82 lines
3.8 KiB
Markdown
82 lines
3.8 KiB
Markdown
We shall only consider finite processes (processes without recursive definitions)
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- a limited handling of recursion is possible
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- deciding bisimilarity for general processes is undecidable
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Inference system = axioms + inference rules
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- soundness: whatever I infer is correct (i.e., bisimiar)
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- completeness: whatever is bisimilar, it can be inferred
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#### Axioms & Rules for Strong Bisimilarity
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just what we've seen so far, literally.
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quite obvious.
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basically we can let the left or the right process evolve, leaving the other unchanged, or they can synchronize.
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- if a process does not perform any action, a restriction won't do anything
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- restriction is distributive
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- if the process does not do the restricted action, the restriction won't do anything
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- what if we restrict on the actual action?
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- well, then the process becomes 0
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- if $P = P \land P = Q => Q = P$ (it's transitive :o)
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- the same happens with contexts
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#### Soundness theorem: $\vdash P=Q \implies P \sim Q$
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*Proof:*
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If $\vdash LHS=RHS$, we need to consider the relation $\{ LHS,RHS \} \cup Id$ and prove it's a bisimulation (spoiler: it is).
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Since bisimilarity is an equivalence and a congruence, the inference rules holds.
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>[!def] Standard form
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$P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ and $\forall_{i}P_{i}$ is in standard form.
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**Lemma:** $\forall P \space \exists \space P'$ in standard form such that $\vdash P = P'$
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*Proof:* by induction on the structure of P.
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**Base case:** $P \triangleq 0$. It suffices to consider $P' \triangleq 0$ and conclude reflexivity.
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**Inductive step:** we have to consider three cases.
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1. $P \triangleq P_{1}|P_{2}$. By induction, we have that $\exists P_{1}',P_{2}'$ in standard form such that $\vdash P_{1}=P_{1}'$ and $P_{2}=P_{2}'$.
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By context closure, $\vdash P_{1}|P_{2}=P_{1}'|P_{2}$ and $\vdash P_{1}'|P_{2}=P_{1}'|P_{2}'$
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replacing one by one every continuation with its standard form, obtaining standard form.
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### Axioms & Rules for Weak Bisimilarity
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#### Example
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A server for exchanging messages, in its minimal version, receives a request for sending messages and delivers the confirmation of the reception
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Specification: $$Spec \triangleq send.\overline{rcv}$$
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The behavior of such a server can be implemented by three processes in parallel:
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- one handles the button for sending
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- another one effectively sends the message (through the restricted action *put*) and waits for the signal of message reception (through the restricted action *go*)
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- the last one gives back to the user the outcome of the sending
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$$S \triangleq send.\overline{put} \quad M \triangleq put.\overline{go} \quad R \triangleq go.\overline{rcv}$$
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$$Impl \triangleq (S|M|R)\setminus\{put, go\}$$
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exercise: prove the weak bisimilarity
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Let us consider the parallel of processes M and R, by using the axiom for parallel, we have $$\vdash M|R=put.(\overline{go}|R)+go.(M|\overline{rcv})$$
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By using the same axiom to the parallel of the three processes, we obtain
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$$\vdash S|(M|R)=send.(\overline{put}|(M|R))+put.(\overline{go}|R|S)+go.(\overline{rcv}|S|M)$$
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By restricting *put* and *go*, and by using the second axiom for restriction, we have that:
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We now apply the third axiom for restriction to the three summands:
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