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

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(x 1,x 2,…,x n )  Define a predicate on the fly  Mixed integers and reals  Lift to reals  Skolem constant  Choice operator

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

Results catetorycasesCVC3Translation provedAve timeprovedAve time simplify Simplify burns ricart piVc Hard casesCVC3Translation No Prep With Prep 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

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