Planeación Financiera

3.5.1 Weak Simulation on States

We first introduce the weak simulation on networks which can be seen as an one direc-tion weak bisimuladirec-tion defined in Definidirec-tion16.

Below follows the definition of weak simulation.

Definition 18 (Weak Simulation). A relation R ⊆ N × N is a weak bisimulation iff E R F implies that for each k and C_E,F,k, whenever

E ∝ C_E,F,k−→ µ,^αk there exists

F ∝ C_E,F,k ==⇒^αk _c µ^′ such that µ ⊑R µ^′.

Let E and F be weakly bisimilar, written as E w^M F , if there exists a weak simulation R such that E R F .

Lemma 14. E k C w^M F k C for any C provided that E w^M F . Proof. Similar with the proof of Lemma11 and is omitted here.

Theorem 16. w^M is a congruence and preorder.

Proof. We first prove that w^M is a preorder. The reflexivity is trivial, we only prove the transitivity here i.e. E w^M F and F w^M G implies that E w^M G. In order to do so, we need another definition of weak simulation, calledw^M₁ . The definition of w^M₁ is almost the same as w^M except that E ∝ C_E,F,k −→ µ is replaced by the weak^αk transition E ∝ C_E,F,k ==⇒^αk _c µ.

It can be proved that w^M = w^M₁ . It is easy to see that E w^M₁ F implies that E w^M F since E ∝ C_E,F,k −→ µ is a special case of E ∝ C^αk E,F,k αk

==⇒_c µ. We prove that E w^M F implies E w^M₁ F , it is enough to show that

R= {(E, F ) ∈ N × N | E w^M F }

is a weak simulation under the new definition. For simplicity we will omit the parameter C_E,F,k in the sequel. Suppose that E R F and E ==⇒^αk _c µ. If α_k = (x,L) ⊳ k, we need to prove that there exists F ==⇒^αk c µ^′ such that µ ⊑R µ^′. We are going to prove by induction on E ==⇒ µ. First assume that E^αk ==⇒ µ, there are two cases to be considered:^αk 1. E −→ µ^τ ₁ ==⇒ µ. Since E R F i.e. E^αk w^M F , there exists F ==⇒^τ _c µ^′₁ such that

µ₁ ⊑R µ^′₁. By induction there exists F ==⇒^τ c αk

==⇒ µ^′ such that µ ⊑R µ^′.

2. E −→ µ^αk 1 τ

=

=⇒ µ. Since E w^M F , there exists F ==⇒ µ^{τ ′}₁ such that µ₁ ⊑R µ^′₁. The following proof is similar with Clause 1, and is omitted here.

If α_k= ν ˜xhx,Li@l, there are also two cases:

1. E−→ µ^τ 1 αk

==⇒ µ. This case is similar with the first clause when α_k= (x,L) ⊳ k.

2. E−−−−−−−→ µ^{ν ˜xhx,L1i@l1} 1

ν ˜xhx,L2i@l2

========⇒ µ such that

(ν ˜xhx,L1i@l1) ⊗ (ν ˜xhx,L2i@l2) = α_k. Since E w^M F , there exists

F ========⇒^{ν ˜xhx,L1i@l1} _c µ^′₁ such that µ1 ⊑R µ^′₁. As a result there exists

F ========⇒^{ν ˜xhx,L1i@l1} _c========⇒^{ν ˜xhx,L2i@l2} _cµ^′ such that µ ⊑R µ^′.

When E ==⇒^αk _c µ, we know there exists {E==⇒^αk _c µ_i}1≤i≤n and {w_i}1≤i≤n such that P

1≤i≤nw_i= 1 and P

1≤i≤nw_i· µi = µ. Since we have proved that for each E==⇒ µ^αk _i, there exists F ==⇒^αk _c µ^′_i such that µ_i ⊑R µ^′_i, thus there exists F ==⇒^αk _c µ^′ such that µ ⊑R µ^′.

Since we have proved that w^M =w^M₁ , in order to show thatw^Mis a preorder, it is equivalent to prove thatw^M₁ is a preorder. Suppose that E w^M₁ F and F w^M₁ G, we prove that E w^M₁ G. According to the definition ofw^M₁ , there exists weak simulations R₁ and R₂ such that E R₁ F and F R₂ G. Therefore whenever E =====⇒^(x,L)⊳k _c µ₁, there

exists F =====⇒^(x,L)⊳k c µ2 and G =====⇒^(x,L)⊳k c µ3 such that µ1 ⊑R₁ µ2 and µ2 ⊑R₂ µ3. In other words, there exists ∆₁ and ∆₂ satisfying the conditions in Definition 7. Let

R= R₁◦ R2 = {(E^′, G^′) | ∃F^′.(E^′ R₁ F^′∧ F^′ R₂ G^′)},

we show that ∆ defined in this way does satisfy the conditions in Definition7. Condition one is easy since ∆(E, G) > 0 implies that there exists F such that ∆₁(E, F ) > 0 and

we prove that the second condition is satisfied too. The third condition is similar as the second one, and is omitted here. Therefore µ1 ⊑R µ3, this completes the proof.

