CTL Model Checking

Prof. P.H. Schmitt

Formal Systems II

Fixed Point Theory

Fixed Points Deﬁnition

Deﬁnition

Let f : P(G) → P(G) be a set valued function and Z a subset of G.
1. Z is called a fixed point of f if f(Z) = Z . 2. Z is called the least fixed point of f is Z is a fixed point and for all other fixed points U of f the relation Z ⊆ U is true.

3. Z is called the greatest fixed point of f is Z is a fixed point and for all other fixed points U of f the relation U ⊆ Z is true.

Finite Fixed Point Lemma

Let G be an arbitrary set, Let P(G) denote the power set of a set G. A function f : P(G) → P(G) is called monotone of for all X, Y ⊆ G

X ⊆ Y ⇒ f(X) ⊆ f(Y )

Let f : P(G) → P(G) be a monotone function on a ﬁnite set G.
1. There is a least and a greatest ﬁxed point of f.

2. _n≥1fⁿ(∅) is the least ﬁxed point of f.

3. _n≥1fⁿ(G) is the greatest ﬁxed point of f.

of part (2) of the ﬁnite ﬁxed point Lemma

Monotonicity of f yields

∅ ⊆ f(∅) ⊆ f²(∅) ⊆ . . . ⊆ fⁿ(∅) ⊆ . . .

Since G is ﬁnite there must be an i such that fⁱ(∅) = fⁱ⁺¹(∅).

Then Z = _n≥1fⁿ(∅) = fⁱ(∅) is a ﬁxed point of f:

f(Z) = f(fⁱ(∅)) = fⁱ⁺¹(∅) = fⁱ(∅) = Z

Let U be another ﬁxed point of f. From ∅ ⊆ U be infer by monotonicity of f at ﬁrst f(∅) ⊆ f(U) = U. By induction on n we conclude fⁿ(∅) ⊆ U for all n. Thus also Z = fⁱ(∅) ⊆ U.

Continuous Functions

Deﬁnition

A function f : P(G) → P(G) is called
1. ∪-continuous (upward continuous), if for every ascending sequence

M₁⊆ M₂⊆ . . . ⊆ M_n⊆ . . .

2. ∩-continuous (downward continuous) , if for every descending

sequence M₁⊇ M₂⊇ . . . ⊇ M_n⊇ . . .

Fixed Points For Continuous Functions

Let f : P(G) → P(G) be an upward continuous functions and g : P(G) → P(G) a downward continuous function.

The for all M, N ∈ P(G) such that M ⊆ f(M) and g(N) ⊆ N the following is true.

1. _n≥1fⁿ(M) is the least ﬁxed point of f containing M,

2. _n≥1gⁿ(M) is the greatest ﬁxed point of g contained in M.

Proof of (1)

By monotonicity we ﬁrst obtain

M ⊆ f(M) ⊆ f²(M) ⊆ . . . ⊆ fⁿ(M) ⊆ . . .

Let P = _n≥1fⁿ(M). This immediately gives M ⊆ P. Furthermore

f(P) = f( _n≥1fⁿ(M))

by continuity since f(M) ⊆ f²(M)

_n≥1fⁿ(M)

Assume now that Q is another ﬁxed point of f satisfying M ⊆ Q. By Monotonicity and the ﬁxed point property f(M) ⊆ f(Q) = Q and furthermore for every n ≥ 1 also fⁿ(M) ⊆ Q.

Thus we obtain P = _n≥1fⁿ(M) ⊆ Q

Knaster-Tarski-Fixed-Points Theorem

Let f : P(G) → P(G) be a monotone function. Then f has a least and a greatest ﬁxed point.
Note, G need not be ﬁnite.

Fixed Point

Let L = {S ⊆ G | f(S) ⊆ S}, e.g., G ∈ L.

Let U = L. Show f(U) = U!

For all S ∈ L by the property of an intersection: U ⊆ S. By monotonicity and deﬁnition of L f(U) ⊆ f(S) ⊆ S.

Thus f(U) ⊆ L = U and we have already established half of our claim. By monotonicity f(U) ⊆ U implies f(f(U)) ⊆ f(U) which yields f(U) ∈ L and futhermore U ⊆ f(U).

We thus have indeed U = f(U).

Least Fixed Point

Now assume W is another ﬁxed point of f,i.e., f(W) = W. This yields W ∈ L= {S ⊆ G | f(S) ⊆ S} and

U = L ⊆ W.

Thus U is the least ﬁxed point of f. Following the same line of argument one can show that

{S ⊆ G | S ⊆ f(S)} is the greatest ﬁxed point of f.

