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CTL Model Checking

Prof. P.H. Schmitt

  • Institut fur Theoretische Informatik
  • ¨

  • Fakultat fur Informatik
  • ¨
  • ¨

  • Universitat Karlsruhe (TH)
  • ¨

Formal Systems II

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 1 / 26

Fixed Point Theory

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 2 / 26

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.

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
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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.

S

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

T

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 4 / 26

Proof

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(∅).

S

Then Z = n≥1 fn(∅) = 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.

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 5 / 26

Continuous Functions

Definition

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

M1 ⊆ M2 ⊆ . . . ⊆ Mn ⊆ . . .

  • [
  • [

  • f(
  • Mn) =
  • f(Mn)

  • n≥1
  • n≥1

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

sequence M1 ⊇ M2 ⊇ . . . ⊇ Mn ⊇ . . .

  • \
  • \

  • f(
  • Mn) =
  • f(Mn)

  • n≥1
  • n≥1

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 6 / 26

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.

S

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

T

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
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Proof of (1)

By monotonicity we first obtain

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

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

S
S

f(P) = f( n≥1 fn(M))

S

n+1

=== P

f

(M)

by continuity since f(M) ⊆ f2(M)

n≥1

S

n≥1 fn(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.

S

Thus we obtain P = n≥1 fn(M) ⊆ Q

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 8 / 26

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.

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 9 / 26

Proof

Fixed Point

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

T

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.

T

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).

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 10 / 26

Proof

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

T

U = L ⊆ W.

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

S

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 11 / 26

CTL Model Checking
Algorithm

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
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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 fAX, fEX : P(S) → P(S) are defined by

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
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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

τ(p)

==========

{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(F1 U F2)) τ(E(F1 U F2)) τ(AFF1) τ(EFF1) lfp[τ(F2) ∪ (τ(F1) ∩ fAX(Z))] lfp[τ(F2) ∪ (τ(F1) ∩ fEX(Z)] lfp[τ(F1) ∪ fAX(Z)] lfp[τ(F1) ∪ fEX(Z)] gfp[τ(F1) ∩ fEX(Z)] gfp[τ(F1) ∩ fAX(Z)] τ(EGF1)

10 τ(AGF1)

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 14 / 26

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) ∩ fAX(τ(AG F1))

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 15 / 26

Correctness Proof

Case 10 AG F1 (continued)

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

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

Thus gn ∈ H.

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 16 / 26

Correctness Proof

Case 6 E(F1 U F2)

By definition

τ(E(F1 U F2)) = {g ∈ S | s |= F2 or s |= F1 and

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

τ(E(F1 U F2)) = τ(F2) ∪ (τ(F1) ∩ fEX(τ(E(F1 U F2)))

Thus τ(E(F1 U F2)) is a fixed point of the function

τ(F2) ∪ (τ(F1) ∩ fEX(Z)).

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

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 17 / 26

Correctness Proof

Case 6 E(F1 U F2) (continued)

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

1. giRgi+1 for all 0 ≤ i < n.

2. gn ∈ τ(F2).

3. gi ∈ τ(F1) for all 0 ≤ i < n.

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

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

g0 = gn ∈ τ(F2).

n − 1 Ã n By induction hypothesis we have g1 ∈ H Since g0 ∈ τ(F1) and g0Rg1 we obtain g0 ∈ (τ(F1) ∩ fEX(H)) ⊆ H and thus also g0 ∈ H.

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 18 / 26

CTL Model Checking
Example

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 19 / 26

Transition System

s0 n1n2

  • s1
  • s5
  • n1t2
  • t1n2

  • s3
  • s8

  • s2 c1n2
  • n1c2 s6
  • t1t2
  • t1t2

  • s4
  • s7

  • t1c2
  • c1t2

  • Prof. P.H. Schmitt
  • CTLMC

  • Summer 2009
  • 20 / 26

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  • Corecursion and Coinduction: How They Relate to Recursion and Induction

    Corecursion and Coinduction: How They Relate to Recursion and Induction

    Corecursion and coinduction: what they are and how they relate to recursion and induction Mike Gordon 1 Contents Preface 3 Summary 4 Data, codata, constructors and destructors ............... 4 Recursion and corecursion ......................... 5 Fixed points, induction, coinduction and bisimulation ......... 7 Overview of the rest of this article .................... 9 Natural numbers, Peano’s axioms and Peano structures 10 Algebras and coalgebras 11 Numbers and conumbers .......................... 12 Example of corecursion for numbers ................... 14 Lists and colists ............................... 15 Example of corecursion for lists ...................... 17 Coinduction and bisimulation 19 Bisimulation coinduction for conumbers ................. 20 Example of bisimulation coinduction for numbers ..... 20 Justification of bisimulation coinduction for numbers ... 21 Bisimulation coinduction for lists ..................... 22 Examples of bisimulation coinduction for lists ....... 22 Justification of bisimulation coinduction for lists ..... 27 F-algebras and F-coalgebras 28 Set theory notation ............................. 29 Numbers and lists ............................. 29 Morphisms ................................. 30 Initial and terminal algebras ........................ 32 Least and greatest fixed points 32 Fixed points, numbers and conumbers .................. 33 Summary of fixed point proof rules for numbers and conumbers .... 34 Fixed points, lists and colists ....................... 34 Summary of fixed point proof rules
  • On the Origins of Bisimulation and Coinduction

    On the Origins of Bisimulation and Coinduction

    On the Origins of Bisimulation and Coinduction DAVIDE SANGIORGI University of Bologna, Italy The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory. Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such as Knaster-Tarski) that underpin the fixed-point theory presented. Categories and Subject Descriptors: F.4.1 [Mathematical logic and formal languages]: Math- ematical Logic—Computational logic; Modal logic; Set theory; F.4.0 [Mathematical logic and formal languages]: General General Terms: Theory, Verification Additional Key Words and Phrases: Bisimulation, coinduction, fixed points, greatest fixed points, history Contents 1 Introduction 112 2 Background 114 2.1Bisimulation................................114 2.2Approximantsofbisimilarity......................116 2.3 Coinduction . 116 3 Bisimulation in Modal Logic 119 3.1Modallogics................................119 3.2Fromhomomorphismtop-morphism..................120 3.3JohanvanBenthem...........................122 3.4Discussion.................................125 4 Bisimulation in Computer Science 125 4.1Algebraictheoryofautomata......................125 4.2RobinMilner...............................128 Author’s address: Dipartimento di Scienze dell’Informazione Universita’ di Bologna Mura Anteo Zamboni, 7 40126 Bologna, ITALY. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc.