2 Binary Decision Diagrams 2. Binary Decision Diagrams Verification - PowerPoint PPT Presentation

2. Binary Decision Diagrams 27 2.1 BDD concepts — Example: which Boolean function is represented by the following BDD? y g a 1 0 b 0 0 1 1 0 0 1 1  The function associated with a node can be determined only if the functions associated with the successor nodes are known bf(v) = var(v)· bf(v 0 ) + var(v)· bf(v 1 )

2. Binary Decision Diagrams 28 2.1 BDD concepts  " Bottom-up procedure": 1. Step a 1 0 b b 0 1 0 1 Functions Functions 0 and 1 0 and 1

2. Binary Decision Diagrams 29 2.1 BDD concepts  " Bottom-up procedure": 2. Step a 1 bf(v) = var(v)· bf(v 0 ) + var(v)· bf(v 1 ) 0 b b 0 1 = b · 0 + b · 1 = b 0 1 Functions Functions 0 and 1 0 and 1

2. Binary Decision Diagrams 30 2.1 BDD concepts  " Bottom-up procedure": 3. Step bf( ) bf(v) = var(v)· bf(v 0 ) + var(v)· bf(v 1 ) ( ) bf( ) ( ) bf( ) a = a · 0 + a · b = a · b 1 1 bf(v) = var(v)· bf(v 0 ) + var(v)· bf(v 1 ) 0 b b 0 1 = b · 0 + b · 1 = b 0 1 Functions Functions 0 and 1 0 and 1

2. Binary Decision Diagrams 31 2.1 BDD concepts  There are many variants of binary decision diagrams   Most useful and common: OBDD's (Ordered Binary Most useful and common: OBDD s (Ordered Binary Decision Diagrams, Bryant 1986)  OBDD properties: OBDD properties:  Ordered : The variables appear in a fixed ordering on all paths — Technically, an index (a positive integer) is associated with each variable index(var(v)) — For each node v with successors v 0 and v 1 we have: index(var(v)) < index(var(v 0 )) und index(var(v)) < index(var(v )) index(var(v)) < index(var(v 1 ))

2. Binary Decision Diagrams 32 2.1 BDD concepts     a a b b c c variable i bl order a, b, c index(a) = 1 a 0 1 index(b) = 2 index(b) = 2 b b 1 1 0 0 index(c) = 3 index(c) 3 c c c c 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1

2. Binary Decision Diagrams 33 2.1 BDD concepts  OBDD properties (cont'd.):  Reduced :  Reduced : — The function represented by one node is different from the functions of all other nodes — The two successors of each node are distinct

2. Binary Decision Diagrams 34 2.1 BDD concepts — Reduction example: a a 1 0 1 0 b b b b 0 1 0 1 0 0 1 1 1 1 1 0 1 0 Several representations of 1 a 1 0 b 0 0 1 1 Identical Identical successors 1 0

2. Binary Decision Diagrams 35 2.1 BDD concepts  Simplified representations exist, e.g.,  1-edges to the right 0-edges to the left  1-edges to the right, 0-edges to the left  Edges to 0 omitted  etc  etc. a — Example: (a  b) · (c  d) · (e  f)  or: 0 edges are dashed lines  or: 0 edges are dashed lines b b b b c 1 d d 0 all e e others f f 1 0

2. Binary Decision Diagrams 36 2.1 BDD concepts Now: Boolean function  OBDD   Example above: (a  b) · (c  d) · (e  f)  Example above: (a  b) · (c  d) · (e  f)  Let          F ( (a b) (c ) ( d) (e ) ( f) ) (a ( b) r )  Variable ordering a, b, c, d, e, f  Following Boole s expansion theorem, we have the  Following Boole's expansion theorem, we have the following cofactors of F w.r.t. a a 1 1 0 0 a       a       F F (1 (1 b) r b) r b r b r F F (0 (0 b) r b) r b r b r

2. Binary Decision Diagrams 37 2.1 BDD concepts  More expansions: a 1 0 a      a      F (1 b) r b r F (0 b) r b r a 1 0 0 b b b b 1 0 0 1 ab  ab    F 0 F 0 F F r ab ab etc.

