Clasificación de los presupuestos

For now we only consider finite PAs i.e. only contain finite states. In this section we will show that these results also apply for PAs with countable states. Assume S is a countable set of states S. We adopt the method used in (56) to deal with strong branching bisimulation since all the other cases are similar. First we recall some standard notations from topology theory. Given a metric space (S, d) where d is a metric, a sequence {s_i | i ≥ 0} converges to s iff for any ǫ > 0, there exists n such that

PCTL

Figure 4.7: Relationship of different preorders in strong scenario.

d(s_m, s) < ǫ for any m ≥ n. A metric space is compact if every infinite sequence has a convergent subsequence.

Below follows the definition of metric over distributions from (56).

Definition 38 (Metric). Given two distributions µ, ν ∈ Dist(S), the metric d is defined by

d(µ, ν) = Sup_C∈S|µ(C) − ν(C)| .

Since the metric is defined over distributions while in Definition 28 we did not consider distributions explicitly, thus we need to adapt the definition of Prob_π,s(C, C^′, n) in the following way: s==⇒ µ iff either^n,C

w^b PCTL_{\ X}

w PCTL^∗

\ X

P

\ \

\

\

Figure 4.8: Relationship of different preorders in weak scenario.

• µ = δ_s, or

• s −→ ν such that

X

∀r∈Supp(ν).r====⇒^n−1,C νr

ν(r) · ν_r = µ.

It is obvious that for each π, C, C^′, and n, there exists s==⇒ µ such that µ(C^{n,C ′}) = Prob_π,s(C, C^′, n).

Now we can define the compactness of probabilistic automata as in (56) with a slight difference.

Definition 39 (Compactness). Given a probabilistic automaton P, P is i-compact iff {µ | s==⇒ µ}^i,C

is compact under metric d for each s ∈ S and ∼^b_i closed set C.

As mentioned in (56,87), the convex closure does not change the compactness, thus we can extend==⇒ to allow combined transitions in a standard way without changing^n,C anything, but for simplicity we omit this. A probabilistic automaton is compact iff it is i-compact for any i ≥ 1.

We introduce the definition of capacity as follows.

Definition 40 (Capacity). Given a set of states S and a π-algebra B, a capacity on B is a function Cap : B → (R⁺∪ {0}) such that¹:

1R⁺ is the set of positive real numbers.

1. Cap(∅) = 0,

2. whenever C₁ ⊆ C₂ with C₁, C₂∈ B, then Cap(C₁) ≤ Cap(C₂),

3. whenever there exists C₁ ⊆ C2 ⊆ . . . such that ∪i≥1C_i = C, or C₁ ⊇ C2 ⊇ . . . such that ∩_i≥1C_i= C, then

i→∞lim Cap(C_i) = Cap(C).

A capacity Cap is sub-additive iff

Cap(C₁∪ C₂) ≤ Cap(C₁) + Cap(C₂) for any C1, C2 ∈ B.

Different from (56), the value of Prob_π,s(C, C^′, n) depends on both C and C^′. Let PreCap^C_s,n(C^′) = Sup_πProb_π,s(C, C^′, n),

PostCap^C_s,n^′(C) = Sup_πProb_π,s(C, C^′, n)

i.e. given a C^′, PreCap^C_s,nwill return the maximum probability from s to C^′ in at most n steps via only states in C, similar for PostCap^C_s,n^′. The following lemma shows that both PreCap^C_s,n and PostCap^C_s,n^′ are sub-additive capacities.

Lemma 25. PreCap^C_s,n and PostCap^C_s,n^′ are sub-additive capacities on B where B is the π-algebra only containing ∼^b_i closed sets.

Proof. Refer to the proof of Lemma 5.2 in (56).

Now we can show that the following results are still valid as long as the given probabilistic automaton is compact even when it contains infinitely countable states.

Theorem 33. Given a compact probabilistic automaton, 1. ∼^b_n = ∼_PCTL−

n,

2. there exists n ≥ 0 such that ∼^b_n = ∼PCTL.

Proof. 1. The proof of ∼^b_n ⊆ ∼PCTL⁻_n is similar with the proof of Theorem 21, and is omitted here. We prove that ∼PCTL⁻

n ⊆ ∼^b_n in the sequel following the proof of Theorem 6.10 in (56). Let

R= {(s, r) | s ∼_PCTL− n r},

we need to prove that R is a strong i-depth branching bisimulation. In order to do so, we need to prove that for any (s, r) ∈ R,

PreCap^C_s,n(C^′) = PreCap^C_r,n(C^′)

for each R closed sets C and C^′. Since both C and C^′ may be countable union of equivalence classes while each equivalence class can only be characterized by countable many formulas, therefore we have

C = ∪^∞_i=1(∩^∞_j=1C_i,j) and C^′ = ∪^∞_i=1(∩^∞_j=1C_i,j^′ )

where ∩^∞_j=1Ci,j corresponds the i-th equivalence class in C, and Ci,j corresponds the set of states determining by the j-th formula satisfied by i-th equivalence class, similar for ∩^∞_j=1C_i,j^′ and C_i,j^′ . Similar as (56), let

B_k = ∩^∞_j=1(∪^k_i=1C_i,j), A^l_k= ∩^l_j=1(∪^k_i=1C_i,j), B_k^′ = ∩^∞_j=1(∪^k_i=1C_i,j^′ ), A^′_k^l= ∩^l_j=1(∪^k_i=1C_i,j^′ ).

