Towards Game-Based Predicate Abstraction

1. Towards Game-BasedPredicate Abstraction experiments in compositional model checking Adam Bakewell, University of Birmingham

2. GAMEBASEDMODELCHECKING

3. MAGE • Compositional (game-based) MC’ing works, has benefits, and can be efficient. proof: • Take the regular language game model of IA • Reformulate model as tree of automata • Make model branches be function/model application • Be lazy: make transitions when demanded by checker • Be symbolic: don’t build the automata • Do CEGAR: approximate, certify, refine

4. COMO • Compositional MC’ing can verify real code with competitive efficiency proof: • Take a C program and process with CIL • Transform to ALGOL-like compositional CFG language • Do MAGE better: pre-composed interaction transitions; bdd; cbv; state compression+random access; etc

5. compositional CFGs Program ::= (τ f(xi:τi){new yi:τ’i.M})+ M ::= expr | M;M | lval:=expr | goto φ | case expr of {ki->Mi} φ = path to target from enclosing function root [[goto φ1]] = jump from state φ0·φ2 to φ0·φ1 where φ2 is the local path to this goto

6. still ill • Still far too slow • Data approximation is easily outwitted • Other model checkers do much better with a different kind of approximation….

7. e.g. xx1.c e.g. cc1.c #include<assert.h> extern int y; extern int main () { int x=y; assert(x!=x+1); return(0); } #include<assert.h> extern char d; extern int main () { char c=d; assert(c!=c+1); return(0); }

8. e.g. evenloop.c #include <assert.h> extern int y; int main() { int x = 0; int i = y; while(i-- >= 0) x += 2; assert(x%2==0); return(0); }

9. PREDICATEABSTRACTION

10. evenloop.c CFG model 2y,0 x:=0;i=y i--;x+=2 (i--;x+=2)(y-1) 0,y 2,y-1 ┴ i < 0 i < 0 i < 0 0,y 2,y-1 2y,0 assert state: x,i UNSAFE SAFE

11. predicate abstraction (PA) • P is a seq of predicates over program variables • PA model is a CFG where • States are #P bit-vectors (P-valuations) • Transitions exist where some basic block can change state sat. source P-valuation to state sat. target P-valuation • If #P <#(program state) then checking PA model is much quicker than checking program model

12. e.g. CFG PA model i--;x+=2 x:=0;i=y i--;x+=2 T F ┴ i<0 i<0 T F assert assert SAFE UNSAFE P=[x%2==0]

13. PA issues • P • must be sufficient to prove (un)safety • must be compact for feasibility (minimization) • derive from program/annotations • infer (e.g. by interpolation) • modelling and checking • which p’s to update/test (scope) • when to update/test (laziness)

14. PA needs games? • CFG PA Is not compositional! • Issues may be addressed elegantly with a compositional, semantic-direct, formulation • Let’s try it…

15. GAMEBASEDPREDICATEABSTRACTION

16. GBPA v1 • Take COMO architecture • Do a simple p. extraction (assert invariants!) • Locate each p. in model tree • Replace state model with p. model: (:=) sets p. state (case) tests p. state

17. evenloop.c GBPA model T,T x+=2 tmp>0 tmp:=i;i-- x:=0;i=y SAFE T,T T,T ┴ assert tmp≤0 T,F T,F T,F P=[x%2==0,tmp>0] UNSAFE

18. PA model formalization • move: arena + value + p.valuation • [[CELL(x,P)]] • Binds vars x, abstracted as preds P • states are P-valuation vectors • write move: (x := v|p) goes to state2 if SAT(state2[x’/x] & x’=v & p & state) • read move: (x.x|state) preserves state; composition with [[case]] does SAT, e.g. SAT(x=v & p)

19. laziness is free • COMO architecture is built to make model transitions on-demand: • (:=) SAT calls can be suspended when a valid next state is found • (case) SAT calls suspend while an alternative is explored

20. lo PREDICATE LOCATION

21. mind gym

22. mind gym: verify your answer #include <assert.h> int main() { int x = 19; assert(x==19); x=((((((((x+x)*2)+5)/3)+3)/2)+6)/7)+2; assert(x==??); return(0); }

23. e.g. multi assignment int main() { int x = 19; x=x+x; x=x*2; x=x+5; x=x/3; x=x+3; x=x/2; x=x+6; x=x/7; x=x+2; assert(x==5); return(0); } x!=5 here x=5 or x!=5 here assert has spurious failure

24. e.g. manual interpolation x=x+3; assert(x==30); x=x/2; assert(x==15); x=x+6; assert(x==21); x=x/7; assert(x==3); x=x+2; assert(x==5); return(0); } int main() { int x = 19; assert(x==19); x=x+x; assert(x==38); x=x*2; assert(x==76); x=x+5; assert(x==81); x=x/3; assert(x==27);

25. e.g. manual interpolation • 10 assignments • 10 predicates • 10 assertions • how many SAT calls to prove safety?

26. p location • GBPA CELL automatically restricts p scope • Program transform with tighter CELLs will need less SAT calls

27. e.g. manual relocation int gym3(z) { int y=z+6; assert(y==21); y=y/7; assert(y==3); y=y+2; assert(y==5); return(y); } int main() { int x = 19; assert(x==19); x=gym1(x); assert(x==81); x=gym2(x); assert(x==15); x=gym3(x); assert(x==5); return(0); } #include <assert.h> int gym1(z) { int y=2*z; assert(y==38); y=y*2; assert(y==76); y=y+5; assert(y==81); return(y); } int gym2(z) { int y=z/3; assert(y==27); y=y+3; assert(y==30); y=y/2; assert(y==15); return(y); }

28. e.g. manual relocation • in main: • 4 p’s to update: 16 SAT per (:=) • 4 asserts test on state size 4 p’s • in gymi functions: • 3 p’s to update: 8 SAT per (:=) • local state size 3 p’s mostly sufficient • global state size 7 p’s sometimes used • total SATs: 68 + 3 * 27 = 149

29. delayed SAT • An alternative to interpolation: • Allow state updates to be delayed until the kth • And delay state tests • Generalise CELL strategy s.t. states also record seq of delayed changes • Refine by increasing k

30. the position • Game based model checking needs PA • PA fits the games framework • We get lazy PA for free • Tight CELL location offers minimization • delayed/incremental solving needs to be accommodated