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```									VERIFICATION OF
PARAMETERIZED SYSTEMS
MONOTONIC ABSTRACTION IN
PARAMETERIZED SYSTEMS
Parosh Aziz Abdullah, Giorgio Delzanno, Ahmed Rezine

NAVNEETA NAVEEN PATHAK
AGENDA
   INTRODUCTION

 PARAMETERIZED                 SYSTEMS

 TRANSITION          SYSTEMS

 ORDERING

 MONOTONIC ABSTRACTION

Monotonic Abstraction in Parameterized Systems   2
INTRODUCTION

Monotonic Abstraction as a simple and effective
method to prove safety properties for
Parameterized Systems with linear topologies.

Main idea : Monotonic Abstraction for
considering a transition relation that is an over-
approximation of the one induced by the
parameterized system.

Monotonic Abstraction in Parameterized Systems   3
MODEL CHECKING + ABSTRACTION

Infinite-State
Abstraction
System

Model                                  Finite-State
Checking                                  System

Monotonic Abstraction in Parameterized Systems   4
AGENDA
   INTRODUCTION

 PARAMETERIZED                 SYSTEMS

 TRANSITION          SYSTEMS

 ORDERING

 MONOTONIC ABSTRACTION

Monotonic Abstraction in Parameterized Systems   5
PARAMETERIZED SYSTEMS

P1          P2                  P3                  ..........        PN

P2                                 .........

P1                           P4                                  PN

.........
P3

AIM : To verify correctness of the systems for the
whole family of Parameterized Systems.
Monotonic Abstraction in Parameterized Systems                 6
DEFINITION

A parameterized system P is a triple (Q,X, T ),
Q - set of local states,
X - set of local variables,
T - set of transition rules.

A transition rule t is of the form:
t: [ q | grd → stmt | q´ ]
where q, q´ ϵ Q
grd → stmt is a guarded command
grd ϵ B(X) U G(X U Q)
stmt : set of assignments
Monotonic Abstraction in Parameterized Systems   7
Parameterized System, P = (Q,T)
A process Q = {Green, Black, Blue, Red} and T = {t t t t t t }
1, 2, 3. 4, 5, 6
moves from where t t t – Local transition rules – Initially all
Idle State
2, 5, 6
Idle to Black                              processes are in this
t1, t4 – Universal Rules
state when it        t3 – Existential Rule        state
wants to
access its
critical
section.
V LR          t1              t6
Once a process
moves from Black
to Blue state, it                                                    Critical State –
“closes the door”                                                    Eventually a process
on all processes in            t2                                    will enter this state
Idle state                                             t5

t4
∃L     t3                       VL
Monotonic Abstraction in Parameterized Systems                      8
AGENDA
   INTRODUCTION

 PARAMETERIZED                 SYSTEMS

 TRANSITION          SYSTEMS

 ORDERING

 MONOTONIC ABSTRACTION

Monotonic Abstraction in Parameterized Systems   9
TRANSITION SYSTEMS
A transition system T is a pair (C,⇒)
where,
C - (infinite) set of configurations ,
⇒ - binary relation on C,
⇒* - reflexive transitive closure of ⇒

A configuration c ϵ C is a sequence u1 , ...... , un of process
states.
i.e. corresponding to an instance of the system with n
processes.

Monotonic Abstraction in Parameterized Systems    10
The word below represents a configuration in
an instance of system with 5 processes.

Valid Transitions
t3

Invalid Transitions
t3

Monotonic Abstraction in Parameterized Systems   11
Initial Configuration

All configurations that have atleast 2 RED processes

AIM : Init       *           Bad ?

Monotonic Abstraction in Parameterized Systems   12
AGENDA
   INTRODUCTION

 PARAMETERIZED                 SYSTEMS

 TRANSITION          SYSTEMS

 ORDERING

 MONOTONIC ABSTRACTION

Monotonic Abstraction in Parameterized Systems   13
ORDERING
c1, c2 – configurations
c1 ≤ c2 - c1is a subword of c2

e.g.            ≤
Upward Closed Configurations
Set U of configurations is upward closed, if
whenever c ϵ U and c ≤ c´ then c´ϵ U.

c – configuration,
ĉ – denotes upward closed set U:= {c´ | c ≤ c´}
ĉ contains all configurations larger than c w.r.t. ordering ≤.
i.e. c is the generator of U
Monotonic Abstraction in Parameterized Systems   14
Why Upward Closed Sets ?

