178 lines
No EOL
8.5 KiB
Markdown
178 lines
No EOL
8.5 KiB
Markdown
- Up to now, we have considered non-deterministic processes
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- Two main features are missing for modeling a concurrent system:
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- Simultaneous execution of processes
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- Inter-process interaction
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- Solutions adopted:
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- Parallel composition, with interleaving semantics
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- Producer/consumer paradigm
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>[!note] Suggested reading
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>https://en.wikipedia.org/wiki/Calculus_of_communicating_system
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Given a set of names N (that denote events)
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- $a (\in N)$ denotes consumption of event a
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- $\bar{a}$ (for $a \in N$) denotes production of event a
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- $a$ and $\bar{a}$ are complementary actions: they let two parallel processes synchronize on the event a
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When two processes synchronize, an external observer has no way of understanding what is happening in the system
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- synchronization is not observable from the outside; it produces a special ‘silent’ action, that we denote with τ
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The set of actions we shall consider is: $N \cup \bar{N} \cup \{\tau\}$
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It is also useful to force some processes of the system to synchronize between them (without the possibility of showing to the outside some actions)
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The restriction operator P\a restricts the scope of name a to process P (a is visible only from within P)
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This is similar to local variables in a procedure of an imperative program
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The set of CCS processes is defined by the following grammar:
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$$P::=\sum_{i \in I} \alpha _{i}.P_{i} \space |\space A(a_{1}\dots a
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_{n})\space|\space P|Q \space | \space P\setminus a$$
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Dove:
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- **$\alpha _{i}.P_{i}$:** the process $P_{i}$ can perform the action $a$ and continue as the process $P_{i}$
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- $A(a_{1}\dots a_{n})$ parametric call to a process
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- $P|Q$ parallel composition: P and Q are in a concurrent execution and can communicate to each-other
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- $P\setminus a$ as described before, is the process P without $a$ (the action is hidden / visible only internally by P)
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**Recap:**
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- `P ⟶ᵃ P'` means: _Process `P` can perform action `a` and become `P'`_
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- `α` is an action: input (`a`), output (`ā`), or internal (`τ`)
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- `{a₁/x₁, ..., aₙ/xₙ}` is a **substitution** (used in process definitions with parameters)
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#### 📜 Inference rules
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##### 1. **Choice (Summation Rule)**
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$${\sum_{i \in I} \alpha_i . P_i \xrightarrow{\alpha_j} P_j} \quad \text{for all } j \in I$$
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> A process that has multiple options (`α₁.P₁ + α₂.P₂ + ...`) can non-deterministically choose any one of them.
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##### 2. **Process Definition (Call/Expansion)**
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$$\frac{P\{a_1/x_1, \dots, a_n/x_n\} \xrightarrow{\alpha} P'}{A(a_1, \dots, a_n) \xrightarrow{\alpha} P'} \quad A(x_1, \dots, x_n) \triangleq P$$
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> If a process `A(x₁...xₙ)` is defined as `P`, then invoking `A(a₁...aₙ)` behaves like `P` with the arguments substituted.
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##### 3. **Restriction (Hiding)**
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$$\frac{P \xrightarrow{\alpha} P'}{P \setminus a \xrightarrow{\alpha} P' \setminus a} \quad \text{if } \alpha \notin \{a, \bar{a}\}$$
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> If process `P` can do action `α` and it’s **not** the hidden action (`a` or `ā`), then `P \ a` can also do it, and the resulting process remains restricted.
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##### 4. **Parallel Composition – Independent Actions**
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There are three rules here for different situations in `P₁ | P₂`.
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###### a. **Left process makes a move:**
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$$\frac{P_1 \xrightarrow{\alpha} P_1'}{P_1 \mid P_2 \xrightarrow{\alpha} P_1' \mid P_2}$$
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> If `P₁` makes a move alone, the parallel composition can reflect it.
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###### b. **Right process makes a move:**
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$$\frac{P_2 \xrightarrow{\alpha} P_2'}{P_1 \mid P_2 \xrightarrow{\alpha} P_1 \mid P_2'}$$
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> Symmetrical to the previous: `P₂` makes the move.
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###### c. **Synchronization rule (communication):**
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$$\frac{P_1 \xrightarrow{a} P_1' \quad P_2 \xrightarrow{\bar{a}} P_2'}{P_1 \mid P_2 \xrightarrow{\tau} P_1' \mid P_2'}$$
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> If one process can **send** (`a`) and the other can **receive** (`ā`), they **synchronize**, and the result is an **internal (τ)** action. This models communication.
