From 91a6dd8f0dd2201a6b5618466902cbb9e7f7526a Mon Sep 17 00:00:00 2001 From: Quentin Quadrat Date: Tue, 14 May 2024 01:41:37 +0200 Subject: [PATCH] Update doc and use light screenshots --- data/examples/Simple.json | 35 ++++++ doc/petri.md | 221 +++++++++++++++++++------------------ doc/pics/911.png | Bin 105154 -> 38547 bytes doc/pics/EventGraph01.png | Bin 11120 -> 14898 bytes doc/pics/EventGraphBar.png | Bin 15820 -> 0 bytes doc/pics/Graph01.png | Bin 19530 -> 7145 bytes doc/pics/TimedPetri01.png | Bin 15240 -> 10547 bytes 7 files changed, 149 insertions(+), 107 deletions(-) create mode 100644 data/examples/Simple.json delete mode 100644 doc/pics/EventGraphBar.png diff --git a/data/examples/Simple.json b/data/examples/Simple.json new file mode 100644 index 0000000..4c62e4d --- /dev/null +++ b/data/examples/Simple.json @@ -0,0 +1,35 @@ +{ + "revision": 3, + "type": "Timed Petri net", + "nets": [ + { + "name": "Timed Petri net", + "places": [ + { "id": 0, "caption": "P0", "tokens": 1, "x": 105, "y": 130 }, + { "id": 1, "caption": "P1", "tokens": 2, "x": 485, "y": 131 }, + { "id": 2, "caption": "P2", "tokens": 0, "x": 488, "y": 344 }, + { "id": 3, "caption": "P3", "tokens": 1, "x": 69, "y": 342 } + ], + "transitions": [ + { "id": 0, "caption": "T0", "x": 280, "y": 130, "angle": 0 }, + { "id": 1, "caption": "T1", "x": 353, "y": 227, "angle": 0 }, + { "id": 2, "caption": "T2", "x": 602, "y": 239, "angle": 0 }, + { "id": 3, "caption": "T3", "x": 237, "y": 342, "angle": 0 }, + { "id": 4, "caption": "T4", "x": 13, "y": 233, "angle": 0 } + ], + "arcs": [ + { "from": "P0", "to": "T0" }, + { "from": "T0", "to": "P1", "duration": 3 }, + { "from": "P1", "to": "T1" }, + { "from": "T1", "to": "P2", "duration": 5 }, + { "from": "P1", "to": "T2" }, + { "from": "T2", "to": "P2", "duration": 4 }, + { "from": "P2", "to": "T3" }, + { "from": "T3", "to": "P0", "duration": 5 }, + { "from": "T3", "to": "P3", "duration": 5 }, + { "from": "P3", "to": "T4" }, + { "from": "T4", "to": "P0", "duration": 4 } + ] + } + ] +} diff --git a/doc/petri.md b/doc/petri.md index a02a6f0..a3009b1 100644 --- a/doc/petri.md +++ b/doc/petri.md @@ -1,84 +1,99 @@ # What are Petri nets, timed Petri nets, timed event graph? -## Petri nets and timed Petri nets +## Petri nets and derived types Petri nets are one of several mathematical modeling languages for the description of distributed systems and synchronization of their processes. It is a class of discrete event dynamic systems. -A Petri net is a directed bipartite graph that has two types of nodes: places +- A Petri net is a directed bipartite graph that has two types of nodes: places and transitions respectively depicted as circles and rectangles. A place can contain zero or any positive number of tokens, usually depicted either as black -circles or as numbers. Arcs, depicted as arrows. An arc allows directing two -nodes of different types: either from a place to a transition or from a -transition to a place. In timed Petri nets, "place -> transition" arcs are -evaluated with a strictly positive duration value which simulates the time needed -by the place to perform the associated action. A transition is activated if all -places connected to it as inputs contain at least one token. When a transition -is activated a token is burnt from each of these places and added to each place -connected as outputs to this transition. - -Usually, the numbers of tokens in places depict the states of the systems, -tokens represent the resources of the system and transitions describe the -synchronization of resources (rendez-vous). Petri is another way to model state -machines. - -In above figure 1, there are 5 transitions (`T0, T1, T2, T3, T4`) and 4 -places (`P0, P1, P2, P3`). Places `P0` and `P1` have 1 token each, the place -`P1` has 2 tokens and the place `P3` has no token. The arc `T0 -> P1` has 3 -units of times to simulate the fact that `P1` will need this duration to perform -its action. Transitions `T0, T1, T2, T4` are activated but the transition `T3` is -not activated. - -**Note to developers concerning unique identifiers** - -In this editor, for a given type (place and transitions), unique identifiers are -unsigned integers `0 .. N`. Numbers shall be consecutive and without -"holes". This is important when generating graphs defined