Finally we prove that w^M is a congruence, it is enough to show that R= {(ν ˜x(E k G), ν ˜x(F k G)) | E w^M F }

is a weak simulation. The following proof is similar with Theorem 13, and is omitted here.

Since weak simulation is one direction weak bisimulation, and is strictly coarser than weak bisimulation, therefore there exists networks which are not weakly bisimilar with each other, but one network is able to simulation the other one, refer to the following example.

Example 33. Consider the networks E and F in Example 31 where we have shown that E 6≈ F , but it holds that F w^M E. The only non-trivial case is when

this can be simulated by the following weak transition of E: that the node at k can receive messages from other locations with positive probability i.e. k 7−→ m is not always equal to 0 for m 6= l.

3.5.2 Weak Simulation on Distributions

In this section we give the definition of weak simulation on distributions. Based on Definition 17, the weak simulation can be given in a straightforward way as follows:

Definition 19 (Weak Simulation on Distributions). A relation R ⊆ ND × ND is a weak simulation iff µ₁ R µ₂ implies that for each k and C_µ1,µ2,k, whenever Proof. Similar to the proof of Lemma 12, and is omitted here.

Lemma 16. If µ wd µ^′ where Supp(µ) = {Ei}1≤i≤n, then there exists

Proof. Similar to the proof of Lemma13, and is omitted here.

Similar with w^M, we can also show that wd is a congruent preorder.

Theorem 17. wd is a congruence and preorder.

Proof. We first prove thatwd is a preorder. The reflexivity is trivial, and we only prove the transitivity here i.e. E wd F and F wd G implies that E wd G. Similar as Theorem 16, we define another weak simulation, denoted as w^′_d, which is almost the same aswd except that (µ₁ ∝ Cµ1,µ2,k) −→^αk ρµ^′₁ is replaced by (µ₁ ∝ Cµ1,µ2,k)==⇒^αk _ρµ^′₁

is a weak simulation under the new definition. Again we will omit the parameter C_µ1,µ2,k for simplicity. Suppose that µ₁ Rµ₂ and µ₁==⇒^αk _ρµ^′₁, we need prove that there

such that

(ν ˜xhx,Mii@l) ⊗ (ν ˜xhx,Nii@l) = αk

for each 1 ≤ i < n, X

1≤i≤n

ρ_i= ρ and X

1≤i≤n

(ρ_i

ρ · µ^′_1i) = µ^′₁. The following proof is similar with Clause 1,and is omitted here.

3. α_k = τ . This case is similar with Clause 1, and is omitted here.

Since we have proved that wd = w^′_d, it is enough to show that w^′_d is a preorder.

The proof is similar with Theorem 16. The congruence of wd can also be proved in a similar way as Theorem 14 based on Lemma 15 and 16. These proofs are omitted here.

We give an example of weak simulation over distributions as follows:

Example 34. Suppose that we have two networks:

E = ⌊2 · p + 2 · (x) · q + 3 · q{y/x}⌋_l k ⌊0⌋_k, F = ⌊2 · p + 5 · (x) · q⌋_lk ⌊hy ⊲ li⌋_k.

Assume that in the given SMF M, the node at l can receive messages from k with probability 0.6. Let C = {{(0.6, k)} 7−→ l}, then we have

E k C wd F k C.

Intuitively, in E the node at k has received the message y, while in F the message y has not been received by l. Since (F k C)(k, l) = 0.6, the node at k can also receive y and evolve into ⌊q{y/x}⌋_k with probability ⁵₇ · 0.6 = ³₇, which is the same as in E, similarly for other cases.

As in the bisimulation setting, wd is strictly coarser thanw^M. Theorem 18. w^M ⊂wd.

Proof. It is easy to see that E w^M F implies E wd F given Definition 18 and 19. To show that E wd F does not always imply E w^M F , it is enough to give a counterexample. Note that Example31also works here, for the same reason E 6w^M F , but we can show E wd F .