1. Verifying Clausal Proofs, DRUPing and Interpolants SAT/SMT Seminar
Yael Meller
April 23rd 2017

2. Outline Validating unsat claim of SAT solver
Resolution proof
Clausal proof (RUP)1
DRUP- Optimized clausal proof2
Extracting interpolants from clausal proofs3
Extracting “simpler” interpolants3
Verification of Proofs of Unsatisfiability for CNF Formulas
Evgueni Goldberg andYakov Novikov, DATE 2003
Trimming while Checking Clausal Proofs
Marijn Heule, Warren Hunt and Nathan Wetzler, FMCAD 2013
Druping for Interpolants
Arie Gurfinkel and Yakir Vizel, FMCAD 2014

3. Check satisfiability of a CNF formula Basic steps:
CDCL SAT Solvers
Check satisfiability of a CNF formula
Basic steps:
Arbitrary decisions for un-assigned variables
Propagate values (BCP)
Analyze conflicts, learn clauses, and change decisions
If UnSAT, SAT solvers can generate refutation proofs and unsat core
Some uses of the above:
validate unsatisfiability claim
extract interpolants

4. Refutation Proof – Validate UnSAT claim
Goal: Validate the SAT solver. I.e., check, based on the output of the solver, that the solver’s result is correct
Easy to check given a counterexample
Harder to validate an UnSAT claim
Two methods for validating UnSAT claim:
Resolution proof
Clausal proof

5. Resolution step: (a,b), (c, b)  (a,c) Resolution proof:
A DAG that tracks resolution steps leading from the original clauses to the empty clause
Roots – original clauses
Intermediate nodes – derived clauses

6. Resolution Proof Resolution proof for:
F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1) g3 g3 g2
g2,g3 a1
a1,g2,g3 g4
a1,g2,g3
g2,g4 g1 a1
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3 g1

7. Resolution from Conflict Clause
The Resolution proof can be constructed during clause learning

8. Resolution from Conflict Clause
The Resolution proof is constructed when a clause is learnt
F = a  (a  b)  (b  c  d)  (b  d)
(a  c)
(b  c )
(c) d a b b c
Decision d a
Learnt clause

9. Resolution proof - pros and cons:
Easy to validate
Hard to obtain
Can be huge in size

10. Consider the clauses learnt by the SAT solver:
Clausal Proof
Consider the clauses learnt by the SAT solver:
Given a formula F, and a sequence of learnt clauses (C1,….,Cn) where Cn is the empty clause – check that indeed F derives the learnt clauses.

11. Verification of Proofs of Unsatisfiablity for CNF Formulas
Goldberg and Novikov, DATE 2003

12. Main observation: each learnt clause can be validated using BCP
Clausal Proof Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
Let F be a formula, and let <C1,….,Cn> be a sequence of learnt clauses where Cn is the empty clause.
Main observation: each learnt clause can be validated using BCP
If prior to learning clause C the CNF is F’, then if BCP(F’,C) derives a conflict, then F’C
Clausal proof: validate the leant clauses using the simple bcp procedure

13. The clausal proof: <(g2,g3), (g3), ()>
Clausal Proof - Example Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
CNF original formula: F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1)
The clausal proof: <(g2,g3), (g3), ()>
BCP(F,(g2,g3)): apply unit propagation (UP) on g2 and g3
a1 g1
Clause is false - conflict
BCP((F,(g2,g3)), (g3))
BCP (F,(g2,g3),g3), true)
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3 g1

14. CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
Clausal Proof Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
For every i in (1,…,n):
Execute BCP((F,C1,….,Ci-1), Ci)
If did not reach a conflict, then proof is invalid.
Important note: all learnt clauses are part of the proof. I.e., we cannot ignore learnt clauses that were deleted

15. CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
Clausal Proof Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
For every i in (1,…,n):
Execute BCP((F,C1,….,Ci-1), Ci)
If did not reach a conflict, then proof is invalid.
validate learnt clauses in the reverse order that they were learnt (RUP – reverse unit propagation)
Can mark clauses that were used in the BCP checks. If we reach an unmarked clause – we can skip it.
Produces unsat core

16. CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
Clausal Proof Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
CNF original formula: F, The learnt clauses: <C1,C2,….,Cn>
For every i from n to 1:
Mark Cn
If Ci is marked: execute BCP((F,C1,….,Ci-1), Ci) and mark touched clauses
If did not reach a conflict, then proof is invalid.
validate learnt clauses in the reverse order that they were learnt (RUP – reverse unit propagation)
Can mark clauses that were used in the BCP checks. If we reach an unmarked clause – we can skip it.
Produces unsat core

17. Resolution proof - pros and cons:
Clausal Proof vs. Resolution Proof Goldberg and Novikov, “Verification of proofs of unsatisfiablity for CNF formulas”, DATE 2003
Resolution proof - pros and cons:
Easy to validate
Hard to obtain
Can be huge in size
Clausal Proof - pros and cons:
Emitted with low overhead
Much smaller than resolution proof
Relatively expensive to validate (need to trust the bcp..)