CTL Model Checking
Algorithm

Two Auxiliary Concepts

Let T = (S, R, v) be a transition system and F an CTL formula. The set

τ(F) = {s ∈ S | s |= F}

is called the characteristic region of F in T . The universal and existential next step functions f_AX, f_EX: P(S) → P(S) are deﬁned by

f_AX(Z) = {s ∈ S | for all t with sRt we get t ∈ Z} f_EX(Z) = {s ∈ S | there exists a t with sRt and t ∈ Z}

CTL Model Checking Algorithm

Let T = (S, R, v) be a transition system and F an CTL formula. τ(F) is computed by the following high-level recursive algorithm:

123456789

==========

{s ∈ S | v(s, p) = 1}

τ(F₁) ∩ τ)F₂)

τ(F₁∧ F₂) τ(F₁∨ F₂)

τ(¬F₁) τ(F₁) ∪ τ(F₂)

S \ τ(F₁)

τ(A(F₁U F₂)) τ(E(F₁U F₂)) τ(AFF₁) τ(EFF₁) lfp[τ(F₂) ∪ (τ(F₁) ∩ f_AX(Z))] lfp[τ(F₂) ∪ (τ(F₁) ∩ f_EX(Z)] lfp[τ(F₁) ∪ f_AX(Z)] lfp[τ(F₁) ∪ f_EX(Z)] gfp[τ(F₁) ∩ f_EX(Z)] gfp[τ(F₁) ∩ f_AX(Z)] τ(EGF₁)

10 τ(AGF₁)

Correctness Proof

Case 10 AG F₁

By deﬁnition

τ(AG F₁) = {s ∈ S | s ∈ τ(F₁) and h ∈ τ(AG F₁) for all h with gRh}

Using Deﬁnition of the universal next step function:

τ(AG F₁) = τ(F₁) ∩ f_AX(τ(AG F₁))

So, τ(AG F₁) is a ﬁxed point of the function τ(F₁) ∩ f_AX(Z). It remains to see that it is the greatest ﬁxed point.

Correctness Proof

Case 10 AG F₁(continued)

Let H be another ﬁxed point, i.e., H = τ(F₁) ∩ f_AX(H). We need to show H ⊆ τ(AG F₁)). Consider g₀∈ H with the aim of showing that for all n ≥ 0 and all g_isatisfying g_i−1Rg_ifor all 1 ≤ i ≤ n we obtain g_n∈ τ(F₁).

Observe H = τ(F₁) ∩ f_AX(H) ⊆ τ(F₁). Thus it suﬃces to show for all n ≥ 0 that g_n∈ H. For n = 0 that is true by assumptions. Assume g_n−1∈ H. g_n−1∈ H ⊆ f_AX(H) = {g | for all h with gRh we have h ∈ H}.

Thus g_n∈ H.

Correctness Proof

Case 6 E(F₁U F₂)

By deﬁnition

τ(E(F₁U F₂)) = {g ∈ S | s |= F₂or s |= F₁and

there exists h with gRh and h |= F₂}
Using next step function:

τ(E(F₁U F₂)) = τ(F₂) ∪ (τ(F₁) ∩ f_EX(τ(E(F₁U F₂)))

Thus τ(E(F₁U F₂)) is a ﬁxed point of the function

τ(F₂) ∪ (τ(F₁) ∩ f_EX(Z)).

It remains to show that it is the least ﬁxed point.

Correctness Proof

Case 6 E(F₁U F₂) (continued)

Consider H ⊆ S with H = τ(F₂) ∪ (τ(F₁) ∩ f_EX(H)). We need τ(E(F₁U F₂)) ⊆ H. Fix g₀∈ τ(E(F₁U F₂)), try to arrive at g₀∈ H. There is an n ∈ N and there are g_ifor 1 ≤ i ≤ n satisfying

1. g_iRg_i+1for all 0 ≤ i < n.

2. g_n∈ τ(F₂).

3. g_i∈ τ(F₁) for all 0 ≤ i < n.

We set out to prove g_n∈ H by induction on n.

n = 0 Since H = τ(F₂) ∪ (τ(F₁) ∩ f_EX(H)) ⊇ τ(F₂) and

g₀= g_n∈ τ(F₂).

n − 1 Ã n By induction hypothesis we have g₁∈ H Since g₀∈ τ(F₁) and g₀Rg₁we obtain g₀∈ (τ(F₁) ∩ f_EX(H)) ⊆ H and thus also g₀∈ H.

CTL Model Checking
Example

Transition System

s₀n₁n₂

CTL Model Checking

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support