Compositional reasoning for Parameterized Verification

1. Compositional reasoning for Parameterized Verification Murali Talupur Joint work with Sava Krstic, John O’leary, Mark Tuttle

2. Protocol Verification • Distributed protocols are crucial components of modern computer systems • Cache coherence protocols for example • Designed parametrically • Full validation requires parametric reasoning • Protocol verification involves two main issues • Tightly coded • So standard predicate abstraction and COI reduction will not work • Unbounded parallelism

3. Existing Methods Regular Model Checking Aggregated Trans Counter Abstraction Theorem Proving Invisible Invariants CMP Increasing Manual Effort WS1S Index predicates Automatic methods don’t scale Manual methods require human guidance but scale

4. CMP method • Compositional reasoning based method • Proposed by McMillan, elaborated by Chou et al and further simplified by Krstic • CMP scales to large protocols • This was one of the first techniques to handle Flash protocol • User has to supply “lemmas” • Supplying lemmas is easier than supplying inductive invariants • Easier than pure theorem proving

5. CMP Method True or Real Cex P(N) PA Abstract Model Check spurious cex P#(N) Strengthen Invent Lemma

6. Data Type Reduction • Reduces unbounded range [1..N] to [1,2, o] • Throws away the state spaces of [3..N] • Any condition involving them is conservatively over-approximated 1 2 PA Other 1 2 3 N-1 N P(N)

7. Rules are given in guarded command form: rl: ! a ruleset src : [1,2] do rule "NI_Local_Get_Put" Sta.UniMsg[src].Cmd = UNI_Get & !Sta.Dir.Pending ==> Var NxtSta: State begin NxtSta := Sta; NxtSta.Dir.Dirty := false; NxtSta.UniMsg[src].Cmd := UNI_Put; endrule; endruleset; rule "ABS_NI_Local_Get_Put" true & !Sta.Dir.Pending ==> var NxtSta : STATE; begin NxtSta := Sta; NxtSta.Dir.Dirty := false; NOP endrule; Example ruleset src : NODE do rule "NI_Local_Get_Put" Sta.UniMsg[src].Cmd = UNI_Get & !Sta.Dir.Pending ==> Var NxtSta: State begin NxtSta := Sta; NxtSta.Dir.Dirty := false; NxtSta.UniMsg[src].Cmd := UNI_Put; endrule; endruleset;

8. Data Type Reduction • Data type reduction is syntactic • Very fast • Abstract model has small state space • Behavior of “Other” is not constrained at all • Need to add lemmas constraining the behaviors of “Other”

9. Refinement with Lemmas • User provides relevant lemmas • Parameterized system is strengthened with the lemmas • Data type reduction is applied as usual • The resulting abstract model is more refined than previously • Behavior of Other restricted by the states of processes 1 and 2

10. ruleset src : NODE do rule "NI_Local_Get_Put" Sta.UniMsg[src].Cmd = UNI_Get & !Sta.Dir.Pending & forall dst: NODE do src != dst -> !(Sta.Proc[dst].CacheState = Cache_E) end ==> var NxtSta : STATE; begin NxtSta := Sta; NxtSta.Dir.Dirty := false; NxtSta.UniMsg[src].Cmd := UNI_Put; endrule; endruleset; rule "ABS_NI_Local_Get_Put" true & !Sta.Dir.Pending & forall dst: [1,2]. !(Sta.Proc[dst].CacheState = Cache_E) end ==> var NxtSta : STATE; begin NxtSta := Sta; NxtSta.Dir.Dirty := false; NOP endrule; Murphi Example invariant "Lemma" forall src : NODE do forall dst : NODE do dst != src -> (Sta.Proc[dst].CacheState = Cache_E -> Sta.UniMsg[src].Cmd != UNI_Get) end end;

11. CMP Method 8 i,j. (i,j) P(N) ² P(N) ² (1,2) Strengthening Circular Reasoning Ps(N) ² (1,2) Abstraction DTR is conservative PA ² (1,2)

12. Circular Reasoning Principle • System P consists of guarded rules ! a • Let Ri stand for all states reachable within i steps in P (8 s 2 Ri s ²)) (8 s 2 Ri s ²) Ps: Æ! a Ps²) P ²

13. Application • McOP is the cache coherence protocol of an experimental system with more than 50 cores • Vastly more complex than Flash • German has 7 msg types, Flash has 16, McOP has 55 • The proof took one month • 25 lemmas • Final count does not include the several wrong/weak lemmas that were used • 5 auxiliary variables

14. Improvements to CMP • Automate as much as possible • Reduce the burden on human user • Derive lemmas automatically • Instead of data type reduction use richer abstraction

15. Deriving Lemmas from Flows • We can use message flows to derive powerful lemmas automatically

16. Dir Dir i j i j ReqS ReqS RecvReqS RecvReqS SendInv SendInvAck GntS RecvGntS RecvInvAck GntS RecvGntS Flows: Examples Process i intiates a Request Shared transaction: Case 1 Process i intiates a Request Shared transaction: Case 2

17. Deriving Lemmas from Flows • We can use message flows to derive powerful lemmas automatically • Advantages: • Message flows are readily available in design documents • Easy to understand • Flows are local involving two agents unlike system wide invariants • Valuable validation collateral

18. Dir i j ReqS RecvReqS GntS RecvGntS Constraints from Flows ReqShare(i) SendReqS(i),RecvReqS(i),SendGntS(i),RecvGntS(i) Precedence between rules: For process i, action RecvReqS(i) must happen before SendGntS(i) Using lemmas from flows cut down the number of manual lemmas by 75%! Our FMCAD’08 paper has more details

19. O1 On O2 Using Richer Abstractions Use lightweight environment abstraction to track processes [3..N] instead of completely throwing away their states 1 2 PA Other 1 2 3 N-1 N P(N)

20. Conclusion • CMP works very well in practice • The idea of complementing model checkers with user supplied lemmas works quite well • Perhaps the only method that really works • CMP is not just for cache coherence verification • Applicable to other distrbuted algorithms/concurrent software as well