18. Trimming while Checking Clausal Proofs
Heule, Hunt and Wetzler, FMCAD 2013

19. CNF original formula: F, The clausal proof: <C1,C2,….,Cn>
Optimizations on validating clausal proofs Heule, Hunt and Wetzler, “Trimming while Checking Clausal Proofs”, FMCAD 2013
CNF original formula: F, The clausal proof: <C1,C2,….,Cn>
For every i from n to 1:
Execute BCP((F,C1,….,Ci-1), Ci)
If did not reach a conflict, then proof is invalid.
Forward checking: validate each learnt clause in the order that they were learnt.
Easy to parallelize.
Can start when a clause is learnt.
May check clauses that are not required to validate the proof

20. DRUP – extends a clausal proof by tracking deleted clauses:
Optimizations on validating clausal proofs Heule, Hunt and Wetzler, “Trimming while Checking Clausal Proofs”, FMCAD 2013
What about deleted learnt clauses? RUP assumes all learnt clauses are in the clausal proof.
DRUP – extends a clausal proof by tracking deleted clauses:
The clausal proof is now <C1,C2,C3,C1d,C4,….,C3d,….,Cn>
For every i from n to 1:
Execute BCP((F,AC), Ci) where AC includes all the non-deleted clauses from C1,…..,Ci-1
If did not reach a conflict, then proof is invalid.

21. Summary up to now Validating Check unsat claim of SAT solver
Resolution proof
Resolution is constructed during conflict learning
Clausal proof (RUP)
DRUP- Optimized clausal proof

22. DRUPing for Interpolants
Gurfinkel and Vizel, FMCAD 2014

23. Given an unsatisfiable pair (A,B) of propositional formulas
Interpolants
Given an unsatisfiable pair (A,B) of propositional formulas
A(X,Y)  B(Y,Z) is unsatisfiable
There exists a formula I (the interpolant of (A,B)) such that:
A I
I  B is unsatisfiable
I is over the common variables of A and B

24. A B Interpolants A-local variables: a1
Global variables: g1, g2, g3, g4 A B
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

25. Let (A,B) be an unsatisfiable pair of propositional formulas
Calculating Interpolants from Resolution Proofs McMillan, “Interpolation and SAT-Based Model Checking”, CAV 2003
Let (A,B) be an unsatisfiable pair of propositional formulas
For a clause C, g(C) denotes the disjunction of the shared variables in C.
Given a proof of unsatisfiability for (A,B), define itp(C) for every node C in the proof as follows:
If C is a root, then
If CA then itp(C) = g(C)
Else itp(C) is constant TRUE
else let C1 and C2 be the antecedents of C, and let v be their resolution variable
If v is local to A, then itp(C) = itp(C1)  itp(C2)
Else itp(C) = itp(C1)  itp(C2)
The interpolant for (A,B) is itp().

26. Interpolants from Resolution Proofsg3 g3 g2
g2,g3 a1
a1,g2,g3 g4
a1,g2,g3
g2,g4 g1 a1
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

27. Interpolants from Resolution Proofs
The interpolant on F is I I I g3 g3
I = [(g1  g2)  (g1  g3)] 
[(g2  g3  g4)  (g2  g4)] g2
g2,g3 a1
(g2  g3  g4)  (g2  g4)
a1,g2,g3 g4
(g1  g2)  (g1  g3)
g2  g4
a1,g2,g3
g2,g4 g1 a1
g1  g2
g1  g3
g2  g3  g4 g2 g4 T T
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

28. Extracting interpolants is efficient, given a resolution proof
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
Extracting interpolants is efficient, given a resolution proof
Drawback: the SAT solver has to log the resolution proof
Extra time and memory for such logging
Proof is not targeted for “best” interpolant
Main idea: extract interpolants from a DRUP proof instead of a resolution proof

29. Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF
SAT
Clausal
Proof
BCP
DRUP*
core
proof
BCP +
Learning
Replay
Interpolant

30. Main idea: execute 2 phases:
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
The DRUP process traverses the resolution graph top-down, where the interpolation calculation is done bottom-up.
Intuitively, we should construct the interpolants as part of the forward checking
Main idea: execute 2 phases:
Phase 1: Create a core proof via DRUP (i.e., find the relevant learnt clauses)
Phase 2: Replay the proof forward and construct the interpolant

