1. Lifting Propositional Interpolants to the Word-Level
Daniel Kroening and Georg Weissenbacher
supported by and SRC
© ETH Zürich

2. Kaput

3. Concise reason for malfunction
looking at the actual circuit

4. “Word-Level” reason for malfunction

5. Craig-Interpolant Explains inconsistency of two formulas
in terms of the symbols shared by both formulas
x == y
x == y
&& z != x
x > y
Can be extracted from cut-free proofs
Applications in verification:
image approximation in model checking
refinement predicates for predicate abstraction

6. Proof-constructing decision procedures
Efficient decision procedures available
often based on bit-blasting (eager or lazy)
we use range encoding (Pnueli et al., 2002)
does not provide interpolants
SAT solver provides resolution proof
compute bit-level interpolant in polynomial time

7. Example: Bit-blasting of equality logic
proof space
resolution proof

8. Lifting propositional resolution proofs

9. Lifting Leaf Nodes
Introduce word-level facts based on structure of bit-level encoding
Replace vector-variables by word-level expressions

10. Lifting inner nodes Based on structure of resolution proof
Uses decision procedure for equality logic
proof rules used must allow for interpolation (cut-free)
McMillan’s interpolating theorem prover complete for equality logic combined with propositional logic
McMillan’s technique to compute WL-Interpolant

11. Conclusion Extract interpolant from lifted word-level proof
Performance mainly determined by bit-blasting decision procedure
Future work: Extend technique to bit-vector logic