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Bebop : A Symbolic Model Checker for Boolean Programs

Bebop : A Symbolic Model Checker for Boolean Programs. Thomas Ball Sriram K. Rajamani. http://research.microsoft.com/slam/. Outline. Boolean Programs and Bebop What? Why? Results Demo Semantics of Boolean Programs Technical details of algorithm Evaluation Related Work.

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Bebop : A Symbolic Model Checker for Boolean Programs

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  1. Bebop: A Symbolic Model Checker for Boolean Programs Thomas Ball Sriram K. Rajamani http://research.microsoft.com/slam/

  2. Outline • Boolean Programs and Bebop • What? • Why? • Results • Demo • Semantics of Boolean Programs • Technical details of algorithm • Evaluation • Related Work

  3. Boolean Programs: What • Model for representing abstractions of imperative programs in C, C#, Java, etc. • Features: • Boolean variables • Control-flow: sequencing, conditionals, looping, GOTOs • Procedures • Call-by-value parameter passing • recursion • Control non-determinism

  4. Boolean programs: Why bool x,y; [1] while (true) { [2] if(x == y) { [3] y = !x; } else{ [4] x = !x; [5] y = !y; } [6] if (?) break; } [7] if(x == y) [8] assert (false); Representation of program abstractions, a la Cousots Each boolean variable represents a predicate: • (i < j) • (*p==i) && ( (int) p == j) • (p  T), where T is recursive data type • [Graf-Saidi]

  5. Bebop - Results • Reachability in boolean programs reduced to context-free language reachability • Symbolic interprocedural dataflow analysis • Adaptation of [Reps-Horwitz-Sagiv, POPL’95] algorithm • Complexity of algorithm is O(E  2n) • E = size of interprocedural control flow graph • n = max. number of variables in the scope of any label

  6. Bebop - Results • Admits control flow + variables • Existing pushdown model checkers don’t use variables (encode variable values explicitly in state) [Esparaza, et al.] • Analyzes procedures separately • exploits procedural abstraction + locality of variable scopes • Uses hybrid representation • Explicit representation of control flow graph, as in a compiler • Implicit representation of reachable states via BDDs • Generates hierarchical trace

  7. Bebop Demo!

  8. Outline • Boolean Programs and Bebop • Semantics of Boolean Programs • “stackless” semantics using context-free grammar • Technical details of algorithm • Evaluation • Related Work

  9. Stackless Semantics • State  = <p,> • p = program counter •  = valuation to variables in scope at p • No stack! • (B): finite alphabet over boolean program B • <call,p,> • Call (with return to p),  a valuation to Locals(p) • <ret,p,> • Return to p,  a valuation to Locals(p)

  10. ’(x) = (x), x in Locals(c) • ’(g) = (g), g a global <c:Pr(), > <d: Proc Pr(),’> <e, ’> <r, >  = <call,e,>  = <ret,e,> State transition <p,> --> <p’,’> (x) = (x), x in Locals(c) ’(g) = (g), g a global

  11. Trace Semantics • Context-free grammar L(B) constrains allowable traces • M -> <call,q,> M <ret,q,> • M -> M M • M ->  • 0 -1-> 1 -2-> … m-1 -m-> m is a trajectory of B iff • i -i+1-> i+1 is a state transition, for all i • 1 2 … m  L(B)

  12. Outline • Boolean Programs and Bebop • Semantics of Boolean Programs • Technical details of reachability algorithm • Binary Decision Diagrams (BDDs) • Path edges • Summary edges • Example • Preliminary Evaluation • SLAM Project

  13. x y y z z z z 1 1 0 0 0 0 1 1 Binary Decision Diagrams • Acyclic graph data structure for representing a boolean function (equivalently, a set of bit vectors) • F(x,y,z) = (x=y)

  14. x y y x z y y x 0 0 y y 1 z z z z z 1 1 1 0 0 0 0 1 1 Hash Consing + Variable Elimination

  15. Path Edges • <e,p>  PE(p), iff • Exists initialized trajectory ending in <e,e>, where e = entry(Proc(p)) • Exists trajectory from <e,e> to <p,p> • PE(p) is a set of pairs of valuations to boolean variables in scope in Proc(p) • Can be represented with a BDD!

