SAT-Based Decision Procedures for Subsets of First-Order Logic: Part II: Separation Logic

1. SAT-Based Decision Procedures for Subsets of First-Order Logic Part II: Separation Logic Randal E. Bryant Carnegie Mellon University http://www.cs.cmu.edu/~bryant

2. Outline • Background • SAT-based Decision Procedures • Equality with Uninterpreted Functions • Translating to propositional formula • Exploiting positive equality and sparse transitivity • Separation Logic • Translating to propositional formula • Hybrid encoding techniques

3. Suitable for verifying wider class of systems Terms (T ) Integer Expressions ITE(F, T1, T2)If-then-else Fun (T1, …, Tk) Function application T + 1 Increment T – 1 Decrement Formulas (F )Boolean Expressions F, F1F2, F1F2 Boolean connectives T1 = T2 Equation T1 < T2 Inequality Pred(T1, …, Tk) Predicate application Separation Logic with Uninterpreted Functions (SUF)

4. Eliminate function and predicate applications using fresh variables and ITE expressions [Bryant, German, Velev, CAV’99] f(x) v1andf(y) ITE(x = y, v1, v2) v Integer variable Formulas (F )Boolean Expressions F, F1F2, F1F2 Boolean connectives T1 = T2 Equation T1 < T2 Inequality Pred(T1, …, Tk) Predicate application Separation Predicate b Boolean variable SUF  Separation Logic Terms (T ) Integer Expressions ITE(F, T1, T2) If-then-else Fun (T1, …, Tk) Function application T + 1 Increment T - 1 Decrement

5. Boolean Formula SAT Solver satisfiable/unsatisfiable Eager Boolean Encoding Methods for Separation Logic Separation Logic Formula Small Domain Encoding (SD) Per-Constraint Encoding (EIJ)

6. x x x+1 x+1 0x1x00y1y00y1y00z1z00z1z00x1x0 + 1 y y z z Values increase Small Domain Encoding (SD) [Bryant, Lahiri, Seshia, CAV’02] x  y  y  z  z  x+1 Observation: To check satisfiability, need to consider all possible relative orderings of finitely-many expressions • Can use Boolean encoding of finite range of values • 4 values in this case, so 2-bit encoding

7. e1 x  y y  z e2 e1 e2 e3 e3 z  x+1  Overall Boolean Encoding e1 e2 e4 New Separation Predicate  e4 x  z e4 e3 Transitivity Constraints Per-Constraint Encoding (EIJ) [Strichman, Seshia, Bryant, CAV’02] x  y  y  z  z  x+1

8. c3 + c4 c3 + c2 c1 + c4 c1 + c2 c4 c3 Enforcing Transitivity Constraints xy + c1 • Graph Representation of Separation Constraints • Directed multigraph where edges labeled by constants • Fourier-Motzkin Elimination • Eliminate nodes in succession • Possibly exponential growth in edges x c1 x y z c1 c2 y

9. c3 + c4 c3 + c2 c1 + c4 c1 + c2 c4 c3 Introducing New Predicates xy + c1 x c1 x y z Sample Predicates c1 c2 y Sample Transitivity Constraint Sample Ordering Constraint (for c1 < c2)

10. Comparing Eager Encoding Methods • Of SD and EIJ encoding methods, which one is better? • Comparison with respect to • Size of resulting Boolean formula • Performance of SAT solver

11. Example: N = 6813 • Method • Boolean Encoding Size • EIJ • > 1000000 • SD • 54465 Size of Boolean Encoding: SD better than EIJ • Let N be size of original separation logic formula • Size of a directed acyclic graph representation • SD encoding size is worst-case O(N2) • EIJ encoding size is worst-case O(2N) • Can generate O(2N) transitivity constraints

12. Impact on SAT problem: SD vs EIJ • Experimentally compared zChaff performance on SD and EIJ encodings of several unsatisfiable formulas • Sample result: EIJ better than SD for zChaff