2. Binary Decision Diagrams 38 2.1 BDD concepts  The problem of reduction:  In the example above it was easy to detect F  In the example above it was easy to detect   F F F r r ab ab and to merge the nodes for and F F ab ab  Redundant nodes have to be removed Redundant nodes have to be removed  Redundant nodes — Either represent the same function Either represent the same function — Or have identical successors (easy to detect)   How to know that two nodes represent the same function? How to know that two nodes represent the same function?

2. Binary Decision Diagrams 39 2.1 BDD concepts  Two functions     f af a f g ag a g a a a a a a are equal iff they have identical cofactors b b f a g a 1 1 1 1     f a g a a 1 a 1 1 1 f g c c a a 0 1 0 1 g a g a f a f a

2. Binary Decision Diagrams 40 2.1 BDD concepts  If we presume that all successor nodes of two nodes represent distinct functions then the two nodes represent p p identical functions iff the direct successor nodes are pairwise identical a a 0 0 1 1 0 0 1 1 g a g g g a f a f f f a  This results in a simple bottom-up -procedure: redundant nodes are eliminiated in the redundant nodes are eliminiated in the bottom-level first, the in the second level, etc.

2. Binary Decision Diagrams 41 2.1 BDD concepts a 0 1 b b 1 1 0 0 0 c c c c 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1. Step: 1. Step: 0/1 leafs a 0 1 b b 1 1 0 0 all all c c c c others 1 1 0 0 0 1

2. Binary Decision Diagrams 42 2.1 BDD concepts 2. Step: a 1 c nodes d b b 1 1 0 0 all c c c c 1 others 1 0 0 0 1 a 0 1 b b 1 1 0 0 all others c c 0 1 0 1

2. Binary Decision Diagrams 43 2.1 BDD concepts a 0 1 3. Step b nodes b nodes b b 1 0 1 0 all ll c c others 0 1 1 0 1  We can decide that the two b nodes do not  We can decide that the two b-nodes do not represent the same function by means of the c- nodes

2. Binary Decision Diagrams 44 2.1 BDD concepts  Example: derive the OBDD for the following function, variable order r e g variable order r,e,g p = eg + rg + reg r e g p 0 0 0 0 0 0 1 1 p r = eg + g = g 0 1 0 0 1 0 1 1 0 1 1 0 2 3 = 8 cases p r = eg + eg 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 r Traffic- e p Light Checker g

2. Binary Decision Diagrams 45 2.1 BDD concepts p r = eg + g = g p = eg + rg + reg p g g g r r p r = eg + eg + 1 0 0 e g g g 0 0 1 1

2. Binary Decision Diagrams 46 2.1 BDD concepts  Reduction was necessary in the original concept by R. Bryant (1986), but can be avoided completely (s. Sect. 2.3) y ( ), p y ( )

2. Binary Decision Diagrams 47 2.1 BDD concepts  OBDD´s can be implemented easily by means of 2:1-Multiplexors p  1 1 x x 0 1 0 1 _ & & fx fx _ fx fx x _ fx fx x x

2. Binary Decision Diagrams 48 2.1 BDD concepts — Example: a 0 0 1 1 a 0 1 c b b 0 1 0 1 c c c 0 0 0 0 1 0 1

2. Binary Decision Diagrams 49 2.1 BDD concepts  Given a certain variable ordering, OBDD´s are canonical representations of Boolean functions, i.e., there exists p , , exactly one OBBD-representation for each Boolean function  Two circuits implementing the same function have a identical OBDD's 0 1 b b a 1 1 =1 0 0 b 0 1 = a a a a 0 1 & a b b 1 1 & & 0 0 0 0 b b & 0 1 b

2. Binary Decision Diagrams 50 2.1 BDD concepts Ci Circuit 1: it 1 OBDD´ OBDD´s S 3 S 2 S & 1 S G & 0 D 1 3 & & 1 B & & 3 c n+4 E & =1 & & 3 3 P & 1 A & 3 & F =1 3 =1 D & 2 & 1 B Q & 3 2 & 1 E & & 2 & 1 A & 2 & F =1 =1 2 2 =1 1 D 1 & & Q 1 2 B & & 1 1 A=B & & E & 1 & 1 A 1 & F =1 1 =1 D Q & 0 1 & 1 1 B & & 0 & & E E F 0 =1 0 & 1 =1 A 0 Q 0 & & M = c n Circuit 2: S 3 S 2 S S & & 1 S & G 0 D 1 3 & & 1 B & & 3 c n+4 E & =1 & 3 P & 1 A & 3 & F =1 3 =1 D & 2 & 1 1 B Q Q & & 3 2 1 & E & 2 & 1 & A 2 & F =1 2 =1 D 1 & & Q 1 2 B & & 1 1 A=B & & E & 1 & & 1 1 A A 1 & F =1 1 =1 D Q & 0 & 1 1 1 B & & 0 & E F 0 =1 0 & 1 =1 A 0 Q 0 & & M c n