by adjacency -matrices: indices of the matrix will directly match unique identifiers and -therefore no lookup table is needed. To distinguish the type of node a `T` or -`P` char is also prepended to the number. Arcs have no identifier because their -relation to directing nodes is unique since this editor does not manage multi -arcs. Therefore, when deleting a place (or a transition) the latest place (or -latest transition) in the container gets the unique identifier of the deleted -element to guarantee consecutive identifiers without "holes". - -## Dynamic simulation - -When running the simulation with this editor, tokens will transit along the arcs of -the net. When transitions are activated, tokens are instantaneously burnt from -places and directly "teleported" to transitions. Then, they will move along the -arc from the transition to the place. This "travel" will hold the number of -seconds indicated by the arc. This symbolized the time needed for the Place to -perform its action. A fading effect will help to show you which arcs and -nodes are activated. +circles or as numbers. Arcs, depicted by arrows, allows directing two nodes of +different types: either from a place to a transition or from a transition to a +place. A transition is activated if all places connected to it as inputs contain +at least one token. When a transition is activated a token is burnt from each of +these places and added to each place connected as outputs to this +transition. Usually, the numbers of tokens in places depict the states of the +systems, tokens represent the resources of the system and transitions describe +the synchronization of resources (rendez-vous). + +- Coloured Petri net is a backward compatible extension. They allow tokens to have +a data value attached to them. This attached data value is called the token +color. + +- A Petri net is called a state machine if each transition has exactly one +upstream and one downstream place. + +- A Petri net is called an event graph if each place has exactly +one upstream and one downstream transition. + +- In timed Petri nets or timed event graphs, "place -> transition" arcs are evaluated with strictly positive duration values which simulate the times needed by places to perform their associated action. + +- GRAFCET (GRAphe Fonctionnel de Commande Etapes-Transitions in French, aka SFC +(Sequential Function Chart in English) are derived from Petri net for engineers +allowing the representation and the analysis of automata. They are particularly +well suited for systems with sequential evolution (which are decomposable into +steps). Places have zero or one tokens. As long as a place has a token, discrete +actions associated to it are performed (i.e. close the door, switch on the light +after a delay of X ms ...). Contrary to Petri net where transition can be see +with a boolean experssion always set to true, GRAFCET transitions have boolean +expressions associated to them (i.e. button pressed and door is closed) and when +this expression is true it allows burning tokens of upstream places as descrived +for classic Petri nets. GRAFCET offers some extra features like reseting tokens +to a given configuration (forçage in French). + +### Example + +The above figure is a timed Petri net. + +![EventGraph](pics/TimedPetri01.png) + +*Fig 1 - Timed Petri net (made with this editor).* + +Tthere are 5 transitions (`T0, T1, T2, T3, T4`) and 4 places (`P0, P1, P2, P3`). Places `P0` and `P1` have 1 token each, the place `P1` has 2 tokens and the place `P2` has 0 token. Transitions `T0, T1, T2, T4` are activated (green) but the transition T3 is not activated (no color) since all of their upstream places have at least one token. +The arc `T0 -> P1` has 3 units of times to simulate the fact that `P1` will need this duration to perform its action. In [literature](https://www.rocq.inria.fr/metalau/cohen/SED/book-online.html) +timed event graphs are displayed with vertical bars inside places: they indicate +holding times of these places (in time units). In our net editor bars are displayed +as numbers on arcs going to places. -Notes: -- Duration of animations is not yet normalized, so a short duration will make - the GUI blink. -- To simulate a classic Petri net, simply set all duration in arcs to 0. -- Inputs generating