To Split or to Conjoin: The Question in Image Computation

1. To Split or to Conjoin:The Question in Image Computation 1 {mooni, fabio}@colorado.eduUniversity of Colorado at Boulder 2 kukula@synopsys.comSynopsys Inc. 3 kravi@cadence.comCadence Inc. In-Ho Moon1, James Kukula2Kavita Ravi3, Fabio Somenzi1

2. Outline • Introduction • Image Computation Methods • Transition Relation Method • Transition Function Method • Transition Relation vs. Function Methods • Hybrid Image Computation • Experimental Results • Conclusions

3. Introduction • Model Checking • The most widely used method in formal verification • Does the system (implementation) satisfy the property (specification)? • State space explosion • BDD explosion in symbolic model checking • The explosion occurs mostly in intermediate BDDs during conjunctions in image/preimage computations. • Image/Preimage Computations • Finding all successor/predecessor states from the given states at once, respectively • The key steps in symbolic model checking

4. Contribution Symbolic Reachability Analysis Model Checking Image/Preimage Computations BDD Operations

5. Image Computation • Two approaches • Transition Relation Method [ICCAD90, DAC91] • Conjunctions • Transition Function Method [IFIP89, ICCAD90] • Recursive splitting • Transition relation method is superior to transition function method in most cases • In some cases, transition function method is more efficient than transition relation method. • Especially, in most cases of approximate reachability analysis. • Questions • Why is that? • What if we combine the two methods?

6. Transition Relation Method • Image Computation • Img(T(x,w,y), C(x)) =  x,w. ( Ti(x,w,y)  C(x)) • Preimage Computation • Pre(T(x,w,y), C(y)) =  y,w. ( Ti(x,w,y)  C(y)) • Early Quantification • u. ( f(u, v)  g(v) ) = ( u. f(u, v) ) g(v) • Img(T, C) =  v1. (T1  ···   vk. (Tk  C)) 1  i  k 1  i  k

7. Transition Function Method • Image Computation [IFIP89, ICCAD90] • Input Splitting • Output Splitting • Preimage Computation • Simultaneous Substitution [CAV91] • Sequential Substitution [PhD92] • Domain Cofactoring [ICCAD98]

8. Transition Function Method (Cont’d) • Input Splitting • Img(f(x,w), C(x)) = Img(fv, Cv) + Img(fv’, Cv’) • f = (f1, …, fm) : function vector • v : splitting variable (x or w) • Occurs most frequently in the supports [Cho96] • Constant Functions • Img((f1=1, …, fm), C) = y1 Img((f2, …, fm), C) • Img((f1=0, …, fm), C) = y1’  Img((f2, …, fm), C) • Terminal Cases • Img(f, 0) = 0 • Img(|f|1, C) = 1 where f is non-constant & C  0 • From the implementation point of view, we don’t need y variables in the transition function method.

9. Transition Function Method (Cont’d) • Domain Cofactoring • Pre(f, C) = v • Pre(fv, C) + v’ • Pre(fv’, C) • v : splitting variable (x) • Constant Functions • Pre((f1=1, …, fm), C) = Pre((f2, …, fm), Cy1) • Pre((f1=0, …, fm), C) = Pre((f2, …, fm), Cy1’) • Terminal Cases • Pre(f, 1) = 1 • Pre(f, 0) = 0 • Pre(|f|=0, C) = C • Optimization • Drop fj if yj  support(C(y))

10. Transition Relation vs. Function Methods • Transition Function Methods • Based on splitting • Needs one set of state variables • Good : takes much less memory in most cases • Bad : may have too many recursive calls • Transition Relation Methods • Based on conjunction • Needs two sets of state variables • Good : much faster in most cases • Bad : intermediate BDDs may grow very large Question : Can we combine the merits of both methods?

11. Hybrid Image Computation Split Split Conjoin Conjoin Dynamic Hybrid Static Hybrid

12. Dependence Matrix Quantify Conjunction • Average Variable Lifetime  = 1 j  n(m - ij + 1) m  n From • Dependence Matrix • m : the number of functions • n : the number of variables • dij = 1 : i-th function depends on j-th variable d1 d2 d3 dm m n  = (4+4+3+1) / (4 x 4) = 12 / 16 = 0.75

13. Examples (32-bit rotator & multiplier) Good quantification schedule May be easy for conjunctions No good quantification schedule Needs splitting

14. Example (hw_top & one submachine) • Explains why splitting is better than conjunction in approximate reachability.

15. To Split or to Conjoin • Variable lifetime  • Conjoin if   0.5 +  • Split otherwise • Min/Max decision depth • Min : splitting may help for even small  • Max : to avoid too deep recursions • Decide only between min and max depth

16. Experimental Results - 1 • Time in Reachability Analysis

17. Experimental Results - 2 • Time in Approximate Reachability Analysis

18. Experimental Results - 3 • Time in Model Checking • Without Reachability Analysis

19. Conclusions • We have presented a hybrid image method • Combining the conjunction and splitting approaches • Dynamic decision whether to split or to conjoin based on variable lifetime from the dependence matrix • Much more robust than either pure method • The analysis of dependence matrix explains why splitting is better than conjunction in approximate reachability • Future Work • Improve decision strategy • Analyze why the results for preimage were not as good as those for image