ECE 697B (667) Spring 2006 Synthesis and Verification of Digital Systems

1. ECE 697B (667)Spring 2006Synthesis and Verificationof Digital Systems Binary Decision Diagrams (BDD)

2. Outline • Background • Canonical representations • BDD’s • Reduction rules • Construction of BDD’s • Logic manipulation of BDD’s • Application to verification and SAT • Reading: read one of the BDD tutorials available on class web site • Anderson, or • Somenzi ECE 667 - Synthesis & Verification - Lecture 10

3. Common Representations • Boolean functions ( f : B  B ) • Truth table, Karnaugh map • SoP, PoS, ESoP • Reed-Muller expansions (XOR-based) • Decision diagrams (BDD, ZDD, etc.) • Each minimal, canonical representation is characterized by • Decomposition type • Shannon, Davio, moment decomposition, Taylor exp., etc. • Reduction rules • Redundant nodes, isomorphic sub-graphs, etc. • Composition method (“Apply”, compose rule) • What they represent • Boolean functions (f : B  B) • Arithmetic functions (f : B  Int ) • Algebraic expressions (f : Int Int ) ECE 667 - Synthesis & Verification - Lecture 10

4. Binary Decision Diagrams (BDD) • Based on recursive Shannon expansion f = x fx + x’ fx’ • Compact data structure for Boolean logic • can represents sets of objects (states) encoded as Boolean functions • Canonical representation • reduced ordered BDDs (ROBDD) are canonical • essential for verification ECE 667 - Synthesis & Verification - Lecture 10

5. ROBDD’s • Directed acyclic graph (DAG) • one root node, two terminals 0, 1 • each node, two children, and a variable • Shannon co-factoring tree, except reduced and ordered (ROBDD) • Reduced: • any node with two identical children is removed • two nodes with isomorphic BDD’s are merged • Ordered: • Co-factoring variables (splitting variables) always follow the same order along all paths xi1 < xi2 < xi3 < … < xin ECE 667 - Synthesis & Verification - Lecture 10

6. Example Two different orderings, same function. root node a f = ab+a’c+bc’d a 1 c+bd b c c+bd b b 0 c+d d d+b c c c d b b d 0 1 0 1 ECE 667 - Synthesis & Verification - Lecture 10

7. ROBDD Ordered BDD (OBDD) Input variables are ordered - each path from root to sink visits nodes with labels (variables) in ascending order. not ordered ordered order = a,c,b a a c c c b b b c 1 1 0 0 Reduced Ordered BDD (ROBDD) - reduction rules: • if the two children of a node are the same, the node is eliminated: f = vf + vf • if two nodes have isomorphic graphs, they are replaced by one of them These two rules make it so that each node represents a distinct logic function. ECE 667 - Synthesis & Verification - Lecture 10

8. f v 0 1 fv fv Efficient Implementation of BDD’s • BDDs is a compressed Shannon co-factoring tree: • f = v fv + v fv • leafs are constants “0” and “1” • Three components make ROBDDs canonical (Proof: Bryant 1986): • unique nodes for constant “0” and “1” • identical order of case splitting variables along each paths • hash table that ensures: • (node(fv) = node(gv)) Ù (node(fv) = node(gv)) Þ node(f) = node(g) • provides recursive argument that node(f) is unique when using the unique hash-table ECE 667 - Synthesis & Verification - Lecture 10

9. Onset is Given by all Paths to “1” F = b’+a’c’ = ab’+a’cb’+a’c’all paths to the 1 node Notes: • By tracing paths to the 1 node, we get a cover of pair wise disjoint cubes. • The power of the BDD representation is that it does not explicitly enumerate all paths; rather it represents paths by a graph whose size is measures by its nodes and not paths. • A DAG can represent an exponential number of paths with a linear number of nodes. • BDDs can be used to efficiently represent sets • interpret elements of the onset as elements of the set • f is called the characteristic function of that set f a 0 1 fa = cb’+c’ c 1 fa= b’ b 0 0 1 0 1 ECE 667 - Synthesis & Verification - Lecture 10

