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Concurrent Systems/notes/12 - CCS (che non so cosa voglia dire).md
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@ -39,36 +39,22 @@ Dove:
#### 📜 Inference rules #### 📜 Inference rules
##### 1. **Choice (Summation Rule)** ##### 1. **Choice (Summation Rule)**
$${\sum_{i \in I} \alpha_i . P_i \xrightarrow{\alpha_j} P_j} \quad \text{for all } j \in I$$
**$$∑_{i ∈ I} αᵢ.Pᵢ ⟶^{αⱼ} Pⱼ\space\space\spac, for \space all \space j \in I$$ > A process that has multiple options (`α₁.P₁ + α₂.P₂ + ...`) can non-deterministically choose any one of them.
> A process that has multiple options (`α₁.P₁ + α₂.P₂ + ...`) can nondeterministically choose any one of them.
---
### 2. **Process Definition (Call/Expansion)**
**`P{a₁/x₁ ... aₙ/xₙ} ⟶ᵅ P'`
`A(x₁...xₙ) ≜ P`
——————————————
`A(a₁...aₙ) ⟶ᵅ P'`**
##### 2. **Process Definition (Call/Expansion)**
$$\frac{P\{a_1/x_1, \dots, a_n/x_n\} \xrightarrow{\alpha} P'}{A(a_1, \dots, a_n) \xrightarrow{\alpha} P'} \quad A(x_1, \dots, x_n) \triangleq P$$
> If a process `A(x₁...xₙ)` is defined as `P`, then invoking `A(a₁...aₙ)` behaves like `P` with the arguments substituted. > If a process `A(x₁...xₙ)` is defined as `P`, then invoking `A(a₁...aₙ)` behaves like `P` with the arguments substituted.
--- ##### 3. **Restriction (Hiding)**
$$\frac{P \xrightarrow{\alpha} P'}{P \setminus a \xrightarrow{\alpha} P' \setminus a} \quad \text{if } \alpha \notin \{a, \bar{a}\}$$
### 3. **Restriction (Hiding)**
**`P ⟶ᵅ P'` and `α ∉ {a, ā}`
——————————————
`P \ a ⟶ᵅ P' \ a`**
> 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. > 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.
--- ##### 4. **Parallel Composition – Independent Actions**
$$\frac{P_1 \xrightarrow{\alpha} P_1'}{P_1 \mid P_2 \xrightarrow{\alpha} P_1' \mid P_2}$$
### 4. **Parallel Composition – Independent Actions** $$\frac{P_2 \xrightarrow{\alpha} P_2'}{P_1 \mid P_2 \xrightarrow{\alpha} P_1 \mid P_2'}$$
There are three rules here for different situations in `P₁ | P₂`. There are three rules here for different situations in `P₁ | P₂`.
--- ---