To Split or to Conjoin: The Question in Image Computation 1 {mooni, University of Colorado at Boulder 2 Synopsys Inc. 3 Cadence Inc. In-Ho Moon 1, James Kukula 2 Kavita Ravi 3, Fabio Somenzi 1

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 Model Checking Reachability Analysis Image/Preimage Computations BDD Operations Symbolic

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. (  T i (x,w,y)  C(x)) Preimage Computation  Pre(T(x,w,y), C(y)) =  y,w. (  T i (x,w,y)  C(y)) Early Quantification   u. ( f(u, v)  g(v) ) = (  u. f(u, v) )  g(v)  Img(T, C) =  v 1. ( T 1  ···   v k. (T k  C)) 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(f v, C v ) + Img(f v’, C v’ ) f = (f 1, …, f m ) : function vector v : splitting variable (x or w) Occurs most frequently in the supports [Cho96]  Constant Functions Img((f 1 =1, …, f m ), C) = y 1  Img((f 2, …, f m ), C) Img((f 1 =0, …, f m ), C) = y 1 ’  Img((f 2, …, f m ), 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(f v, C) + v’ Pre(f v’, C) v : splitting variable (x)  Constant Functions Pre((f 1 =1, …, f m ), C) = Pre((f 2, …, f m ), C y 1 ) Pre((f 1 =0, …, f m ), C) = Pre((f 2, …, f m ), C y 1 ’ )  Terminal Cases Pre(f, 1) = 1 Pre(f, 0) = 0 Pre(|f|=0, C) = C  Optimization Drop f j if y j  support(C(y))

10 Transition Relation vs. Function Methods 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 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 Question : Can we combine the merits of both methods?

11 Conjoin Hybrid Image Computation Static Hybrid Dynamic Hybrid Split

12 Dependence Matrix  m : the number of functions  n : the number of variables  d ij = 1 : i-th function depends on j-th variable n m = ( ) / (4 x 4) = 12 / 16 = 0.75 d1d2d3dmd1d2d3dm Quantify Conjunction From Average Variable Lifetime =  1  j  n (m - i j + 1) m  n

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

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    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

20 Range Computation Converting Image to Range Computation  Img(f, C) = Img(f  C, 1) = Img(f  C)   : constrain operator [CMD89b] Optimization Techniques  Decomposition due to disjoint support Img(f) = Img(f A ) Img(f B ) if support(f A )  support(f B ) = Ø  Identical and complementary components Img((f 1,f 2 )) = y1  y2 iff f 1 = f 2 = y1  y2 iff f 1 = f 2 ’  Identical subproblems Image cache

21 To Split or to Conjoin Variable lifetime  Conjoin if    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 Other considerations  Keep splitting only with improvement  Conjoin with big

22 Optimizations in Hybrid Method Essential Variables  C = e  C where e is a cube  Guarantee BddSize(T  e) < BddSize(T)  Dynamic turning on/off Combining Input and Output Splitting  Input splitting by default  Output splitting only when a function is a cube or the complement of a cube. Converting image to range computation  BddSize(T  C)  BddSize(T)  N  Dynamic turning on/off

23 Implementation of Hybrid Method Keeps only Transition Function  Build relations when to switch to conjoin  Overhead on building relations Keeps only Transition Relation  Splitting on transition relation  Cannot use the optimization techniques  Good for non-determinism Keeps both Transition Function and Relation  Splitting on both at the same time  Utilize the optimization techniques  Performs the best in most cases