1.  2001 CiesielskiBDD Tutorial1 Decision Diagrams Maciej Ciesielski Electrical & Computer Engineering University of Massachusetts, Amherst, USA ciesiel@ecs.umass.edu

2.  2001 CiesielskiBDD Tutorial2 Common Representations Boolean functions ( f : B  B ) –Truth table, Karnaugh map –SoP, PoS, ESoP –Reed-Muller expansions (PPRM, FPRM, GRM, etc.) –Decision diagrams (BDD, ZDD, KFDD, *BMD, etc.) Arithmetic functions ( f : B  Int ) –Binary Moment Diagrams (*BMD, K*BMD, *PHDD) –Algebraic Decision Diagrams (ADD) Arithmetic functions (f : Int  Int ) –Taylor Expansion Diagrams (TED)

3.  2001 CiesielskiBDD Tutorial3 Canonical Representations Each minimal, canonical representation is characterized by –Decomposition type Shannon, Davio, moment decomposition, Taylor, etc. –Reduction rules Redundant nodes, isomorphic sub-graphs, etc –Composition method (“Apply” rule) They can represent –Boolean functions (f : B  B) –Arithmetic functions (f : B  Int ) –Algebraic expressions (f : Int  Int )

4.  2001 CiesielskiBDD Tutorial4 Decomposition Types Shannon expansion f = x f x + x’ f x’ Positive Davio (moment decomposition): replace x’=1-x f = x f x + (1-x) f x’ = f x’ + x f  x where f  x = f x - f x’ Negative Davio f = f x + (1-x) f  x’

5.  2001 CiesielskiBDD Tutorial5 Binary Decision Diagrams ( BDD ) Based on recursive Shannon expansion f = x f x + x’ f x’ 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

6.  2001 CiesielskiBDD Tutorial6 Sannon Expansion  BDD f a’ = f(a=0) = bc b 0 f = ac + bc a f f a = f(a=1) = c + bc g b’ = (bc) |b=0 = 0 g= bc h= c + bc g b = (bc) |b=1 = c h b’ = (c+bc) |b=0 = c h b = (c+bc) |b=1 = c b c 1

7.  2001 CiesielskiBDD Tutorial7 BDD Reduction Rules -1 1.Eliminate redundant nodes (with both edges pointing to same node) f = a’ g(b) + a g(b) = g(b) (f a + f a’ = 1) b g a b f g

8.  2001 CiesielskiBDD Tutorial8 BDD Reduction Rules -2 2. Merge duplicate nodes (isomorphic subgraphs) Nodes must be unique f 1 = a’ g(b) + a h(c) = f 2 f = f 1 = f 2 aa bc h g f1f1 f2f2 a bc g h f

9.  2001 CiesielskiBDD Tutorial9 BDD Construction Reduced Ordered BDD 1 edge 0 edge a b c f 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 Truth table f = ac + bc Decision tree 1 0 001010 a b c b ccc f

10.  2001 CiesielskiBDD Tutorial10 BDD Construction – cont’d 10 a b c b ccc ff 10 a b c b c 10 a b c f = (a+b)c 2. Merge duplicate nodes 1. Merge terminal nodes 3. Remove redundant nodes

11.  2001 CiesielskiBDD Tutorial11 Logic Manipulation using BDDs Useful operators ¬ 1 0 0 1 FF’ 0 1 F(x,y) x=b 0 1 F(y) Restrict – Restrict: F| x=b = F(x=b) where b = const – Complement ¬ F = F’ (switch the terminal nodes)

12.  2001 CiesielskiBDD Tutorial12 APPLY Operator Apply: F G, any Boolean operation (AND, OR, XOR,  ) =  F G 0 1 0 1 0 1 F G   Useful in constructing BDD for arbitrary Boolean logic Any logic operation can be expressed using Restrict, Apply Efficient algorithms, work directly on BDD graphs

13.  2001 CiesielskiBDD Tutorial13 BDD: APPLY Operation Basic operator for efficient BDD manipulation (structural) Based on recursive Shannon expansion F OP G = x (F x OP G x ) + x’(F x’ OP G x’ ) where OP = OR, AND, XOR, etc Works directly on BDD

14.  2001 CiesielskiBDD Tutorial14 BDD: APPLY Operation - AND 10 a c ac a AND c 10 a 2 c 10 3 0.3 2.3 a c 1.3 1.11.0 AND = =

15.  2001 CiesielskiBDD Tutorial15 BDD: APPLY Operation - OR OR ac 10 a c 4 5 bc 10 b c 6 7 = = 10 a b c f = ac+bc c 4+6 0+0 a 7+5 1 0+6 b 6+5 0+5 0 0+7

16.  2001 CiesielskiBDD Tutorial16 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 ) 10 a b c F = a’bc + abc +ab’c G = ac +bc 10 a b c 

17.  2001 CiesielskiBDD Tutorial17 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 in function variables, gives a satisfying solution (test vector) ab ab’c H 0 1 a b c