Equivalence Checking Using Cuts and Heaps Andreas Kuehlmann Florian Krohm IBM Thomas J. Watson Research Center Presented by: Zhenghua Qi

2 Previous approaches —BDD Equivalence checking in combinational verification BDD based approaches The functions of the two circuits to be compared are converted into canonical forms which are then structurally compared. The functions of the two circuits to be compared are converted into canonical forms which are then structurally compared. –Advantages : Efficient –Disadvantages: Exponential memory complexity

3 Previous approaches —Cutpoint Cutpoint-based verification Three phases: –Choose cut-points –The overall verification task is partitioned along these cutpoints into a set of smaller verification problems which are solved independently –Eliminate false negatives PI PO

4 Previous approaches —False Negatives False Negatives: Two functions are equivalent, but the verification algorithm declares them as different. Two functions are equivalent, but the verification algorithm declares them as different. Methods to handle false negatives –Based on re-substitution –Based on cut frontiers –Based on ATPG (Automatic Test Pattern Generation) technique F f2f2 f1f1 z x y G g2g2 g1g1 z x y Let f 1 (x)=g 1 (x)  x –if f 2 (z,y)  g 2 (z,y),  z,y then f 2 (f 1 (x),y)  g 2 (f 1 (x),y)  F  G –if f 2 (z,y)  g 2 (z,y),  z,y  f 2 (f 1 (x),y)  g 2 (f 1 (x),y)  F  G

5 Presented Approach The verification technique presented in this paper, utilizes BDDs, circuit graph hashing, cutpoint guessing and false negative elimination. Differences from previous approaches: –The processing of BDDs is prioritized by their size and limited to an upper bound –The BDD construction is not stopped at cutpoints.

6 Verification Overview Implemented as a Boolean reasoning engine. Construct circuit graph Identify equivalent parts using hash table F G Compute BDDs Identify equal functions Mark potential cutpoints Inject new BDDs using cutpoints Check false negatives Equal/Not EqualUndecided Boolean functions

7 Basic Algorithm for Equivalence Checking Basic procedure –Construct circuit model using two- input AND gates and inverters. –Perform actual comparison

8 Circuit graph manipulation—example (a) Two functionally identical circuits (b) Original graph for both circuits (c) BDDs are computed for vertices 1, 2, 3, 4, 5

9 Circuit graph manipulation—example (a) Two functionally identical circuits (d) BDD is computed for 6 which causes 6 and 2 to merge (e) Forward hashing causes 7 and 8 to merge and solves the verification problem

10 Advanced Algorithm Using Cut Frontiers All vertices that have been merged are now used as cutpoints to inject new BDD variables onto the heap All cutpoints with identical cut levels are assigned to a cut frontier

11 Elimination of False Negatives Cutpoint variables need to be resubstituted by their driving functions The elimination process is controlled by a heap –Initialize heap with all BDDs –Take the BDD with smallest size, re- substitute its topmost cut variable

12 Practical Experiments Number of functionally equivalent vertices versus total number of vertices in typical circuit graphs Validate the assumptions that many industrial circuits are structurally similar The technique was measured for a number of IBM internal circuits

13 Practical Experiments Verification performance for selected circuits

14 Conclusions The paper presents a new method to perform functional comparison of combinational circuits using BDDs, circuit graph hashing, cutpoint guessing and false negative elimination. The presented approach performs efficiently for a wide variety of designs with some degree of structural similarity.