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##### 🧠 Summary Table:
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| Rule Type | Description |
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| ---------------------- | ----------------------------------------------------------- |
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| Choice | Select one summand to act |
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| Process Call | Expand definition by substituting parameters |
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| Restriction | Only allows transitions not involving the restricted action |
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| Parallel (independent) | A single component acts, the other stays the same |
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| Parallel (sync) | Two components synchronize to produce τ |
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#### Example
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#### Example, but explained
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##### 📘 Definitions
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We start with the following **process definitions**:
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- $A \triangleq a.A'$
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- $A' \triangleq \bar{b}.A$
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- $B \triangleq b.B'$
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- $B' \triangleq \bar{c}.B$
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Our **initial process** is: $A \mid B$
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##### ▶️ Step 1: Transition from A
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From the definition:
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- $A \triangleq a.A'$
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- So we can do: $A \xrightarrow{a} A'$ (A consumes a and becomes A')
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Using the **parallel rule** for the left-hand side:
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$$\frac{A \xrightarrow{a} A'}{A \mid B \xrightarrow{a} A' \mid B}$$
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✅ **First transition:** $A \mid B \xrightarrow{a} A' \mid B$
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##### ▶️ Step 2: Synchronization: $\bar{b}$ and $b$
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We now have:
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- Left process: $A' \triangleq \bar{b}.A$
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- Right process: $B \triangleq b.B'$
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From this, we can do:
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- $A' \xrightarrow{\bar{b}} A$
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- $B \xrightarrow{b} B'$
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These actions **complement each other**, so we can apply the **synchronization rule**:
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$$\frac{A' \xrightarrow{\bar{b}} A \quad B \xrightarrow{b} B'}{A' \mid B \xrightarrow{\tau} A \mid B'}$$
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✅ **Second transition:**
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$$A' \mid B \xrightarrow{\tau} A \mid B'$$
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##### ▶️ Step 3: Transition from B'
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From the definition:
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- $B' \triangleq \bar{c}.B$
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- So: $B' \xrightarrow{\bar{c}} B$
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Now use the **right parallel rule**:
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$$\frac{B' \xrightarrow{\bar{c}} B}{A \mid B' \xrightarrow{\bar{c}} A \mid B}$$
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✅ **Third transition:**
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$$A \mid B' \xrightarrow{\bar{c}} A \mid B$$
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##### Synchronization
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In the image above, we can observe a τ-transition due to synchronization between two complementary actions, in fact:
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- A' produces b and becomes A
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- B consumes b and becomes B'
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Since $\bar{b}$ and $b$ are complementary actions, they can **synchronize**, resulting in a **silent action** $\tau$ (synchronization rule):
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$$\frac{A' \xrightarrow{\bar{b}} A \quad B \xrightarrow{b} B'}{A' \mid B \xrightarrow{\tau} A \mid B'}$$
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### LTS and parallelism
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In the construction of the LTS we loose the consciousness of the parallel
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- it is indeed possible, by having the new set of actions, to obtain the previous LTS through the syntax we considered last class.
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The usefulness of the parallel is two-fold:
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- it is the fundamental operator in concurrency theory
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- it allows for a compact and intuitive writing of processes.
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### Restriction
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What if we restrict on b?
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##### Example 1:
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since $a$ is not in $\{b, \bar{b}\}$, we will see the transitions:
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$$
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\frac{\frac{\frac{a.A' \xrightarrow{a} A' \quad A \triangleq a.A'}{A \xrightarrow{a} A'}}
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{{A \mid B \xrightarrow{a} A' \mid B}}a \not \in \{b, \bar{b}\}}{{(A \mid B) \setminus b \xrightarrow{a} (A' \mid B) \setminus b}}
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$$
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Split in separate peaces for better readability:
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$$\frac{a.A' \xrightarrow{a} A' \quad A \triangleq a.A'}{A \xrightarrow{a} A'}
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\quad
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\frac{A \xrightarrow{a} A'}{A \mid B \xrightarrow{a} A' \mid B}
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\quad
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\frac{A \mid B \xrightarrow{a} A' \mid B \quad a \notin \{b, \bar{b}\}}{(A \mid B) \setminus b \xrightarrow{a} (A' \mid B) \setminus b}$$
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##### Example 2:
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since $b$ is in $\{b, \bar{b}\}$, we will NOT see the transitions:
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$$\frac{\frac{\frac{b.B' \xrightarrow{b} B' \quad B \triangleq b.B'}{B \xrightarrow{b} B'}}
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{{A \mid B \xrightarrow{b} A \mid B'}}b \in \{b, \bar{b}\}}{{(A \mid B) \setminus b}}$$
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This time I won't split it in separate pieces since I don't feel like doing it, just buy glasses.
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###### So... What if we restrict on b?
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- The effect of the restriction on b is that we have deleted the transitions involving b (hide all transitions labelled with $b$ and $\bar{b}$).
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- Notice that the τ, even if it has been generated by synchronizing on b, it is still present after applying the restriction on b!
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- the purpose of the τ is exactly to signal that a synchronization has happened but to hide the event on which the involved processed synchronized.
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- In general, it is possible that whole states disappear upon restriction of some names: this would be the case, e.g., if we consider the LTS arising from $(A’ | B)\setminus a,b$ (restricting on both a and b): |