periodically tokens are currently not implemented. For the -moment, during the simulation, you can place your mouse cursor to any place and -with `+` key you can add a token this will mimic input generating tokens. -- For outputs, simply add transitions without successors. Tokens will be -definitively deleted from the simulation. +Inputs of nets are transitions with no upstream places. Outputs of event +graphs are transitions with downstream places. In figure 1, there are no inputs +and no outputs. -## What decision is made when several transitions can burn tokens from the same place? +### What decision is made when several transitions can burn tokens from the same place? Question: In above figure 1, the place `P1` has two tokens and has two leaving arcs `P1 -> T1` and `P1 -> T2`. How transitions `T1` and `T2` will burn tokens in `P1` and therefore in which arc tokens will transit to? -Answer: Unless you want to simulate a system with concurrences between resources, +Answer: This is named `or-divergent branching`. Unless you want to simulate a system with concurrences between resources, this kind of net is a bad design and should be avoided when architecturing real systems since the execution of this kind of Petri net is nondeterministic: when multiple transitions are enabled at the same time, they will fire in any order. Therefore you should adapt your Petri net to define uniquely -the trajectory of the token. +the trajectory of the token. With GRAFCET, designers add mutual exclusion expression in receptivities (i.e. `T1` will be `"button1 pressed"` and `T2` will `"(button2 pressed) and (not button1 pressed)"`). + +## Type of net managed by this editor + +This editor allows to edit and simulate (timed) Petri nets, (timed) event graph +and partially GRAFCET. It does not manage coloured Petri nets (probably one day) neither +state-machines (because to fire we need to click on transitions and in this mode transitions will not be shown). If you are interested by state machines, you may be more +interested by a more modern form [Harel's +statecharts](https://www.itemis.com/en/products/itemis-create/documentation/user-guide/overview_what_are_state_machines) +or better +[synccharts](http://www-sop.inria.fr/members/Charles.Andre/CA%20Publis/Cesa96/SyncCharts_Cesa96.pdf) +a graphical subset of the +[Esterel](https://www.college-de-france.fr/sites/default/files/documents/gerard-berry/UPL8106359781114103786_Esterelv5_primer.pdf) +language (i.e. used in Airbus). + +When running timed Petri net simulations with this editor, tokens will transit along the arcs of the net. When transitions are activated, tokens are instantaneously burnt from places and directly "teleported" to transitions. Then, they will move along the +arc from the transition to the place. This "travel" will hold the number of +seconds indicated by the arc. This symbolized the time needed for the Place to +perform its action. A fading effect will help to show you which arcs and +nodes are activated. -A nondeterministic execution policy is made at the discretion of the developer of -the editor and you will have different behavior depending on how the editor has -been developed. Currently in our case, initially, the order when iterating on +A non-deterministic execution policy is made at the discretion of editor developers and you will have different behavior depending on how the editor has been developed. Currently in our case, initially, the order when iterating on transitions and arcs only depends on their order of creation but now we randomize. When several places are fired, the maximum possible of tokens will be burned within a single step of the animation cycle but, internally, we iterate @@ -88,7 +103,20 @@ Therefore, in this particular example, since `T1` has been created before `T2` 1st token will go to `T1` and the second will go to `T2`. If `P1` had a single token, in early `T1` will always be chosen but in the latest version, we randomize. -# What are Timed Graph Events? +### Note concerning node identifiers + +In this editor, for a given type (place and transitions), unique identifiers are +unsigned integers `0 .. N`. Numbers shall be consecutive and without +"holes". This is important when generating graphs defined by adjacency +matrices: indices of the matrix will