  16. Representing Path Edges with BDDs • Example PE(p) for boolean variables x,y and z: • PE(p) = F(x,y,z,x’,y’,z’) = (x’=x)^(y’=y)^(z’=x^y) • BDDs also used to represent transfer functions for statements • Transfer(z := x^y) = F(x,y,z,x’,y’,z’) = (x’=x)^(y’=y)^(z’=x^y)

  17. decl g; void main() begin decl h; h := !g; A(g,h); skip; A(g,h); skip; if (g) then R: skip; fi end 1 g'=0^h'=1 |g'=1^h'=0 g=g’=0^a1=a1’=0^a2=a2’=1 | g=g’=1^a1=a1’=1^a2=a2’=0 g=g’=0^a1=a1’=0^a2=a2’=1 g=0^g’=1^a1=a1’=0^a2=a2’=1 Join(S,T) = { <1,2> | <1,J>S, <J,2>T } void A(a1,a2) begin if (a1) then A(a2,a1); skip; else g := a2; fi end

  18. Summary Edges<1,2> = Lift(<d,r>,Pr) • 1(x) = 2(x), x in Locals(c) • Locals don’t change • 1(g) = d(g) and r(g) = 2(g), g global • Propagation of global state <1,2> c: Pr() d: Proc Pr() e r <d,r>

  19. decl g; void main() begin decl h; h := !g; A(g,h); skip; A(g,h); skip; if (g) then R: skip; fi end g=0^g’=1^h=h’=1 g=0^g’=1^h=h’=1 g=0^g’=1^a1=a1’=1^a2=a2’=0 g=0^g’=1^a1=a1’=0^a2=a2’=1 void A(a1,a2) begin if (a1) then A(a2,a1); skip; else g := a2; fi end

  20. decl g; void main() begin decl h; h := !g; A(g,h); skip; A(g,h); skip; if (g) then R: skip; fi end g'=0^h'=1 |g'=1^h'=0 1 g=0^g’=1^h=h’=1 g’=h’=1 g=g’=a1=a1’=a2=a2’=1 g=g’=a1=a1’=a2=a2’=1 void A(a1,a2) begin if (a1) then A(a2,a1); skip; else g := a2; fi end

  21. Worklist Algorithm • while PE(v) has changed, for some v • Determine if any new path edges can be generated • New path edge comes from • Existing path edge + transfer function • Existing path edge + summary edge (transfer function for procedure calls) • New summary edges generated from path edges that reach exit vertex

  22. Generating Error Traces • Partition reachable states into “rings” • A ring R at stmt S is numbered N iff there is a shortest trace of length N to S ending in a state in R • Hierarchical generation of error trace • Skip over or descend into called procedures

  23. Outline • Boolean Programs and Bebop • Semantics of Boolean Programs • Technical details of algorithm • Preliminary Evaluation • Linear behavior if # vars in scope remains constant • Self application of Bebop • Related Work

  24. void level<i>() begin decl a,b,c; if (g) then while(!a|!b|!c) do if (!a) then a := 1; elsif (!b) then a,b := 0,1; elsif (!c) then a,b,c := 0,0,1; else skip; fi od else <stmt>; <stmt>; fi g := !g; end decl g; voidmain() begin level1(); level1(); if(!g) then reach: skip; else skip; fi end

  25. Application: Analysis Validation • Live variable analysis (LVA) • A variable x is live at s if there is a path from s to a use of x (with no intervening def of x) • Used to optimize bebop • Quantify out variables as soon as they become dead • How to check correctness of LVA? • Analysis validation • Create a boolean program to check results of LVA • Model check boolean program (w/out LVA)

  26. Analysis Validation • Output of LVA: { (s,x) | x is dead at s } • Boolean program • Two variables per original program var x: • x_dead (initially 0) • x_defined (initially 0) • For each fact (s,x): • x_dead, x_defined := 1, 0; • For each def of x: • x_defined := 1; • For each use of x • if (x_dead && !x_defined) LVAError(); • Query: is LVAError reachable?

  27. Results • Found subtle error in implementation of LVA • Was able to show colleague that there was another error, in his code • Analysis validation now part of regression test suite

  28. Related Work • Pushdown Automata (PDA) decidability results • [Hopcroft-Ullman] • Model checking PDAs • [Bouajjani-Esparza-Maler] [Esparza-Hansel-Rossmanith-Schwoon] • Model checking Hierarchical State Machines • [Alur, Grosu] • Interprocedural dataflow analysis • [Sharir-Pnueli] [Steffen] [Knoop-Steffen] [Reps-Horwitz-Sagiv]

  29. Related Work • Reps-Horwitz-Sagiv (RHS) algorithm • Handles IFDS problems • Interprocedural • Finite domain D • Distributive dataflow functions (MOP=MFP) • Subsets of D • Dataflow as CFL reachability over “exploded graph” • Our results • RHS algorithm can be reformulated as a traditional dataflow algorithm over original control-flow graph with same time/space complexity • Reformulated algorithm is easily lifted to powersets of D using BDDs • Arbitrary dataflow functions • Path-sensitive

  30. Summary • Bebop: a model checker for boolean programs • Based on interprocedural dataflow analysis using BDDs • Exploits procedural abstraction • Admits many traditional compiler optimizations • Hierarchical trace generation + DHTML user interface • Release at end of year • SLAM project • Iteratively refine boolean program models of C programs • Use path simulation to discover relevant predicates (simcl) • Automated predicate abstraction (c2bp)

  31. Software Productivity Tools Microsoft Research http://research.microsoft.com/slam/

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