vault backup: 2025-04-14 15:25:00
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"title": "11 - LTSs and Bisimulation"
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"title": "12 - bastaaaaaaaa"
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"title": "class 11"
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"title": "class 11"
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@ -56,14 +56,14 @@ To let p0 be simulated by q0, we should have that p1 is simulated by q1 or q2.
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If S contained one among (p1,q1) or (p1,q2), then it would not be a simulation: indeed, p1 can perform both a c (whereas q1 cannot) and a b (whereas q2 cannot).
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If S contained one among (p1,q1) or (p1,q2), then it would not be a simulation: indeed, p1 can perform both a c (whereas q1 cannot) and a b (whereas q2 cannot).
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**Remark:** for proving equivalence, it is NOT enough to find a simulation of p by q and a simulation of q by p
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**Remark:** for proving equivalence, it is NOT enough to find a simulation of p by q and a simulation of q by p
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p0 is simulated by q0: $$S = {(p0, q0), (p1, q2), (p2, q3)}$$
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p0 is simulated by q0: $$S = {(p0, q0), (p1, q2), (p2, q3)}$$
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q0 is simulated by p0: $$S′ ={(q0,p0),(q1,p1),(q2,p1),(q3,p2)}$$
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q0 is simulated by p0: $$S′ ={(q0,p0),(q1,p1),(q2,p1),(q3,p2)}$$
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However, p0 and q0 are not bisimilar: the transition q0 -> a -> q1 is not bisimulable by any transition from p0 (ndeed, p0 –a–> p1 does not suffice, because p1 can perform a b and so cannot be bisimilar to q1).
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However, p0 and q0 are not bisimilar: the transition q0 -> a -> q1 is not bisimulable by any transition from p0 (ndeed, p0 –a–> p1 does not suffice, because p1 can perform a b and so cannot be bisimilar to q1).
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#### Bisimulation is NOT isomorfism
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#### Bisimulation is NOT isomorfism
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$S = \{(p0,q0), (p1,q1), (p2,q1), (p0,q2)\}$
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$S = \{(p0,q0), (p1,q1), (p2,q1), (p0,q2)\}$
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$S^{−1} = \{(q0,p0), (q1,p1), (q1,p2), (q2,p0)\}$
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$S^{−1} = \{(q0,p0), (q1,p1), (q1,p2), (q2,p0)\}$
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@ -101,10 +101,10 @@ So, (p', q') ∈ ∼
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Let (p,q) ∈ S. Then, there exists a bisimulation that contains the pair (p, q); thus, (p, q) ∈ ∼.
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Let (p,q) ∈ S. Then, there exists a bisimulation that contains the pair (p, q); thus, (p, q) ∈ ∼.
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## A syntax for LTSs
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## A syntax for LTSs
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The only ingredients we used to write down an LTS are:
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The only ingredients we used to write down an LTS are:
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- sequential composition (of an action and a process)
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- sequential composition (of an action and a process)
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@ -117,7 +117,7 @@ For every identifier (denoted with capital letters A,B,..), we shall assume a un
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Let us denote with P{b1/x1 . . . bn/xn} the process obtained from P by replacing name xi with name bi, for every i = 1,..., n.
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Let us denote with P{b1/x1 . . . bn/xn} the process obtained from P by replacing name xi with name bi, for every i = 1,..., n.
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>[!def]
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>[!def]
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>The set of non-deterministic processes is given by the following grammar:
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>The set of non-deterministic processes is given by the following grammar:
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@ -133,7 +133,9 @@ We shall usually omit tail occurrences of ‘.0’ and, for example, simply writ
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From the syntax to the LTS We have shown how it is possible, starting from an LTS, to generate a corresponding process
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From the syntax to the LTS We have shown how it is possible, starting from an LTS, to generate a corresponding process
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It is also possible the inverse translation and then the two formalisms do coincide; the rules that have to be used in this translation are:
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It is also possible the inverse translation and then the two formalisms do coincide; the rules that have to be used in this translation are:
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#### Examples
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examples
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@ -15,7 +15,7 @@ Given a set of names N (that denote events)
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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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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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- 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: 
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The set of actions we shall consider is: 
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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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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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@ -23,8 +23,8 @@ The restriction operator P\a restricts the scope of name a to process P (a is vi
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This is similar to local variables in a procedure of an imperative program
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This is similar to local variables in a procedure of an imperative program
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