PROOF TRANSLATION AND SMT LIB CERTIFICATION

Yeting Ge Clark Barrett

SMT 2008

July 7 Princeton

SMT solvers are more complicated

CVC3 contains over 100,000 lines of code Are SMT solvers correct?

Quest for correct SMT solvers?

To verify a SMT solver is correct? To develop a correct SMT solver?

Good news: we have proofs

Some SMT solvers could produce proofs Proof checking should be easier than

proving the correctness of a SMT solver A proof could be represented as a proof

tree

1| ba 1| b

0| a

Bad news: Proof checking for SMT solvers is not so easy

Theory proof rules require the proof checker to have theory reasoning ability a/2 = b

Choice of proof rules A small set of simple proof rules?

Good for proof checking Large set of complex proof rules?

Good for performance (CVC3 has 298 rules) The correctness of the proof checker becomes

questionable SMT solvers are in constant change

The idea

Use a second prover to check the proof Translate the proof into the second prover The benefits

Could easily handle both simple and complex proof rules Flexible

The challenges A suitable second prover

The correctness is reduced to the second prover Efficiency Translation

This is feasible!

SMT LIB certification

SMT LIB A collection of over 40,000 SMT benchmarks,

most of which from industry applications Each file contains a status field

Some files are incorrectly labeled The proof in the second prover is a certificate A certified SMT LIB will be beneficial to SMT

community Prove as many unsatisfiable cases as possible

(benchmark tmp:source {piVC} :status unsat :category { industrial } :difficulty { 0 } :logic AUFLIA :extrafuns ((V_6 Int))

CVC3

A proof is a tree A proof rule maps a set of proofs to a

proof

Some proof rules are rather complex

The second prover: HOL Light

Simple The core:

430 lines of Ocaml, 10 inference rules, 3 axioms Definitional extension guarantees

correctness Except equality, all logic symbols are defined

All proofs in HOL Light can be broken down into the 10 rules and 3 axioms, if needed

“it sets a very exacting standard of correctness” Efforts to verify the correctness of the core

HOL Light

Powerful Capable of formalizing most mathematics (up to

axiom of choice) Flexible

Programmable Ocaml as meta-language

A number of built-in theories Reals, integers

A lot of useful tools Decision procedures for first-order logic, propositional

logic Decision procedures for reals, integers, …

Translation of terms

HOL Light and CVC3 are connected through C API functions of CVC3

distinct(x1,x2,…,xn) Define a predicate on the fly

Mixed integers and reals Lift to reals

Skolem constant

Choice operator (@x.P)

)()(. skoPxPx

Translation of proof rules

An Ocaml function for each proof rule Naïve method

call HOL Light’s decision procedure Exploit HOL Light’s capability of higher

order reasoning Prove a meta-theorem off-line During the translation, instantiate the meta-

theorem Engineering the translation of a proof rule

Propositional reasoning

SAT solvers can dump a resolution proof

Sequent representation

Definitional CNF and ITE

hole5 Time(s)

Try 1 255

Try 2 155

Seq 37

Sorted

2.8

Results

catetory cases CVC3 Translation

proved Ave time proved Ave time

simplify1 833 833 0.98 833 19.51

Simplify2 2329 2306 1.11 2164 8.85

burns 14 14 0.02 14 1.38

ricart 14 13 0.07 13 17.60

piVc 41 41 0.12 41 1.45

Hard cases

CVC3 Translation

No Prep 5 47.25 5 41.49

With Prep 4 48.91 4 64.27

Hard cases in simplify1: CVC3 spent more than 20 seconds

Results

Found one proof rule that does not preserve validity in CVC3

Found one faulty proof rule in CVC3 Found two mis-labled SMT LIB cases in

AUFLIA

Discussion

Instantiating a meta-theorem in HOL Light is almost like rewriting

Most proof rules can be converted into some meta-theorem

Other methods to improve efficiency Compiling

HOL Light

Conclusion

It is feasible to translate proofs from CVC3 into HOL Light

It is possible to certify many SMT LIB cases in HOL Light

Future works

Prove more SMT LIB cases Improve the translation of arithmetic

proof rules Support more proof rules Support more theories Improve the proof rules of CVC3

Thanks

John Harrison for help with HOL Ligh Sean McLaughlin for writing the first

version of the translator

Reference

C. Barrett and C. Tinelli. CVC3. In W. Damm and H. Hermanns, editors, Proceedings of the 19th International Conference on Computer Aided Verification (CAV ’07), LNCS 4590, pages 298–302. Springer-Verlag, July 2007. Berlin, Germany.

J. Harrison. Hol light: A tutorial introduction. In M. K. Srivas and A. J.Camilleri, editors, FMCAD, LNCS 1166, pages 265–269. Springer, 1996.

S. McLaughlin, C. Barrett, and Y. Ge. Cooperating theorem provers: A case study combining HOL-Light and CVC Lite. In A. Armando and A. Cimatti, editors, Proceedings of the 3rd Workshop on Pragmatics of Decision Procedures in Automated Reasoning (PDPAR ’05), volume 144(2) of Electronic Notes in Theoretical Computer Science, pages 43–51. Elsevier, Jan. 2006. Edinburgh, Scotland.

M. Moskal. Rocket-fast proof checking for smt solvers. In K. Jesen and A. Podelski, editors, TACAS, LNCS 4963, pages 486–500. Springer, 2008.

T. Weber. Efficiently checking propositional resolution proofs in isabelle/hol. volume 212 of CEUR Workshop Proceedings, 2006.