Model Checking and Range-equivalent Circuits

1. Model Checking and Range-equivalent Circuits Date: 2013. 6. 3 Speaker: Chih-Chung Wang

2. Outline • Model Checking • State Explosion Problem and Solutions • Range-equivalent Circuit Minimization • Future Work and Discussion

3. Model Checking • a technique for automatically verifying correctness properties of finite-state systems

4. Model Checking • The Model-Checking Process • Modeling • Running • Running the Model Checker • Analysis • Analyzing the Results

5. Transition System • Transition System (TS) • (S, Act, → ,I ,AP ,L) • finite: S, Act, and AP are finite. State Transition Graph (STG) S0 S1 States of Combinational Part S3 S2

6. Transition System • Reachable State • Induction S4 S5 State Transition Graph (STG) S0 S1 States of Combinational Part S3 S2

7. Induction • Foundation of verification for 40+ years (Floyd, Hoare) • To prove that S : (I, T) has safety property P, prove: • Base case: • I ⟹ P • Inductive case: • P ∧T ⟹ P’

8. Induction Fails • Use a stronger assertion, or • Construct an incremental proof, using previously established invariants • Temporal Verification of Reactive Systems: Safety

9. State Explosion Problem • The size of transition system representations grows exponentially in various components, such as the number of variables in a program graph or the number of components in a concurrent system. • A combinatorial blow up of the state-space • State Explosion Problem or Combinatorial explosion

10. State Explosion Problem • BDD/SAT-based reachability • exact computation of strongest/weakest strengthening • k-induction • unrolling to strengthen • Interpolation-based model checking • property-focused abstract post-condition • Interpolation and SAT-based Model Checking, Kenneth L. McMillan, CAV 2003 • Incremental Proof • IC3 • SAT-based Model Checking without Unrolling

11. State Explosion Problem • Symbolic algorithm • avoid ever building the graph for the FSM • represent the graph implicitly using a formula in quantified propositional logic • ex. BDD • Bounded model checking algorithms • unroll the FSM for a fixed number of steps and check whether a property violation can occur in or fewer steps • typically involving encoding the restricted model as an instance of SAT

12. Symbolic Model Checking • Considering large numbers of states at a single step • binary decision diagrams (BDDs) • avoid ever building the graph • represent the graph implicitly

13. Symbolic Model Checking • Symbolic Model Checking without BDDs • bounded model checking(BMC) • for the Linear Temporal Logic (LTL) • BSEC

14. Linear Temporal Logic • Encoding formulae about the future of paths • ◇ “eventually” (eventually in the future) • □“always” (now and forever in the future) • ¬ ,∧ ,∨ , ∃, ∀……

15. Linear Temporal Logic • LTL model checking

16. Linear Temporal Logic • safety properties • state that something bad never happens • counterexample • SAT-Based Model Checking Without Unrolling • liveness properties • state that something good keeps happening

17. Computation Tree Logic • Computation tree logic (CTL): a branching-time logic • ∃, ∀ • model of time is a tree-like structure in which the future is not determined

18. Fixed-point • : the set of all reachable states at the i-th iteration • The sets of the reachable states in two consecutive iterations are identical • i.e., = initial state fixed-point … reachable states

19. IC3 • Given: • Over-approximations of stepwise reachability: • , , , . . . , • Approximate “onion skins”: • ⇒ • ∧⇒ • Counterexample to stepwise-relative induction : • ∧∧⇒

20. IC3 • Find: • Weakest stepwise assumption (maximum ) • ∃⊆¬ such that ∧∧ ∧⇒ • Guaranteed to exist. • Prefer small (ideally minimal) clause . • Proof that is unreachable for + 1 steps.

21. IC3 • Push forward: • Strengthening stepwise knowledge up to step + 1: • := ∧ for ∈{1, . . . , + 1} • Recuringon the property ¬(at step + 1).

23. Bounded Sequential Equivalence Checking • A special case of Bounded Model Checking (BMC) • Checking two circuits in different timeframe • Linear Temporal Logic • Bound • Sequential depth • Optimization • Resyn2 • NAR, Node Merging

24. Bounded Sequential Equivalence Checking • Unrolling unroll

25. Bounded Sequential Equivalence Checking

26. Bounded Sequential Equivalence Checking

27. Range-equivalent Circuit Minimization • Using range to minimize the circuit optimizes the bounded model checking • In model checking, we reduced the repeated reached states from different paths • Range remained – state set remained • Partial Order Reduction • ex. BDD

28. Range-equivalent Circuit Minimization • Algorithm • Splitting PIs • Removing redundant PIs • Computing RMA • Computing type2 assignments by RMA • Logic implementation • Considering the value assignments • Merging PIs • For each PI, exponential time complexity

29. Range-equivalent Circuit Minimization • Removing redundant PIs

30. Range-equivalent Circuit Minimization • Failure in BSEC • Making range-equivalent circuit wastes too much time • about O(n^3) • n: number of PI

31. Future Work and Discussion • How can we use range in model checking • Which area of model checking? • Discussion • Performance of range • Restriction • Modern techniques in BMC

32. Discussion • Performance of range • Computing range • Modifying range • Partial range-equivalent circuit • not processing all PI • Optimizing range-equivalent circuit • Structural analysis • Restriction • Output set • Circuit Size • Temporal logic

33. Discussion • Modern techniques in BMC • Interpolation, IC3, FAIR, IICTL, … • Reachable after k • Finding a (k + 1) counterexample • IC3 • Stepwise induction • Lemma: clause • Functional representation • compromise between the incremental and monolithic strategies

35. Reference • Website • http://theory.stanford.edu/~arbrad/ • Principles of Model Checking • SAT-based Model Checking without Unrolling • Temporal Verification of Reactive Systems: Safety • Checking Safety by Inductive Generalization of Counterexamples to Induction

38. Temporal Logic • ◇ • “eventually” (eventually in the future) • □ • “always” (now and forever in the future) • ○ • “next” • U • “until”