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Symbolic Algorithms for Infinite-state Systems

Symbolic Algorithms for Infinite-state Systems. Rupak Majumdar (UC Berkeley) Joint work with Luca de Alfaro ( UC Santa Cruz) Thomas A. Henzinger (UC Berkeley). Closed Reactive Systems. Transition systems: S Set of states (possibly infinite)  Set of actions

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Symbolic Algorithms for Infinite-state Systems

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  1. Symbolic Algorithms forInfinite-state Systems Rupak Majumdar (UC Berkeley) Joint work with Luca de Alfaro (UC Santa Cruz) Thomas A. Henzinger (UC Berkeley)

  2. Closed Reactive Systems • Transition systems: • S Set of states (possibly infinite) •  Set of actions • post: S X   S Successor function

  3. Lifted Transition Systems • S Set of states •  Set of actions • Post: 2S X   2SSuccessor function • Post(R) = {t|  s R  a . t = (s,a)} • Pre: 2S X   2SPredecessor function • Pre(R) = {s| a . (s,a)  R}

  4. Observables • Group interesting sets of states as observables • Example: • “Processor 1 is in critical section” • “Thermostat temperature is between 32 and 40” • Observable transition system = • Transition system + • Set of observables  = {O1,O2,…}, OiS

  5. Symbolic Transition Systems • S,, Pre, Post,  • Set of regions R={R1,R2,…}, RiS •  R • Pre, Post : R X R • ,,\ : RXRR •  : RXR  {T,F} Computable Symbolic semi-algorithm: Start with regions in  and compute new regions using the operations above

  6. Example: Rectangular Hybrid Automata • General class: polyhedral hybrid systems [Alur et al] • Other classes: Petri nets, FIFO automata, ...

  7. Verification Questions • Q1 : Reachability • Is an unsafe state reachable? EF unsafe • Q2 : Linear Temporal Logic (regular properties) • Is progress being made? E(GF fair  F goal) • Q3 : ½ Branching temporal logic(ECTL,ACTL) • Nested reachability EF (unsafe  EF err1  EF err2) • Q4 : Branching temporal logic (CTL) • Is progress possible? AG(tick -> EXEF tick)

  8. Q1 : Reachability EF • Is there a trajectory to an unsafe state? R = final loop if R  init then “yes” if Pre(R)  R then “no” R := R  Pre(R) end . . . init final final Pre(final) Similar algorithm by iterating Post’s Operations used: Pre, 

  9. Q2 : LTL Model Checking • Example: Repeated Reachability EGF • Can a set of states be reached infinitely often? • EGF final init final R . . . . Operations: Pre,,  with observables R2 = EXEF R1 R1 = EXEF final

  10. Q3 : ECTL model cecking • ECTL: nested reachability • EF(goal1 /\ EF(goal2) /\ EF(goal3)) • Operations : Pre, ,  EF (goal1 /\ EF goal2 /\ EF goal3) EF goal3 EF goal2 goal1 /\ EF goal2 /\ EF goal3

  11. Q4 : CTL model checking • CTL: can all trajectories from init to goal1 be extended to goal2? • AG(goal1 -> EF goal2) = ~ EF (goal1 /\ ~EF goal2) • Operations : Pre, , , \ EF (goal1 /\ ~EF goal2) EF goal2

  12. Three Specification Logics • L1 : CTL (or, mu calculus) • L2 : ECTL or ACTL • L3 : LTL

  13. Three Symbolic Semi-Algorithms • A1 : Close  under pre, , , \ • A2 : Close  under pre, ,  • A3 : Close  under pre, , obs • (intersection with observables) P0 =  for i = 1,2,3, … Pi = Pi-1 {pre(R) | R  Pi-1 }  {R1  R2 | R1,R2  Pi-1}  {R1  R2 | R1,R2  Pi-1}  {R1 \ R2 | R1,R2  Pi-1} until Pi = Pi-1

  14. Three State Equivalences • E1 : Bisimilarity • E2 : Similarity (mutual simulation) • E3 : Trace Equivalence

  15. Similarity • Similarity: moves can be matched • Bisimilarity = Symmetric similarity • Trace equivalence = same languages  

  16. Triad Symbolic algorithms State equivalences Logics L1: CTL L2: ECTL L3: LTL A1: Pre+Boolean A2: Pre +Positive Boolean A3: Pre +Positive Boolean with  only with observables E1: Bisimilarity E2: Similarity E3: Trace equivalence

  17. Ai Symbolic semi-algorithm Li State Logic Model-checks i = 1,2,3 computes induces Ei State Equivalence All regions definable by Li are generated by Ai If Ai terminates, then symbolic model checking of Li terminates

  18. Ai Symbolic semi-algorithm Li State Logic Model-checks i = 1,2,3 computes induces Ei State Equivalence States s and t are Ei equivalent iff for all regions R generated by Ai, sR iff tR Ai terminates iff Ei has finite index

  19. Ai Symbolic semi-algorithm Li State Logic Model-checks i = 1,2,3 computes induces Ei State Equivalence States s and t are Ei equivalent iff for all formulas  of Li, s satisfies  iff t satisfies  If Ei has finite index, then Li can be model checked on a finite quotient

  20. Classification of systems [STACS00] • STS1 : • A1 terminates, finite bisimilarity, can model check CTL • Ex: Timed automata, O-minimal systems • STS2 : • A2 terminates, finite similarity, can model check CTL • Ex: 2D rectangular automata • STS3 : • A3 terminates, finite trace equivalence, can model check LTL • Ex: initialized rectangular automata

  21. Summary • The triad (algorithm, equivalence, logic) provides a useful tool to prove decidability and provide symbolic algorithms for infinite-state systems • The characterization provides a symbolic model checking algorithm, given some structural property of the system

  22. Summary • The symbolic approach shows how to engineer a model checker: • Export a Region interface implementing the symbolic operations • The model checking algorithm is independent of the front end syntax and region representation • E.g., BLAST toolkit for software

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