13. Impact on SAT: Why is EIJ better than SD? • Conjecture: For SD, SAT solver has to “discover” transitivity constraints as conflict clauses • Violation of transitivity constraint might be discovered only after assigning bits of several bit-vectors • EIJ adds all such constraints a priori • Less learning and backtracking required by the SAT solver

14. Eager Encoding Tradeoffs • SD encoding • Polynomial size encoding • Worse for SAT solvers • EIJ encoding • Worst-case exponential size encoding • Better for SAT solvers • Can we automatically select between SD and EIJ based on the input formula?

15. Selection Strategy Seshia, Lahiri, Bryant, DAC ‘03 • Problem: • Computationally hard to estimate number of transitivity constraints • Can we use a different metric? • Idea: Identify feature of the input formula that varies monotonically with run-time of EIJ (but not with run-time of SD) Estimate number of transitivity constraints, C NO YES C > T ? Use SD encoding Use EIJ encoding

16. A Good Formula Feature: Number of Separation Predicates

17. A Good Formula Feature: Number of Separation Predicates

18. Revised Selection Strategy Easy to count number of separation predicates Very approximate measure of # of transitivity constraints • Constraints only relate predicates that share variables • Also need to automate setting of threshold T • Statistically estimate from “training” set of benchmarks Count number of separation predicates, m NO YES m > T ? Use SD encoding Use EIJ encoding

19. {x,y,z} shared Identifying Variable Classes Æ Ç Ç u¸v Æ z¸x+1 u= v-2 y¸z x¸y {u,v} shared Assignments to {u,v} are independent of those to {x,y,z}

20. Compute 1. Variable classes based on predicates 2. Number of separation predicates for each class {u,v}, mk {x,y,z}, m1 mk > T ? m1 > T ? YES YES NO NO SD SD EIJ EIJ Encode each class using SD or EIJ based on local decision Encoded Boolean Formula Hybrid Encoding Technique Separation Logic Formula

21. Automatically Selecting a Threshold Value: Intuition EIJ run time increases drastically beyond a certain number of separation predicates

22. Automatically Selecting a Threshold Value using Clustering Cluster total time (Y-axis) values, minimizing variance of each cluster

23. Experimental Evaluation Setup • Compared Hybrid against • SD and EIJ encodings • Cooperating Validity Checker (CVC) based on lazy encoding method [Stump et al.’02] • Stanford Validity Checker (SVC) – non SAT-based [Barrett et al. ’96] • CVC & SVC can handle more expressive logics than SUF • Benchmarks • 49 unsatisfiable SUF formulas • Load-store unit, out-of-order unit, device driver code, compiler validation, DLX pipeline • Threshold value calculated from subset of 16 benchmarks • Worked well for 39 out of the 49 benchmarks • Setup • Used zChaff SAT solver • Imposed timeout of 1800 sec. on total time (Encoding+SAT)

24. Hybrid vs. SD (39/49 benchmarks) Hybrid better SD better

25. Hybrid vs. EIJ (39/49 benchmarks) Hybrid better EIJ better

26. Hybrid vs. Lazy Encoding (CVC) (39/49 benchmarks) Hybrid better CVC better

27. Hybrid vs. Non-SAT-based Procedure (SVC) (39/49 benchmarks) Hybrid better SVC better

28. SD outperforms Hybrid on 10/49 benchmarks Hybrid better SD better

29. Conclusions & Ongoing Work • Hybrid combination of EIJ and SD encodings • is robust to formula variations • outperforms lazy encoding methods (CVC) • outperforms non-SAT-based methods (SVC) • Ongoing & Future work • Alternate estimators for number of transitivity constraints • Threshold setting technique based on clustering applies to other CAD problems too • Combination of lazy and eager encoding techniques might perform well on satisfiable formulas? • More on UCLID project webpage http://www.cs.cmu.edu/~uclid