directly match unique identifiers and +therefore no lookup table is needed. To distinguish the type of node a `T` or +`P` char is also prepended to the number. Arcs have no identifier because their +relation to directing nodes is unique since this editor does not manage multi +arcs. Therefore, when deleting a place (or a transition) the latest place (or +latest transition) in the container gets the unique identifier of the deleted +element to guarantee consecutive identifiers without "holes". + +## Timed Graph Events A timed event graph is a subclass of Petri net in which all places have a single input arc and a single output arc. This property allows to forbid, for example, @@ -104,33 +132,34 @@ systems as explained in the next sections. *Fig 2 - Event Graph (made with this editor).* -Inputs of event graphs are transitions with no predecessor. Outputs of event -graphs are transitions with no successor. In figure 2, there are no inputs -and no outputs. +### Compact form of timed event graphs -## Other look of event graph in literature +Thanks to the property of event graphs in which places have a single input arc and single arc, another way to represent event graphs in a more compact form, is to merge places with their unique incoming and unique out-coming arcs. From figure 2, we obtain the following figure 3, which is a more compact graph but equivalent. For example, the arc `5.00, P0(2)` means the place `P0` with 2 tokens and 2 units of time for the arc `T0 T2`. -In [literature](https://www.rocq.inria.fr/metalau/cohen/SED/book-online.html) -timed event graphs are displayed with vertical bars inside places: they indicate -holding times of these places (in time units). In our net editor bars are displayed -as numbers on arcs going to places. For example, this net: +![Graph](pics/Graph01.png) -![EventGraph](pics/EventGraphBar.png) +*Fig 3 - A compact form of figure 2 (made with this editor).* -*Timed Event Graph with vertical bars.* +Since, graphs can be represented by adjacency matrices, and since, arcs hold two information (duration and tokens), event graphs can be represented by two matrices (generally sparse): one matrix for duration `N` and the second matrix for tokens `T`. And since, event graphs have good properties with the (max,+) algebra, this editor can generate this kind of matrix directly in this algebra which can be used by the [MaxPlus](https://github.com/Lecrapouille/MaxPlus.jl) +Julia package. -is the same than this net: +``` + | . . 2 . | | . . 5 . | + | 0 . . . | | 5 . . . | +T = | . 0 . 0 |, N = | . 3 . 1 | + | 0 . . . | | 1 . . . | +``` -![EventGraph](pics/EventGraph.png) +Let suppose that matrix indices start from `0`, then `T[0,2]` holds 2 tokens and `N[0,2]` holds the duration 5. The `[0,2]` means the arc `T0 -> T2` in the compact form (or arcs `T0 -> P0` and `P0 -> T2` in the classic Petri form). -*Timed Event Graph (made with this editor).* +Note that origin and destination are inversed, this is because the matrix convention is generally the following: `M . x` with `x` a column vector. This editor can generate some Julia script. Note that `ε` in (max,+) algebra means that there is no existing arc. -## Dater and Counter Form +### Dater and Counter Form Event graphs represent when system events are occurring. We have two different way to represent them: the counter form, and the dater form. -- The counter form of the event graph in figure 2 is: +The counter form of the event graph in figure 2 is: ``` T0(t) = min(0 + T3(t - 1), 0 + T1(t - 5)) T1(t) = min(0 + T2(t - 3)) @@ -140,7 +169,7 @@ T3(t) = min(0 + T2(t - 1)) where `t - 5`, `t - 3` and `t - 1` are delays implied by duration on arcs and `min(2 +` and `min(0 +` implied by tokens from incoming places. `0 +` are just here to highlight the absence of tokens. -- The dater form of the event graph in figure 2 is: +The dater form of the event graph in figure 2 is: ``` T0(n) = max(1 + T3(n - 0), 5 + T1(n - 0)) T1(n) = max(3 + T2(n - 0)) @@ -170,7 +199,7 @@ more friendly than the counter form for two reasons: - thanks to the canonical form (explained in the next section) delays can simplify be either 0 or 1. -- The counter form of the event graph in figure 2 is: +Therefore, the counter form of the event graph in (max,+) algebra is: ``` T0(t) = T3(t - 1) ⨁ T1(t - 