© 2014 carnegie mellon university synthesizing safe bit-precise invariants arie gurfinkel (sei /...

© 2014 Carnegie Mellon University

Synthesizing Safe Bit-Precise Invariants

Arie Gurfinkel (SEI / CMU)Anton Belov (UCD / Synopsys)Joao Marques-Silva (UCD)

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University3

Inductive Invariants: Turing / Floyd / Hoare

A. M. Turing, Checking a Large Routine. In Report of a Conference on High Speed Automatic Calculating Machines, (1949).

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Programs, Cexs, Invariants

A program P = (V, Init, Tr, Bad)P is UNSAFE if and only if there exists a number N s.t.

P is SAFE if and only if there exists a safe inductive invariant Inv s.t.

Inductive

Safe

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Many conferences, techniques, tools …

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

But Bit-Precise Verification is Hard

Bounded Model Checking•CBMC, Boolector, LLBMC, ESBMC, …•efficient discovery of counter-examples•no invariants!

Propositional Verification (Hardware)• Interpolation, IC3, PDR, ABC, …•efficient synthesis of propositional invariants•does not scale to bit-precise verification of software

Linear Arithmetic Verification (Software)• Impact, UFO, CPAChecker, Duality, Blast, GPDR, …•efficient synthesis of arithmetic invariants•not bit-precise (not sound!)• is often sufficient (e.g., UFO at SV-COMP’13 and ‘14)

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

But aren’t

bit-vectors = bit-blasting?

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Typical Bit-vector Decision Procedure

B2P is satisfiability preserving (only!)

Bit-blast (by itself) is not efficient

SAT

Bit-blast

Simplify

B2P

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Safety Verification by Bit-Blasting

Correct, but does not scale

Bit-blast Verify

propositional verifier

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Safety Verification by B2P

Efficient, but…•B2P only preserves satisfiability•Original circuit is reduced (abstracted) too much•Hard to track correspondence between input and output

B2P Verify

True

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Bit-blasting looses all structure!

Lack of structure makes it difficult to generalize

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Our Key Idea: Use Generate and Check Alg.

Given an input program P with a safety property Bad

1. Generate a candidate invariant Cand by verifying Bad on a “simpler” approximation Psimple of P

2. Compute the Maximal Inductive Subset Inv of Cand relative to P using bit-precise reasoning

3. Strengthen Inv using a bit-precise (but possibly slow) verification engine until (Inv Bad)

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

MISPER in a Nutshell

Adapt unsound arithmetic reasoning to guess bit-precise invariants

ApproximateProgram P

+Property

Program PLA

LA Verifier

Adapt using MIS

Candidate CLA

BIT VerifierInvariant IBIT

Yes +Certificate CBIT No + Cex

No + Cex

Unsound

Needs validation

Sound

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Approximate Bit-Vectors by Arithmetic

Ignore (i.e., over-approximate) all bit-vector-specific operationsUnsound, but simple and efficient

Approximate

BoolBit-vector

ArithmeticBool

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Maximal Inductive Subset

Let L be a set of formulas, P=(V, Init, Tr, Bad) a programA subset X of L is a maximal inductive subset iff it is the largest subset of X such that

A Maximal Inductive Subset is unique•inductive invariants are closed under conjunction

Cormac Flanagan, K. Rustan M. Leino: Houdini, an Annotation Assistant for ESC/Java. FME 2001: 500-517

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Minimal Unsatisfiable Subset

Let be a formula and A = {a1, …, an} be atomic propositions occurring negatively in

Assume Æ a1 Æ Æ an is UNSAT

A minimal unsatisfiable subset (MUS) of is the smallest subset X µ A such that Æ X is UNSAT

There are efficient algorithms for computing MUS (a.k.a. UNSAT core) for propositional formulas