31. Replaying for Interpolation Calculation Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
Input:
CNF original formula: F with unsat core marked
Core learnt clauses from the DRUP proof: <C1,C2,….,Cn>
For every i from 1 to n:
Execute BCP((F,C1,….,Ci-1), Ci)
Replicate conflict learning to construct the resolution tree and incrementally calculate the interpolant

32. The DRUP proof: <(g2,g3), (g3), ()> Execute BCP(F, (g2,g3)).
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF original formula: F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1,g4)
The DRUP proof: <(g2,g3), (g3), ()>
Execute BCP(F, (g2,g3)).
Reach Conflict
Construct resolution
g2,g3 a1
a1,g2,g3 g1
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

33. The DRUP proof: <(g2,g3), (g3), ()>
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF original formula: F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1)
The DRUP proof: <(g2,g3), (g3), ()>
[(g1  g2)  (g1  g3)]  g2
g2,g3 a1
(g1  g2)  (g1  g3)
a1,g2,g3 g1
g1  g2
g1  g3 g2
g2  g3  g4 g4
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

34. The DRUP proof: <(g2,g3), (g3), ()>
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF original formula: F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1)
The DRUP proof: <(g2,g3), (g3), ()>
Execute BCP((F,(g2,g3)), g3)
Reach Conflict
Construct resolution I g3 g2
I=[(g1  g2)  (g1  g3)]  g2
g2,g3
g1  g2
g1  g3
g2  g3  g4 g2 g4 T
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

35. The DRUP proof: <(g2,g3), (g3), ()>
Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF original formula: F=(a1,g1,g2),(a1,g1,g3),(a1,g2,g3,g4),(a1,g2),(a1,g4),(g2,g3),(g3),(g1)
The DRUP proof: <(g2,g3), (g3), ()>
Execute BCP((F,(g2,g3),g3), true)
Reach Conflict
Construct resolution I I g3 g3
I=[(g1  g2)  (g1  g3)]  g2
g2,g3
g1  g2
g1  g3
g2  g3  g4 g2 g4 T T
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

36. Extracting interpolants from clausal proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
CNF
SAT
Clausal
Proof
BCP
DRUP*
core
proof
BCP +
Learning
Replay
Interpolant

37. Main idea: Algorithm for calculating a “simpler” interpolant
Finding “better” Interpolants Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
Observation: the BCP process influences the interpolant found. Different BCPs will produce different interpolants.
The clausal proof presents a set of different resolutions.
Huele at al. optimize BCP for minimal core
Gurfinkel and Vizel optimize BCP for simpler interpolant
Main idea: Algorithm for calculating a “simpler” interpolant
an interpolant that is “more” CNF-like
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g2,g3
a1,g2,g3
g2,g4
a1,g2,g3 g1 a1 g4

38. Given an unsatisfiable pair (A,B) of propositional formulas
Shared Derivable Clauses Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
Given an unsatisfiable pair (A,B) of propositional formulas
A clause C is shared-derivable iff
C is over the common variables of A,B
C is derived using only A clauses
Or, A  C

39. Input: a resolution proof of unsatisfiability of (A,B)
Partial CNF Interpolants Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
Input: a resolution proof of unsatisfiability of (A,B)
Find shared-derivable clauses in the proof and
Log them as a CNF formula g
Treat them as B clauses during the computation
Interpolant is itp() Ù g

40. Interpolants from Resolution Proofs Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
I = (g2  g3)  (g2 g4) …
I = (g2  g3)  (g2 g4) T g3 g3 T g2
g2,g3 a1
(g2  g3  g4)
a1,g2,g3 g4
(g1  g2)  (g1  g3) T
a1,g2,g3
g2,g4 g1 a1
g1  g2
g1  g3
g2  g3  g4 g2 g4 T T
a1,g1,g2
a1,g1,g3
a1,g2,g3,g4
a1,g2
a1,g4
g2,g3
g3
g1,g4

41. The algorithm and proof address sequence interpolants
Partial CNF Interpolants Gurfinkel and Vizel, “DRUPing for Interpolants”, FMCAD 2014
When combined with DRUP+Replay, the bcp during Replay is aimed at favoring shared derivable clauses
Correctness is proved by induction on the graph that for every node C in the graph the following holds:
itp(C)g(C) B  C|vars(B)
A(itp(C)C|vars(A))  g(C)
The algorithm and proof address sequence interpolants

42. Summary Validating unsat claim of SAT solver
Resolution proof
Clausal proof (RUP1 and DRUP2)
Extracting interpolants from clausal proofs3
Extracting more CNF-like interpolants3
Verification of Proofs of Unsatisfiability for CNF Formulas
Evgueni Goldberg andYakov Novikov, DATE 2003
Trimming while Checking Clausal Proofs
Marijn Heule, Warren Hunt and Nathan Wetzler, FMCAD 2013
Druping for Interpolants
Arie Gurfinkel and Yakir Vizel, FMCAD 2014