10. f f v 1 0 mux v 0 1 g h g h Implementation Variables are totally ordered: If v< w then v occurs “higher” up in the ROBDD Top variable of a function f is a variable associated with its root node. Example:f = ab + a’bc + a’bc’. Order is (a < b < c). fa = b, fa = b b is top variable of f f a f b reduced f does not depend on a, since fa = fa . Each node is written as a triple:f = (v,g,h) where g = fv and h = fv . We read this triple as: f = if v then g else h = ite (v,g,h) = vg+v ’ h b 0 1 0 1 v is top variable of f ECE 667 - Synthesis & Verification - Lecture 10

11. f = ac + bc f 1 edge a b c f 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 0 edge a b b c c c c 0 0 0 0 0 1 1 1 Truth table Decision tree BDD Construction • Reduced Ordered BDD ECE 667 - Synthesis & Verification - Lecture 10

12. f a f = fa g(b) + fa’ g(b) = g(b) (fa + fa’ = 1) g g b b BDD Reduction Rules -1 • Eliminate redundant nodes (with both edges pointing to same node) ECE 667 - Synthesis & Verification - Lecture 10

13. f f1 f2 a a a h g h g b c b c f1 = fa’ g(b) + fa h(c) = f2 f = f1 = f2 BDD Reduction Rules -2 • Merge duplicate nodes (isomorphic subgraphs) • Nodes must be unique ECE 667 - Synthesis & Verification - Lecture 10

14. f f f = (a+b)c a a a b b b b b c c c c c c c 0 0 0 1 1 1 • 2. Merge duplicate nodes 3. Remove redundant nodes 1. Merge terminal nodes BDD Construction – cont’d ECE 667 - Synthesis & Verification - Lecture 10

15. F G G F =   0 0 1 1 0 1 APPLY Operator • Useful in constructing BDD for arbitrary Boolean logic • Any logic operation can be expressed using Apply (ITE) • Efficient algorithms, work directly on BDD graphs • Apply: F G, any Boolean operation (AND, OR, XOR, ) ECE 667 - Synthesis & Verification - Lecture 10

16. F F’ ¬ F(y) F(x,y) Restrict x=b 0 0 0 1 1 1 1 0 Logic Manipulation using BDDs • Useful operators • Complement ¬ F = F’ • (switch the terminal nodes) • Restrict: F|x=b = F(x=b) where b = const ECE 667 - Synthesis & Verification - Lecture 10

17. Useful BDD Operators – Apply Operation • Basic operator for efficient BDD manipulation (structural) • Based on recursive Shannon expansion F OP G = x (FxOP Gx) + x’(Fx’OP Gx’) whereOP = OR, AND, XOR, etc ECE 667 - Synthesis & Verification - Lecture 10

18.  F G G F =   0 0 1 1 0 1 Apply Operation (cont’d) • Apply: F G where stands for any Boolean operator (AND, OR, XOR, )  • Any logic operation can be expressed using only Restrict and Apply • Efficient algorithms, work directly on BDDs ECE 667 - Synthesis & Verification - Lecture 10

19. a AND c ac a 2 a a = = c 3 1.3 2.3 AND c c 1.0 0.3 0 0 0 1.1 1 1 1 Apply Operation - AND ECE 667 - Synthesis & Verification - Lecture 10

20. ac f = ac+bc bc 4 a 6 b a a OR b 5 7 = = b 0+7 0+6 0+5 4+6 6+5 7+5 c c c c 0+0 0 0 0 0 1 1 1 1 Apply Operation - OR ECE 667 - Synthesis & Verification - Lecture 10

21. a a F = a’bc + abc +ab’c G = ac +bc b b  c c 0 0 1 1 Application to Verification • Equivalence Checking of combinational circuits • Canonicity property of BDDs: • if F and G are equivalent, their BDDs are identical (for the same ordering of variables) ECE 667 - Synthesis & Verification - Lecture 10

22. H a ab b c 1 0 ab’c Application to SAT • Functional test generation • SAT, Boolean satisfiability analysis • to test for H = 1 (0), find a path in the BDD to terminal 1 (0) • the path, expressed infunction variables, gives a satisfying solution (test vector) • Problem: size explosion ECE 667 - Synthesis & Verification - Lecture 10

23. Efficient Implementation of BDD’s Unique Table: • avoids duplication of existing nodes • Hash-Table: hash-function(key) = value • identical to the use of a hash-table in AND/INVERTER circuits Computed Table: • avoids re-computation of existing results hash value of key collision chain hash value of key No collision chain ECE 667 - Synthesis & Verification - Lecture 10