VERIFICATION OF PARAMETERIZED SYSTEMS

1. VERIFICATION OF PARAMETERIZED SYSTEMS MONOTONIC ABSTRACTION IN PARAMETERIZED SYSTEMS Parosh Aziz Abdullah, Giorgio Delzanno, Ahmed Rezine NAVNEETA NAVEEN PATHAK

2. AGENDA • INTRODUCTION • PARAMETERIZED SYSTEMS • TRANSITION SYSTEMS • ORDERING • MONOTONIC ABSTRACTION Monotonic Abstraction in Parameterized Systems

3. 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

4. MODEL CHECKING + ABSTRACTION Monotonic Abstraction in Parameterized Systems

5. AGENDA • INTRODUCTION • PARAMETERIZED SYSTEMS • TRANSITION SYSTEMS • ORDERING • MONOTONIC ABSTRACTION Monotonic Abstraction in Parameterized Systems

6. PARAMETERIZED SYSTEMS • AIM : To verify correctness of the systems for the • whole family of Parameterized Systems. P1 P2 P3 PN .......... P2 ......... PN P1 P4 P3 ......... Monotonic Abstraction in Parameterized Systems

7. 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 ϵB(X) U G(X U Q) stmt : set of assignments Monotonic Abstraction in Parameterized Systems

8. A process moves from Idle to Black state when it wants to access its critical section. 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 Idle State – Initially all processes are in this state V LR t1 t6 Once a process moves from Black to Blue state, it “closes the door” on all processes in Idle state Critical State – Eventually a process will enter this state t2 t5 t4 ∃ L t3 V L Monotonic Abstraction in Parameterized Systems

9. AGENDA • INTRODUCTION • PARAMETERIZED SYSTEMS • TRANSITION SYSTEMS • ORDERING • MONOTONIC ABSTRACTION Monotonic Abstraction in Parameterized Systems

10. 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 ϵC 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

11. The word below represents a configuration in an instance of system with 5 processes. ValidTransitions t3 InvalidTransitions t3 Monotonic Abstraction in Parameterized Systems

12. Initial Configuration Bad Configuration All configurations that have atleast 2 RED processes AIM : Init * Bad ? Monotonic Abstraction in Parameterized Systems 12

13. AGENDA • INTRODUCTION • PARAMETERIZED SYSTEMS • TRANSITION SYSTEMS • ORDERING • MONOTONIC ABSTRACTION Monotonic Abstraction in Parameterized Systems

14. 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

15. WhyUpwardClosedSets ? All sets of Bad configurations (which are worked upon) are upward closed. Upward closed sets have an efficient symbolic representation.i.e. For an upward closed set U, there are configurations c1, ..... , cn with U = ĉ1U......U ĉn Monotonic Abstraction in Parameterized Systems

16. 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

17. BackwardReachabilityAnalysis For a set of configurations, C Use Pre(C) := {c | ∃c´ϵ C; c → c´} IDEA : Start with set of bad upward-closed configurations. 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

18. 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

19. AGENDA • INTRODUCTION • PARAMETERIZED SYSTEMS • TRANSITION SYSTEMS • ORDERING • MONOTONIC ABSTRACTION Monotonic Abstraction in Parameterized Systems

20. 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

21. Local transitions are monotonic! Consider the local transition, Configuration c2 = c2 = c4 This leads to c4 ≥ c2 and also maintains c3≤ c4. t2 c1 = = c3 t2 Monotonic Abstraction in Parameterized Systems

22. Existential transitions are monotonic! Consider the existential transition: c1 = = c3 Configuration, c2 = c2 = = c4 Leading to c4 ≥ c3 t3 t3 Monotonic Abstraction in Parameterized Systems

23. 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 Monotonic Abstraction in Parameterized Systems

24. Solution! Work with Abstract transition relation →A. →A is an monotonic abstraction (over-approximation) of the concrete relation →. When t is universal, we have: c1 →A c2 iff c1´ →c2 for some c1´ ≤ c1 i.e. →A Since ≤→ t t t4 t4 Monotonic Abstraction in Parameterized Systems

25. 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

26. 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

27. 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. Monotonic Abstraction in Parameterized Systems

28. CONCLUSION Monotonic Abstraction in Parameterized Systems

29. 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

30. Thank you for your attention. Monotonic Abstraction in Parameterized Systems