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Solving MIS via MUS

Reduce MIS to multiple calls to MUS

fresh propositional

variables

fresh propositional

variables

called once

incremental SAT

SAT MUS

incremental SAT

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Var-Equivalence

Let A and B be two formulas

Let X be a subset of propositional variables of A and B

Definition: A and B are var-equivalent relative to X if and only if for any satisfying assignment ¿ of X, A¿ and B¿ are equisatisfiable

Claim

B2P() is var-equivalent to relative to X = {posti, prei}

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Implementation

Misper is implemented in Python and relies on many external tools•LLVM for handling C•UFO-MUZ for LA invariants•Boolector for B2P•MUSer2 for MUS step in MIS •Z3 for SMT and HORN

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Results Summary

214 SAFE benchmarks from SVCOMP’2013• includes all non-trivial SAFE benchmarks

All times are in seconds

bit width

inst. cnt Z3/PDR#sol (avg/med)

Misper#sol (avg/med)

Cand#sol (avg/med)

MIS#sol (avg/med)

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all 214 116 (127/8) 174 (28/0.4) 165 (8/0.4) 9 (392/134)

unsol 98 -- 58 (75/1) 52 (22/0.7) 6 (544/366)

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all 214 165 (176/8) 182 (69/0.4) 165 (8/0.4) 17 (661/399)

unsol 49 -- 18 (624/376) 6 (50/21) 12 (911/1,094)

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Detailed Results (16 bits)

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

FrankenBit: Bit-Precise Verification w/ Many BitsMISPER to synthesize bit-precise invariantsLLBMC to search for counterexamplesSilver and Bronze medals at SV-COMP 2014

http://sv-comp.sosy-lab.org/2014/results/index.php

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Related Work

Cormac Flanagan, K. Rustan M. Leino: Houdini, an Annotation Assistant for ESC/Java. FME 2001.• (the first?) algorithm for computing Maximal Inductive Subset

Randal E. Bryant, Daniel Kroening, Joël Ouaknine, Sanjit A. Seshia, Ofer Strichman, Bryan A. Brady: Deciding Bit-Vector Arithmetic with Abstraction. TACAS 2007.•sound under-approximation of bit-vector formulas by shrinking bit-width

Alberto Griggio: Effective word-level interpolation for software verification. FMCAD 2011.•mostly sound over-approximation of bit-vector formulas by arithmetic•but, also uses unsound approximation followed by a sound check

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Conclusion

Sound reasoning from unsound approximations•Use Linear Arithmetic to guess good invariants

•Use efficient bit-vector decision procedures to validate invariants

•Use efficient propositional Minimal Unsatisfiable Subset extractor to find Maximal Inductive Subset

•Use inefficient bit-precise reasoning to complete the proof

Works well on SV-COMP (non bit-vector specific) benchmarks•probably because the properties are mostly bit-vector agnostic•e.g., API usage in Linux Device Drivers

Integrated in FrankenBit: http://arieg.bitbucket.org/fbit

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Future Work

We have just scratched the surface…

CounterExample Guided Approximation-Refinement Loop•block a counterexample by partial bit-blasting•partially embed bit-vectors into integer arithmetic

Better approximations•such as in related work, e.g., Griggio, and Bryant et al.

Adapt lemmas•account for bit-width, overflow, and upper bound•e.g., replace x > 0 with x > 0 & x <= INT_MAX

Tighter integration with fixedpoint solver

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

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Synthesizing Safe Bit-Precise InvariantsGurfinkel, Belov, Marques-Silva

© 2014 Carnegie Mellon University

Contact Information

Arie GurfinkelSenior ResearcherSEI / CMUTelephone: +1 412-268-5800Email: info@sei.cmu.edu

U.S. MailSoftware Engineering InstituteCustomer Relations4500 Fifth AvenuePittsburgh, PA 15213-2612USA

Webwww.sei.cmu.eduwww.sei.cmu.edu/contact.cfm

Customer RelationsEmail: info@sei.cmu.eduTelephone: +1 412-268-5800SEI Phone: +1 412-268-5800SEI Fax: +1 412-268-6257