1. All sets of Bad configurations (which are worked upon)
are upward closed.

2. Upward closed sets have an efficient symbolic
representation.
i.e. For an upward closed set U, there are
configurations c1, ..... , cn with U = ĉ1 U......U ĉn

Monotonic Abstraction in Parameterized Systems   15
Coverability Problem for Parameterized
Systems
To analyze safety properties.

PAR-COV
Instance
• Parameterized System, P = (Q,X,T)
•CF – upward-closed set of configurations
Question
Init   *   CF ?

Monotonic Abstraction in Parameterized Systems   16
Backward Reachability Analysis

For a set of configurations, C
Use Pre(C) := {c | ∃c´ϵ C; c → c´}

IDEA :
ii. Apply function Pre repeatedly generating sequence U0,
U1, U2,.... where
U0 := Bad, and Ui+1 := Ui + Pre(Ui) for all i ≥ 0
Observation :
set Ui characterizes set of configurations from which
set Bad is reachable within i steps
Monotonic Abstraction in Parameterized Systems   17
MONOTONICITY

Monotonicity implies that upward closedness is preserved
through the application of Pre.

Consider:
U – upward closed set,
c1 – member of Pre(U) and c2 ≥ c1
By Monotonicity, it can be proved that
c2 is also a member of Pre(U)

Monotonic Abstraction in Parameterized Systems   18
AGENDA
   INTRODUCTION

 PARAMETERIZED                SYSTEMS

 TRANSITION          SYSTEMS

 ORDERING

 MONOTONIC ABSTRACTION

Monotonic Abstraction in Parameterized Systems   19
MONOTONIC ABSTRACTION

An abstraction that generates over-approximation of the
transition systems.

The abstract transition system is monotonic.
Hence, allowing one to work with upward closed sets.

c1    ≥          c1´

A
c2
Monotonic Abstraction in Parameterized Systems   20
Local transitions are monotonic!

Consider the local transition,

t2
c1 =                                                 = c3

Configuration c2 =

c2 =                                       t2                    c4
This leads to c4 ≥ c2 and also maintains c3 ≤ c4.

Monotonic Abstraction in Parameterized Systems   21
Existential transitions are monotonic!

Consider the existential transition:
t3
c1 =                                                               = c3

Configuration, c2 =

t3
c2 =                                                                      = c4

Monotonic Abstraction in Parameterized Systems             22
Non-monotonicity of Universal
transitions
Consider the following Universal transition:
t4
c1 =                                                             = c3

t4 can be applied to c1 as all process in the left context of the
active process satisfy the condition of transition.

Now consider c2 =
c1 ≤ c2
But t4 is not enabled from c2 since the left context of the
active process violates the conditions of transition.
Monotonic Abstraction in Parameterized Systems          23
Solution!
1. Work with Abstract transition relation →A.
2. →A is an monotonic abstraction (over-approximation) of
the concrete relation →.
3. When t is universal,
t             t
we have: c1 →A c2 iff c1´ → c2 for some c1´ ≤ c1
t4
i.e.                                     →       A

Since
t4
≤                                        →
Monotonic Abstraction in Parameterized Systems        24
Solution.....
Since,
 c1 ≤ c2
 c1 →A c3 implies c2 →A c3
Hence, Abstract transition relation is Monotonic, w.r.t.
Universal Transitions.

The Abstract transition relation is and over-approximation of
the original transition relation
↓↓
If a safety property holds in the abstract model, then it will
also hold in the concrete model.
Monotonic Abstraction in Parameterized Systems   25
Coverability Problem for Approximate
Systems

APRX-PAR-COV
Instance
• Parameterized System, P = (Q,X,T)
•CF – upward-closed set of configurations
Question
Init   *
A CF ?

Monotonic Abstraction in Parameterized Systems   26
A   =(        U         1)

1   reflects the approximation of universal quantifiers

Since            ⊆          A

A negative answer to APRX-PAR-COV implies a negative

Monotonic Abstraction in Parameterized Systems   27
CONCLUSION

Monotonic Abstraction in Parameterized Systems   28
 Introduction to our topic.

 Overview of Parameterized Systems using a simple
example.

 (Infinite) Transition Systems arising from parameterized
systems.

 Introduced Ordering on the set of configurations.

 Definiton and explanation of Monotomic Abstraction; based
on the parameterized systems example.

Monotonic Abstraction in Parameterized Systems   29