VERIFICATION OF PARAMETERIZED SYSTEMS
MONOTONIC ABSTRACTION IN PARAMETERIZED SYSTEMS
NAVNEETA NAVEEN PATHAK
Parosh Aziz Abdullah, Giorgio Delzanno, Ahmed Rezine
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AGENDA
INTRODUCTION
PARAMETERIZED SYSTEMS
TRANSITION SYSTEMS
ORDERING
MONOTONIC ABSTRACTIONMonotonic Abstraction in Parameterized Systems
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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
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MODEL CHECKING + ABSTRACTION
Infinite-State
System
Abstraction
Finite-State
System
Model Checking
Monotonic Abstraction in Parameterized Systems
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AGENDA
INTRODUCTION
PARAMETERIZED SYSTEMS
TRANSITION SYSTEMS
ORDERING
MONOTONIC ABSTRACTIONMonotonic Abstraction in Parameterized Systems
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PARAMETERIZED SYSTEMS
AIM : To verify correctness of the systems for the whole family of Parameterized Systems.
Monotonic Abstraction in Parameterized Systems
P1 P2 P3 PN..........
P1
P2
P3
P4PN
......
...
......
...
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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 commandgrd ϵ B(X) U G(X U Q)stmt : set of assignments
DEFINITION
Monotonic Abstraction in Parameterized Systems
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V LR
V L
∃ L
t1 t6
t2
t3
t4
t5
Idle State – Initially all
processes are in this state
Critical State – Eventually a process will
enter this state
A process moves
from Idle to Black
state when it wants to access its
critical section.
Once a process moves from Black to Blue
state, it “closes the door” on all processes in
Idle state
Parameterized System, P = (Q,T)Q = {Green, Black, Blue, Red} and T = {t1, t2, t3. t4, t5, t6}where t2, t5, t6 – Local transition rules t1, t4 – Universal Rules t3 – Existential Rule
Monotonic Abstraction in Parameterized Systems
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AGENDA
INTRODUCTION
PARAMETERIZED SYSTEMS
TRANSITION SYSTEMS
ORDERING
MONOTONIC ABSTRACTIONMonotonic Abstraction in Parameterized Systems
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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
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The word below represents a configuration in an instance of system with 5 processes.
t3
Valid Transitions
t3
Invalid Transitions
Monotonic Abstraction in Parameterized Systems
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Initial Configuration
Bad ConfigurationAll configurations that have atleast 2 RED processes
AIM : Init * Bad ?
Monotonic Abstraction in Parameterized Systems
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AGENDA
INTRODUCTION
PARAMETERIZED SYSTEMS
TRANSITION SYSTEMS
ORDERING
MONOTONIC ABSTRACTIONMonotonic Abstraction in Parameterized Systems
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ORDERINGc1, c2 – configurationsc1 ≤ c2 - c1is a subword of c2
e.g. ≤Upward Closed Configurations
Set U of configurations is upward closed, ifwhenever 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
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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
16Monotonic Abstraction in Parameterized Systems
Coverability Problem for Parameterized Systems
To analyze safety properties.
PAR-COV
Instance• Parameterized System, P = (Q,X,T)• CF – upward-closed set of configurations
QuestionInit * CF ?
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Backward Reachability Analysis
Monotonic Abstraction in Parameterized Systems
For a set of configurations, CUse Pre(C) := {c | c´∃ ϵ C; c → c´}
IDEA :i. Start with set of bad upward-closed
configurations.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
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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 thatc2 is also a member of Pre(U)
Monotonic Abstraction in Parameterized Systems
19Monotonic Abstraction in Parameterized Systems
AGENDA
INTRODUCTION
PARAMETERIZED SYSTEMS
TRANSITION SYSTEMS
ORDERING
MONOTONIC ABSTRACTION
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MONOTONIC ABSTRACTION
Monotonic Abstraction in Parameterized Systems
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´≥
c2
A
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c1 = = c3
Local transitions are monotonic!
Monotonic Abstraction in Parameterized Systems
t2
Consider the local transition,
Configuration c2 =
c2 = c4
This leads to c4 ≥ c2 and also maintains c3 ≤ c4.
t2
22Monotonic Abstraction in Parameterized Systems
Existential transitions are monotonic!
t3
t3
Consider the existential transition:
c1 = = c3
Configuration, c2 =
c2 = = c4
Leading to c4 ≥ c3
23Monotonic Abstraction in Parameterized Systems
Non-monotonicity of Universal transitions
Consider the following Universal transition:
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.
t4
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1. Work with Abstract transition relation →A.
2. →A is an monotonic abstraction (over-approximation) of the concrete relation →.
3. When t is universal, we have: c1 →A c2 iff c1´ → c2 for some c1´ ≤ c1
i.e. →A
Since
≤ →Monotonic Abstraction in Parameterized Systems
Solution!
t t
t4
t4
25Monotonic Abstraction in Parameterized Systems
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.
Solution.....
26Monotonic Abstraction in Parameterized Systems
Coverability Problem for Approximate Systems
APRX-PAR-COV
Instance• Parameterized System, P = (Q,X,T)• CF – upward-closed set of configurations
QuestionInit * A CF ?
27Monotonic Abstraction in Parameterized Systems
A = ( U 1)
1 reflects the approximation of universal quantifiers
Since ⊆ A
A negative answer to APRX-PAR-COV implies a negative answer to PAR-COV.
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CONCLUSION
Monotonic Abstraction in Parameterized Systems
29Monotonic Abstraction in Parameterized Systems
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.
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Thank you for your attention.
Monotonic Abstraction in Parameterized Systems
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