigma R is the sigma algebra generated by the binary relation R. This notion of event by simulation is interesting because it has several advantages over state by simulation. First of all, perhaps one of the most important one is that the characterization is no longer required in electricity requirements. For state by simulation, this is necessary in order to prove that state by simulation has transitivity. But for event by simulation, this is no longer a problem. And secondly, from a categorical point of view, this notion of event by simulation arise naturally from cospens. Our notion of event by simulation is also a kind of natural extension of that for labeled marker process by taking into account the evaluation for primitive functions. So, apart from being a labeled marker of process event by simulation, we require that this sub-sigma algebra lambda to be compatible with our evaluation in the sense that the inverse image of any value is still an element in this new sigma algebra. So, in order to find an extension of PBDL, which categorizes this new notion of event by simulation, we still go back to AL0 categorization results. So, the previous AL0 not only categorizes state by simulation, but also categorizes event by simulation for labeled marker processes. So, somehow our PBDL is sharp, so it also expresses the threshold check. But since event by simulation is somehow a relatively weak notion of by simulation, our logic PBDL plus turned out to be too strong to be invariant under such by simulation. So, we need to add this threshold check in a more restricted way. There are more than one solution, but what we did is that we allow only new functions of the form diamond PB bigger than R rather than arbitrary F on the left hand side, where B is a Boolean function. So that we avoid over complicated structure on the left hand side of this constructor. And using some proof techniques of the AL0 categorizing results, we can finally show our third theorem named PBDL sharp categorizes event by simulation for PBDL models. Now, to summarize, in our paper, we first proved that the original PBDL categorizes trace equivalence for PBDL models. After that, we go on to introduce states and event by simulation based on the notions for labeled marker processes. These two notions, I think, are quite natural and reasonable generalization. And finally, we finally extended PBDL by incorporating those threshold checks in the AL0 and give categorization results for these two notions of by simulation for PBDL models. So, after this, we have the future work. First of all, we can have, we can, we do need to check some computational properties of our extended PBDL. Logics, for example, is that they still have small model property. And what is the, what is the, what is the complexity of doing, for example, model checking on it. And second, we have, we have a bigger plan of trying to prove some handsomina style results for generic co-algebraic dynamic logic. So, co-algebraic dynamic logic was proposed partly to prove the completeness of game logic. And actually it is a very generic picture, which includes PDL, PDL with neighborhood semantics, and also this PBDL. We have already some primitive results. For example, if, if certain distribution properties like, like, like this hold in co-algebraic dynamic logic, then our proof for trace equivalence of PBDL, our, our categorization for trace equivalence can actually work to prove that these logics also categorizes trace equivalence. Besides, we have also shown that certain notion of by simulation always preserves a logic logical equivalence of co-algebraic dynamic logic. While the inverse is obviously not true. For example, by simulation between PBDL models. For example, PBDL, like, PBDL equivalence does not imply states nor events by simulation between PBDL models. So this is basically the bigger picture that we try, we try to do. And this paper is definitely the first step towards such a goal. Thank you.