5) T1(t) = T2(t - 3) @@ -178,15 +207,15 @@ T2(t) = 2 T0(t - 5) T3(t) = T2(t - 1) ``` -- The dater form of the event graph in figure 2 is: +The dater form of the event graph in (min,+) algebra is: ``` -T0(n) = 1 T3(n - 2) ⨁ 5 T1(n - 0) +T0(n) = 1 T3(n - 2) ⨁ 5 T1(n - 0) T1(n) = 3 T0(n - 0) T2(n) = 5 T0(n - 2) T3(n) = 1 T2(n) ``` -## Canonical Event Graph +### Canonical Event Graph An event graph is said canonical when all places have at most one token and when places after an input transition (meaning a transition without predecessor) have @@ -202,14 +231,14 @@ been transferred to the newly created `P5` place. ![CanonicalEventGraph](pics/CanoEventGraph01.png) -*Fig 3 - Canonical Event Graph (made with this editor).* +*Fig 4 - Canonical Event Graph (made with this editor).* A canonical event graph can directly be converted into an implicit dynamic linear systems inside the (max,+) algebra (see section after) but since editing canonical net is fastidious and therefore the editor will deal it and let the user can directly manipulate the compact form of event graph. -## Implicit max-plus Dynamic Linear Systems +### Implicit max-plus Dynamic Linear Systems Canonical event graphs are interesting since their dater form can be modeled by an implicit dynamic linear system with the (max,+) algebra, which has the @@ -264,30 +293,8 @@ D = | . . |, A = | 2 4 |, B = | 1 |, C = | 3 . | ``` -These kinds of systems are interesting for real-time systems because they can show to the critical circuit of the system (in duration). The algorihm used is [Howard](http://www.cmap.polytechnique.fr/~gaubert/HOWARD2.html). This editor can show the critical circuit as shown in the next figure 5 where the circuit `T3, T0, T2, T1` will consume 13 units of time (5 + 5 + 3) for 2 tokens (in `P0`) and therefore 6.5 units of time by token (this information is for the moment displayed on the console). In this example there is a single connected components for the optimal policy. +These kinds of systems are interesting for real-time systems because they can show to the critical circuit of the system (in duration). The algorihm used is [Howard](http://www.cmap.polytechnique.fr/~gaubert/HOWARD2.html). This editor can show the critical circuit as shown in the next figure 5 where the circuit `T0, T2, T1` will consume 13 units of time (5 + 5 + 3) for 2 tokens (in `P0`) and therefore 6.5 units of time by token (this information is for the moment displayed on the console). In this example there is a single connected components for the optimal policy. ![Circuit](pics/Circuit01.png) -*Fig 5 - The critical circuit in orange (made with this editor).* - -## Compact form of timed event graphs - -Thanks to the property of event graphs in which places have a single input arc and single arc, another way to represent event graphs in a more compact form, is to merge places with their unique incoming and unique out-coming arcs. From figure 2, we obtain the following figure 3, which is a more compact graph but equivalent. For example, the arc `5.00, P0(2)` means the place `P0` with 2 tokens and 2 units of time for the arc `T0 T2`. - -![Graph](pics/Graph01.png) - -*Fig 3 - A compact form of figure 2 (made with this editor).* - -Since, graphs can be represented by adjacency matrices, and since, arcs hold two information (duration and tokens), event graphs can be represented by two matrices (generally sparse): one matrix for duration `N` and the second matrix for tokens `T`. And since, event graphs have good properties with the (max,+) algebra, this editor can generate this kind of matrix directly in this algebra which can be used by the [MaxPlus](https://github.com/Lecrapouille/MaxPlus.jl) -Julia package. - -``` - | . . 2 . | | . . 5 . | - | 0 . . . | | 5 . . . | -T = | . 0 . 0 |, N = | . 3 . 1 | - | 0 . . . | | 1 . . . | -``` - -Let suppose that matrix indices start from `0`, then `T[0,2]` holds 2 tokens and `N[0,2]` holds the duration 5. The `[0,2]` means the arc `T0 -> T2` in the compact form (or arcs `T0 -> P0` and `P0 -> T2` in the classic Petri form). - -Note that origin and destination are inversed, this is because the matrix convention is generally the following: `M . x` with `x` a column vector. This editor can generate some Julia script. 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