Logic Decomposition

1. Logic Decomposition ECE1769 Jianwen Zhu (Courtesy Dennis Wu)

2. Introduction

3. Functional Decomposition • Re-express a Boolean function as • Where • Free set: • Bound set: • Disjoint decomposition

4. Ashenhurst Decomposition Chart • Classic method • Row: bound set • Column: free set • Disjoint decomposition if column multiplicity = 2

5. Specialized Decompositions • Algebraric Factorization: • F = A*B + C, where supp(A) and supp(B) are disjoint • Boolean Factorization: • F = A*B + C, where supp(A) and supp(B) are not disjoint • Bi Decomposition: • F = A (op) B, where op is any operation

6. Boolean Division • F = QD • Definition: Function D is a Boolean divisor of F if there exists a function Q, called the quotient, such that F = QD • Theorem: Function D is a Boolean divisor of F iff F  D. • On-set of F is subset of the on-set of D • F implies D • Example: e + bd = (e + b)(e + d)

7. Boolean Subtraction • F = D + R • Definition: Function D is a Boolean subtractor of F if there exists a function R, called the remainder, such that F = D + R. • Theorem: A function D is a Boolean subtractor of F iff F  D. • On-set of D is a proper subset of the on-set of F • D implies F

8. BDD Summary • Canonical • Compact • Efficient algorithms for many Boolean operations

9. Cut • A cut (D, V – D) of a BDD is a partition of its nodes V into disjoint subsets D and (V – D) such that • root  D and terminals 0, 1  (V-D). • A cut cannot cross any path more than once.

10. A cut where support of D and (V-D) are disjoint. Bound set Free set Idea: Exploit structures in horizontal cut Cut edge classification 0: pointing to 0 1: :pointing to 1 x :pointing to internal nodes Horizontal Cut

11. Generalized Dominator • Consider a cut partitioning the set of BDD nodes function F into D and (V-D). The portion of the BDD defined by D is copied to form a separate graph. In that graph, an edge e is connected to 0 if e  0 in the original BDD of F, and it is connected to 1 if e  1 in the original BDD of F. All the internal edges e  x are left dangling. The resulting graph is called a generalized Dominator.

12. Example Generalized Dominator

13. Boolean AND Decomposition • Obtain divisor by redirecting dangling edges of generalized dominator to 1. • Obtain quotient by minimizing F with the off-set of the divisor as a don’t care set.

14. Example of Boolean AND Decomposition

15. Boolean OR Decomposition • Obtain subtractor by redirecting dangling edges of generalized dominator to 0. • Obtain remainder by minimizing F using the on-set of the subtractor as a don’t care set.

16. Example of Boolean OR Decomposition

17. Finding Generalized Dominators • Number of BDD cuts is exponential. • Filtering Cuts • Valid Cuts contain at least one edge connected to a terminal node. • 0-Equivalent Cuts (same 0 ) result in the same AND decomposition. • 1-Equivalent Cuts (same 1 ) result in the same OR decomposition.

18. Algebraic Bi-Decomposition

19. A node which belongs to every path leading to a ‘1’ terminal is called a 1-dominator Enables fast algebraic AND decomposition 1-Dominator

20. 1-Dominator

21. A node which belongs to every path leading to a ‘0’ terminal is called a 0-dominator Enables fast algebraic OR decomposition 0-Dominator

22. 0-Dominator

23. A node which is contained in every path is called an x-dominator Enables fast XNOR Decomposition X-Dominator

24. X-Dominator

25. X-Dominator

26. Finding Simple Dominators • All simple dominators common in that all internal edges converge to a single node. • Complexity of O(V) • V is the # of variables. • All Simple Dominators are found at the same time. • The simple dominator closest to the middle height of the BDD is used for decomposition

27. A node which is pointed to by both complement and regular edges is called a generalized x-dominator. F=(f) ! (f ! F) Generalized X-Dominator

28. Generalized X-Dominator

29. Each node in a BDD can be decomposed as a simple MUX Only useful when overlap between its two cofactors is small. Simple MUX Decomposition

30. MUX Decomposition

31. Control Signal is a function Requires 2 nodes which cover all paths. Functional MUX Decomposition

32. Functional MUX Decomposition

33. Functional MUX Decomposition

34. Logic Synthesis

35. Sweep • Constant and Single-Variable Nodes Removal • Removal of Functionally Equivalent Nodes AND AND OR OR x y x 1 x y x y y

36. Eliminate • Balance between Global vrs. Local Scenario • Partial Collapsing

37. Global vrs Local BDDs • Representing large Boolean networks in a global BDD causes serious computational problems. • Leave in multilevel form.

38. BDD Decomposition Engine

39. Order of Decomposition • Simple Dominator • Functional MUX • Single MUX • Generalized Dominator • Generalized X-Dominator

40. Typical Logic Optimization Flow BDDlopt • Boolean Simplification • (Variable Ordering) • Factorization • (Recursive BDD Decomposition) • Logic Sharing • (Detected on Factoring Trees)

41. Factoring Trees • Retains information about the decomposition process • Extract functionally equivalent sub-trees

42. Experimental Results • BDDlopt Vrs SIS-1.2 • AND / OR-Intensive Circuits • Resulting area almost the same • BDDlopt out-performs SIS in CPU time • XOR-Intensive Circuits • BDDlopt uses 23% less gates • BDDlopt uses 14% less area • BDDlopt uses 84.4% less CPU time

43. BDS Vrs SIS Experimental Results • Area is 3% larger • Delay is 13% smaller • Memory Requirement is 30% smaller • Synthesis is 8X faster • Superior performance on large circuits

44. Comparison between BDS and SIS

45. Performance Comparison on Large Circuits

46. Conclusion • Decomposition • Boolean decomposition using cuts and generalized dominators. • Fast algebraic decomposition with simple dominators. • XOR, variable and functional MUX decomposition. • Logic Synthesis • Sweep to remove simple redundancy. • Elimination to remove inter-gate redundancy. • Decomposition to recursively break down large functions in to basic gates.