It is easy to see that B_k and B_k^′ are increasing sequences of R closed sets such that ∪^∞_k=1B_k= C, and ∪^∞_k=1B_k^′ = C^′, while A^l_k and A^′_k^l are decreasing sequences of R closed sets such that ∩^∞_l=1A^l_k = B_k and ∩^∞_l=1A^′_k^l = B_k^′. Both A^l_k and A^′_k^l only contain conjunction and disjunction of finite formulas, thus can be described by PCTL⁻_i . The following proof is straightforward due to s ∼_PCTL−

i r and Lemma25.

2. Suppose that ∼PCTL ⊂ ∼^b_n for any n ≥ 0 which means that there exists s and r such that s ∼^b_n r for any n ≥ 0, but s ≁PCTL r. As a result there exists C, C^′ and π such that

i→∞limProb_π,s(C, C^′, i) > 0, but there does not exist π^′ such that

i→∞lim Prob_π^′_,r(C, C^′, i) ≥ lim

i→∞Prob_π,s(C, C^′, i).

In other words,

i→∞lim Prob_π^′_,r(C, C^′, i) < lim

i→∞Probπ,s(C, C^′, i) for any π^′ which indicates that there exists n ≥ 0 such that

Prob_π^′_,r(C, C^′, n) < Probπ,s(C, C^′, n) for any π^′, therefore s ≁PCTL⁻

i r which contradicts with our assumption.

In a similar way we can extend the results of this section to strong bisimulations and weak bisimulations, we skip their proofs here. For the simulations, we need to do more work, since there may be uncountable many downward closed sets. We prove along the line of (57). The following lemma is similar as Lemma 5.1 in (57) with only slight differences: i) we consider downward closed sets instead of upward closed sets, ii) we do not require R to be a preorder, but these do not change the proof.

Lemma 26 (Lemma 5.1 (57)). Let R ⊆ S × S be a relation, and C ⊆ S be a R downward closed set, then C is a union of equivalence classes of ≡R where ≡R is the largest equivalence relation contained in R.

Given a R downward closed set C, we say C is finitely generated if there exists a finite set of equivalence classes of {C_i ∈ S/ ≡R}_i∈I such that C = ∪_i∈IC_i. Since the set of the equivalence classes in S/ ≡Ris countable, thus the set of finitely generated R downward closed set is also countable (57). The following lemma shows an alternative definition of ≺^b_i in Definition 33 where we only focus on finitely generated downward closed sets:

Lemma 27. A relation R ⊆ S ×S is a strong i-depth branching simulation with i ≥ 1 iff s R r implies that s ≺^b_i−1 r and for any finitely generated R downward closed sets C, C^′, and any scheduler σ, there exists σ^′ such that Prob_σ^′_,r(C, C^′, i) ≥ Prob_σ,s(C, C^′, i).

We write s ≺^b_i r whenever there is a strong i-depth branching simulation R such that s R r.

Proof. The proof is similar as the proof of Lemma 5.2 in (57). Let (≺^b_i)^′ denote the new definition, we need to prove that s ≺^b_i r iff s (≺^b_i)^′ r. Since finitely generated R downward closed sets are special cases of R downward closed sets, it is trivial to see that s ≺^b_i r implies s (≺^b_i)^′ r. We prove that s (≺^b_i)^′ r implies s ≺^b_i r by contradiction. Suppose that for any finitely generated R downward closed sets C, C^′ and σ, there exists σ^′ such that Prob_σ^′_,r(C, C^′, i) ≥ Prob_σ,s(C, C^′, i), but there exists Rdownward closed sets C, C^′ and σ such that Prob_σ^′_,r(C, C^′, i) < Probσ,s(C, C^′, i) for any σ^′. Let σ be a scheduler such that Prob_σ^′_,r(C, C^′, i) < Prob_σ,s(C, C^′, i) for any σ^′ and ǫ = Prob_σ,s(C, C^′, i) − Prob_σ^′_,r(C, C^′, i) > 0. According to Lemma 26, there exists sets of equivalences classes: {Cj ∈ S/ ≡R}j∈J and {C_k ∈ S/ ≡R}_k∈K such that C = ∪_j∈JC_i and C^′ = ∪_k∈KC_k where J, K are (infinite) sets of indexes. Define two sequences of finitely generated R downward closed sets:

{C≤j = ∪_j^′_∈J∧j^′_≤j | j ∈ J},

{C≤k = ∪_k^′∈K∧k^′≤k| k ∈ K}.

Obviously both Prob_σ,s(C, C_≤k, i) and Prob_σ,s(C_≤j, C^′, i) are monotone, non-decreasing and converge to Prob_σ,s(C, C^′, i) for any C and C^′. Therefore there exists j ∈ J and k ∈ K such that

Prob_σ,s(C≤j, C^′, i) > Prob_σ,s(C, C^′, i) − ǫ 4, and Probσ,s(C≤j, C≤k, i) > Probσ,s(C≤j, C^′, i) − ǫ