2. Binary Decision Diagrams 51 2.2 Variable Orderings 2.2 Variable Orderings  The variable ordering has a critical impact on the size of the OBDD (= #nodes)   There are static and d namic proced res to determine There are static and dynamic procedures to determine "good" orderings

2. Binary Decision Diagrams 52 2.2 Variable orderings  The number of nodes of a OBDD depends critically on the variable ordering  Classical example (Bryant 1986): x 1 x 1 f = x 1 x 2 + x 3 x 4 + x 5 x 6 0 0 1 1 0 1 x 3 x 3 x 2 0 0 1 0 1 1 1 x 3 x 5 x 5 x 5 x 5 0 1 0 1 0 1 0 1 0 1 x 2 x x x 2 0 0 x 2 x 2 0 x 4 0 0 1 0 1 1 1 1 x 5 x 5 x 4 x 4 x 4 x 4 1 1 0 0 0 1 x 6 x 6 1 1 0 0 6 1 1 0 0 1 0 1 0

2. Binary Decision Diagrams 53 2.2 Variable orderings  Example: n-bit adder: — Order R 1 : a n , b n , a n-1 , b n-1 ,..., a 0 , b 0 — Order R 2 : a n , a n-1 ,..., a 0 , b n , b n-1 ,..., b 0 n= 8 16 32 64 time time 0 02 0.02 0.03 0 03 0 11 0.11 0 19 0.19 R 1 : #nodes 35 75 155 315 time 0.39 16.34   R 2 : #nodes 750 196574

2. Binary Decision Diagrams 54 2.2 Variable orderings  Calculating the best order may result in exponential run time  For a given circuit, "good" orderings can be heuristically determined — Example: Distribution of a "weight" 1/4 z & 1/ 1/2 1/4 4 x 1/2 1/2 & & & 1 y 1/4 1/2 1/4 & & x 1/ 4 4 Sum of weights: x=1/2, y=1/4, z=1/4,  first use x for expansion

2. Binary Decision Diagrams 55 2.2 Variable orderings — Delete selected variable and distribute weight again weight again 1/4 z z & 1/2 1/4 & 3/4 3/4 & & 1 1 1/2 y 3/4 1/2 & Sum of weights : y=3/4, z=1/4,  next use y for expansion — Order: x, y, z

2. Binary Decision Diagrams 56 2.2 Variable orderings  Sifting: dynamic ordering procedure (Rudell ICCAD´93)  Basic step: exchange two adjacent variables (Fujita  Basic step: exchange two adjacent variables (Fujita et al. EDAC´91) a 0 1 b b 1 1 0 0 c c 1 1 0 0 0 0 1

2. Binary Decision Diagrams 57 2.2 Variable orderings  Principle: exchange 0-1 and 1-0 path b 0 1 c c 0 1 0 1 g 0 g 1 g 2 g 3 c 0 1 b b 0 1 0 1 g 0 g 2 g 1 g 3

2. Binary Decision Diagrams 58 2.2 Variable orderings a 0 0 1 1 b b b b 1 1 1 1 0 0 c c 1 1 0 0 0 0 0 0 1 1

2. Binary Decision Diagrams 59 2.2 Variable orderings a 0 0 1 1 c c c c 1 1 1 1 0 0 b b 1 1 0 0 0 0 0 0 1 1

2. Binary Decision Diagrams 60 2.2 Variable orderings  Sifting-procedure:  Calculate variable with max #nodes (the "thickest"  Calculate variable with max. #nodes (the thickest part of the OBDD)  Shift variable over OBDD by pairwise exchange of y p g adjacent variables  Shift variable to a position where #nodes is minimal Minimum Minimum 0 1

