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SAT, Interpolants and Software Model Checking

SAT, Interpolants and Software Model Checking. Ken McMillan Cadence Berkeley Labs. Applications of SAT solvers. BMC of programs using SAT (e.g., CBMC) SAT solvers in decision procedures Eager approach (e.g., UCLID) Lazy approach (Verifun, ICS, many others) SAT-based image computation

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SAT, Interpolants and Software Model Checking

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  1. SAT, Interpolantsand Software Model Checking Ken McMillan Cadence Berkeley Labs

  2. Applications of SAT solvers • BMC of programs using SAT (e.g., CBMC) • SAT solvers in decision procedures • Eager approach (e.g., UCLID) • Lazy approach (Verifun, ICS, many others) • SAT-based image computation • Applied to predicate abstraction (Lahiri, et al) • ... SAT solvers have been applied in many ways in software verification We will consider instead the lessons learned from solving SAT that can be applied to software verification.

  3. Outline • SAT solvers • How do they work • What general lessons can we learn from the experience • Software model checking survey • How various methods do or do not embody lessons from SAT • A modest proposal • An attempt to apply the lessons of SAT to software verification

  4. SAT solvers • Solvers charactized by • Exhaustive BCP • Conflict-driven learning (resolution) • Deduction-based decision heuristics SATO, GRASP,CHAFF,etc DPLL DP DLL variable elimination backtrack search

  5. Lesson #1: Be Lazy • DP approach • Eliminate variables by exhaustive resolution • Extremely eager: deduces all facts about remaining variables • Essentially quantifier elimination -- explodes. • DPLL approach • Lazy: only resolves clauses when model search fails • Resolution use as a form of failure generalization • Learns general facts from model search failure • Implications: • Make expensive deductions only when their relevance can be justified. • Don't do quantifier elimination.

  6. Lesson #2: Be Eager • In a DPLL solver, we always close deduction under unit resolution (BCP) before making a decision. • Guides decision making model search • Guides resolution steps in failure generalization • BCP updated after decision making and clause learning • Implications: • Be eager with inexpensive deduction. • Deduce all the cheap facts before trying any expensive ones. • Let the expensive deduction drive the cheap deduction

  7. Lesson #3: Learn from the Past • Facts useful in one particular case are likely to be useful in other cases. • This principle is embodied in • Clause learning • Deduction-based decision heuristics (e.g., VSIDS) Implication: Deduce facts that have been useful in the past.

  8. Static Analysis • Compute the least fixed-point of an abstract transformer • This is the strongest invariant the analysis can provide • Inexpensive analyses: • value set analysis • affine equalities, etc. • These analyses lose information at a merge: x = y x = z T

  9. Predicate abstraction • Abstract transformer: • strongest Boolean postcondition over given predicates • Advantage: does not lose information at a merge • join is disjunction x = y x = z x=y Ç x=z • Disadvantage: • Abstract post is very expensive! • Computes information about predicates with no relevance justification

  10. PA with CEGAR loop • Choose predicates to refute cex's • Generalizes failures • Some relevance justification • Still performs expensive deduction without justification • strongest Boolean postcondition • Fails to learn from past • Start fresh each iteration • Forgets expensive deductions Choose initial T# Model check abstraction T# true, done Cex yes, Cex Can extend Cex from T# to T? no Add predicates to T#

  11. Boolean Programs • Abstract transformer • Weaker than predicate abstraction • Evaluates predicates independently -- loses correlations Predicate abstraction Boolean programs {T} x=y; {T} {T} x=y; {x=0 , y=0} • Advantages • Computes less expensive information eagerly • Disadvantages • Still computes expensive information without justification • Still uses CEGAR loop

  12. Lazy Predicate Abstraction • Unwind the program CFG into a tree • Refine paths as needed to refute errors x=y x=y Add predicates along path to allow refutation of error y=0 ERR! • Refinement is local to an error path • Search continues after refinement • Do not start fresh -- no big CEGAR loop • Previously useful predicates applied to new vertices

  13. Lazy Predicate Abstraction x=y x=y Add predicates along path to allow refutation of error y=0 ERR! • Refinement is local to an error path • Search continues after refinement • Do not start fresh -- no big CEGAR loop • Previously useful predicates applied to new vertices

