CVCL Lite: An Efficient Theorem Prover Based on Combination of Decision Procedures

1. CVCL Lite:An Efficient Theorem Prover Based on Combination of Decision Procedures Presented by: Sergey Berezin Stanford University, U.S.A.

2. What is CVC Lite? • Validity Checker: G ² f • First-Order Logic with interpreted theories • Arithmetic, uninterpreted functions, arrays, etc. • Theorem Prover based on multiple DPs • C++ library for use in other tools

3. People Project leaders: • Sergey Berezin, Clark Barrett, David Dill Developers and contributors: • Daniel Wichs • Ying Hu • Mark Zavislak • Jim Zhuang • Deepak Goyal • Jake Donham • Sean McLaughlin • Vijay Ganesh

4. Outline • Theoretical Basis • CVCL from User's Point of View • C++ library • Command line • Theory API • Architecture and Functionality • Important Design Decisions

5. Logic • Many-sorted FOL + Theories x=y ) a[i]+2*y < f(rec.f, 15-3*b[j+1]) • Partial functions (e.g. x/y) • Quantifiers (experimental) • Validity Problem: • Is f valid under the set of assumptions G ? G²f

6. Theoretical Basis: Combination of Decision Procedures • Clark Barret's thesis • Fusion of Nelson-Oppen + Shostak methods • Vijay Ganesh's extension of Ghilardi's method: T1[ T2 ² f T1[ T2 [:f ² ? (T1[G1) [ (T2[G2) ² ? Ti[Gi[ Ck ² Ck+1, i2{1,2} Ck are positive ground clauses

7. Outline • Theoretical Basis • CVCL from User's Point of View • C++ library • Command line • Theory API • Architecture and Functionality • Important Design Decisions

8. CVCL as C++ Library • API: ValidityChecker class • Provides functionality: • Create terms and formulas as CVCL Expr • Solve G²f • Manipulate logical context G

9. CVCL Executable Parser & Command Processor CVCL library CVCL API User Input Command Line Executable • PVS-like input language • Parser and command processor • implemented on top of C++ API

10. Theory API(For New Decision Procedures) • "Hackability" – very important! • All functionality implemented locally in DP • No changes to the Core files CVCL Library CVCL Core Theory API Arith Arrays UF

11. Outline • Theoretical Basis • CVCL from User's Point of View • Command line • C++ library • Theory API • Architecture and Functionality • Important Design Decisions

12. CVCL Core SAT Solver Union-Find DB Fact Queue Notify List Arith Arrays UF CVC Lite Architecture

13. (BCP; DP)* s1 s2 s3 :s2 (BCP; DP)* (BCP; DP)* (BCP; DP)* :s3 SAT Solver + DPs BCP: Unit Clauses DP: Ti[Gi[ Ck² Ck+1 Backtracking Mechanism! ? ?

14. Other Important Features • Efficient backtracking mechanism • Partial Functions and Subtypes • Kleene semantics (most general) • Quantifiers (experimental) • Symbolic Simulator (in progress) • Proof Production

15. Outline • Theoretical Basis • CVCL from User's Point of View • Command line • C++ library • Theory API • Architecture and Functionality • Important Design Decisions

16. Theorems and Proof Rules • Computing with proof rules • Every proven formula is a Theorem object • Theorems are constructed with Proof Rules • Proof rules comprise Trusted Code • Soundness checked on-the-fly • Transparent assumption tracking and proof production • Automatically up-to-date

17. Adding Decision Procedures • Core files need not be modified • All functionality is coded locally in DP • Type checking • TCCs (partial functions) • Specialized expressions • Parsing aid • Pretty-printing • Distribution of responsibility among developers

18. Thank you!