2. Binary Decision Diagrams 61 2.2 Variable orderings  Movie "Sifting" by Stefan Höreth:

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2. Binary Decision Diagrams 70 2.2 Variable orderings  im Detail: V4 V3 0 1 V3 V3 V4 V4 V5 V5 V5 V5 1 1

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2. Binary Decision Diagrams 90 2.3 OBDD Construction 2.3 OBDD Construction a 0 1 b 0 1 a a  1 ? & b c c 0 0 1 1 0 0 1 1

2. Binary Decision Diagrams 91 2.3 OBDD construction  Principle: build OBDD while traversing the circuit from inputs to outputs p p OBDD-Package a 0 1 C-Program Traverser Traverser 0 1 a a  1 C-Program & b C-Program  1 c g b C-Program & 0 1 * 0 1

2. Binary Decision Diagrams 92 2.3 OBDD construction a 0 OBDD-Package a 0 1 b C-Program 1 0 1 Traverser Traverser 0 1 0 1 a a  1 C-Program & b C-Program  1 C-Programm  1 C-program  1 c g p g g b c C-Programm & 0 1 0 1 0 1 0 1

2. Binary Decision Diagrams 93 2.3 OBDD construction a 0 OBDD-Package a 0 1 b C-Program 1 0 1 a Traverser Traverser 0 0 1 1 0 1 a a b b  1 0 1 C-Program & b C-Program  1 c g b c c C-Programm & C-Program & 0 0 1 0 1 1 0 0 1 0 1 1

2. Binary Decision Diagrams 94 2.3 OBDD construction  Orthogonality of Boole's expansion f+g = x*(f x + g x ) + x*(f x + g x ), f*g = x*(f x * g x ) + x*(f x * g x ), f = x*f x + x*f x g f * x x x x 0 1 1 0 * * f x f x g x g x

2. Binary Decision Diagrams 95 2.3 OBDD construction  AND-operation of two OBDD´s  Assumption: nodes are represented as triples  Assumption: nodes are represented as triples (x,v0,v1) var low high access-functions f ti

2. Binary Decision Diagrams 96 2.3 OBDD construction function AND(bdd1, bdd2): IF bdd1 0 OR bdd2 0 THEN IF bdd1=0 OR bdd2=0 THEN return 0; t 0 ELSEIF bdd1=1 THEN return bdd2; ELSEIF bdd2=1 THEN return bdd1; ELSE var1:=var(bdd1);var2:=var(bdd2); ( ); ( ); IF var1=var2 THEN x:=var1; v0:= AND(low(bdd1), low(bdd2)), v1:= AND(high(bdd1),high(bdd2)); ( g ( ), g ( )); ELSEIF index(var1) < index(var2) THEN x:=var1; v0:= AND(low(bdd1), bdd2), ( ( ), ), v1:= AND(high(bdd1), bdd2); ELSEIF ... IF v0 = v1 THEN return v0 ELSE return (x v0 v1); IF v0 = v1 THEN return v0 ELSE return (x,v0,v1); ...

2. Binary Decision Diagrams 97 2.3 OBDD construction 3 a 5  1 & b c c 4 a 0 * * c b 0 1 1 0 1 0 1 0 1 3 4 bdd1 bdd2 var1=a var2=c => index(var1) < index(var2)

2. Binary Decision Diagrams 98 2.3 OBDD construction 3 a 5  1 & b c 4 a 0 0 * * c b 0 1 1 0 0 1 1 0 1 0 1 3 3 4 4 bdd1 bdd2 var1=a var2=c => index(var1) < index(var2) x:=var1 := a x:=var1 := a v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)

2. Binary Decision Diagrams 99 2.3 OBDD construction 3 a 5  1 & b c 4 a 0 * * c b 0 1 1 0 1 0 1 0 1 3 3 4 4 bdd1 bdd2 var1=b var2=c => index(var1) < index(var2)

2. Binary Decision Diagrams 100 2.3 OBDD construction 3 a 5  1 a & 0 b 1 1 c 4 b a 0 1 0 0 * * c c b 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 3 3 4 4 5 5 bdd1 bdd2 var1=b var2=c => index(var1) < index(var2) x:=var1 := b x:=var1 := b v0:= and(low(bdd1),bdd2), v1:= and(high(bdd1),bdd2)