  14. SAT-based BMC • Inherits all the properties of SAT • Deduction limited to propositional logic • Cannot directly infer facts like x · y • Inexpensive deduction limited to BCP Convert to Bit Level Loop Unwinding Program SAT

  15. decision x=y; x=z; x=z SAT-based with Static Analysis • Allows richer class of inexpensive deductions • Inexpensive deductions not updated after decisions and clause learning • Coupling could be tighter • Perhaps using lazy decision procedures? Static Analysis Loop Unwinding Convert to Bit Level Program SAT

  16. Lazy abstraction and interpolants • A way to apply the lessons of SAT to lazy abstraction • Keep the advantages of lazy abstraction... • Local refinement (be lazy) • No "big loop" as in CEGAR (learn from the past) • ...while avoiding the disadvantages of predicate abstraction... • no eager image computation • ...and propagating inexpensive deductions eagerly • as in static analysis

  17. Interpolation Lemma (Craig,57) • Notation: L() is the set of FO formulas over the symbols of  • If A Ù B = false, there exists an interpolant A' for (A,B) such that: A Þ A' A' Ù B = false A' 2L(A) ÅL(B) • Example: • A = p Ù q, B = Øq Ù r, A' = q • Interpolants from proofs • in certain quantifier-free theories, we can obtain an interpolant for a pair A,B from a refutation in linear time. [McMillan 05] • in particular, we can have linear arithmetic,uninterpreted functions, and restricted use of arrays

  18. ... A1 A2 A3 Ak True False ... ) ) ) ) A'1 A'2 A'3 A'k-1 Interpolants for sequences • Let A1...An be a sequence of formulas • A sequence A’0...A’n is an interpolant for A1...An when • A’0 = True • A’i-1Æ Ai) A’i, for i = 1..n • An = False • and finally, A’i2L (A1...Ai) ÅL(Ai+1...An) In other words, the interpolant is a structured refutation of A1...An

  19. True x1= y0 x=y; 1. Each formula implies the next ) x1=y0 y1=y0+1 y++; ) y1>x1 x1=y1 [x=y] ) False Path refinement procedure proof structured proof SSA sequence Path Refinement Prover Interpolation Interpolants as Floyd-Hoare proofs 2. Each is over common symbols of prefix and suffix 3. Begins with true, ends with false

  20. L=0 [L!=0] do{ lock(); old = new; if(*){ unlock; new++; } } while (new != old); L=1; old=new L=0; new++ [new!=old] program fragment [new==old] control-flow graph Lazy abstraction -- an example

  21. T 0 L=0 F T L=0 T [L!=0] 2 1 Label error state with false, by refining labels on path Unwinding the CFG L=0 [L!=0] L=1; old=new L=0; new++ [new!=old] [new==old] control-flow graph

  22. L=1; old=new T 3 L=0; new++ T L=0 4 [new!=old] F T T L=0 [L!=0] 5 6 Covering: state 5 is subsumed by state 1. Unwinding the CFG T 0 L=0 L=0 [L!=0] F [L!=0] L=0 2 1 L=1; old=new L=0; new++ [new!=old] [new==old] control-flow graph

  23. old=new T old=new 8 [new==old] [new!=old] T F T [L!=0] T 10 7 11 9 F T Another cover. Unwinding is now complete. Unwinding the CFG T 0 L=0 L=0 [L!=0] F [L!=0] L=0 2 1 L=1; old=new L=1; old=new T 3 L=0; new++ L=0; new++ [new!=old] L=0 4 [new!=old] [new==old] F L=0 [L!=0] 5 6 control-flow graph

  24. x=y x· y X Covering step • If y(x) )y(y)... • add covering arc x B y • remove all z B w for w descendant of y We restict covers to be descending in a suitable total order on vertices. This prevents covering from diverging.

  25. x=0 y=0 y¹0 X F Refinement may remove covers Refinement step • Label an error vertex False by refining the path to that vertex with an interpolant for that path. • By refining with interpolants, we avoid predicate image computation. T x = 0 T [x¹y] [x=y] T T y++ y=2 T T [y=0] T

  26. refine this path y¹0 Forced cover • Try to refine a sub-path to force a cover • show that path from nearest common ancestor of x,y proves y(x) at y T x = 0 x=0 T [x¹y] [x=y] T T y=0 y++ y=2 T y¹0 T [y=0] T F Forced cover allow us to efficiently handle nested control structure

  27. refine this path [x¹z] [x=z] x=z y=1 y=2 y2{1,2} T x=z y=2 [y=1Æ xz] F y=2 value set refined from value set analysis Incremental static analysis • Update static analysis of unwinding incrementally • Static analysis can prevent many interpolant-based refinements • Interpolant-based refinements can refine static analysis T x = 0 x=0 T [x¹y] [x=y] T T y=0 y++ y=2 T y¹0 T [y=0] T F

  28. Applying the lessons from SAT • Be lazy with epensive deductions • All path refinements justified • No eager predicate image computation • Be eager with inexpensive deductions • Static anlalysis updated after all changes • Refinement and static analysis interact • Learn from the past • Refinements incremental – no “big CEGAR loop” • Re-use of historically useful facts by forced covering

  29. Experiments • Windows device driver benchmarks from BLAST benchmark suite • programs flattened to "simple goto programs" • Compare performance against BLAST, a lazy predicate abstraction tool • No static analysis. Almost all BLAST time spent in predicate image operation.

  30. The Saga Continues • After these results, Ranjit Jhala modified BLAST • vertices inherit predicates from their parents, reducing refinements • fewer refinements allows more predicate localization • Impact also made more eager, using value set analysis

  31. Conclusions • Caveats • Comparing different implementations is dangerous • More and better software model checking benchmarks are needed • Tentative conclusions • For control-dominated codes, predicate abstraction is too "eager“ • better to be more lazy about expensive deductions • Propagate inexpensive deductions can produce substantial speedup • roughly one order of magnitude for Windows examples • Perhaps by applying the lessons of SAT, we can obtain the same kind of rapid performance improvements obtained in that area • Note 2-3 orders of magnitude speedup in lazy model checking in 6 months!

  32. Future work • Procedure summaries • Many similar subgraphs in unwinding due to procedure expansions • Cannot handle recursion • Can we use interpolants to compute approximate procedure summaries? • Quantified interpolants • Can be used to generate program invariants with quantifiers • Works for simple examples, but need to prevent number of quantifiers from increasing without bound • Richer theories • In this work, all program variables modeled by integers • Need an interpolating prover for bit vector theory • Concurrency...

  33. Mv Me 0 L=0 [L!=0] 2 1 L=1; old=new 3 L=0; new++ 4 8 Unwinding the CFG • An unwinding is a tree with an embedding in the CFG L=0 [L!=0] L=1; old=new L=0; new++ [new!=old] [new==old]

  34. Mv Me Expansion • Every non-leaf vertex of the unwinding must be fully expanded... If this is not a leaf... 0 L=0 L=0 ...and this exists... ...then this exists. 1 ...but we allow unexpanded leaves (i.e., we are building a finite prefix of the infinite unwinding)

  35. T F L=0 T L=0 F L=0 T These two nodes are covered. (have a ancestor at the tail of a covering arc) Labeled unwinding • A labeled unwinding is equiped with... • a lableing function y : V !L(S) • a covering relation Bµ V £ V 0 L=0 [L!=0] 2 1 L=1; old=new 3 L=0; new++ ... 4 [new!=old] [new==old] [L!=0] 5 6 7 ...

  36. T F L=0 T L=0 F L=0 T Well-labeled unwinding • An unwinding is well-labeled when... • y(e) = True • every edge is a valid Hoare triple • if x B y then y not covered 0 L=0 [L!=0] 2 1 L=1; old=new 3 L=0; new++ 4 [new!=old] [new==old] [L!=0] 5 6 7

  37. old=new T old=new 8 [new==old] [new!=old] T F T [L!=0] T 9 7 10 9 F T Safe and complete • An unwinding is • safe if every error vertex is labeled False • complete if every nonterminal leaf is covered T 0 L=0 F [L!=0] L=0 2 1 L=1; old=new T 3 L=0; new++ L=0 4 [new!=old] F L=0 [L!=0] 5 6 ... ... Theorem: A CFG with a safe complete unwinding is safe.

  38. Unwinding steps • Three basic operations: • Expand a nonterminal leaf • Cover: add a covering arc • Refine: strengthen labels along a path so error vertex labeled False

  39. Overall algorithm • Do as much covering as possible • If a leaf can't be covered, try forced covering • If the leaf still can't be covered, expand it • Label all error states False by refining with an interpolant • Continue until unwinding is safe and complete

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