Binary Decision Diagrams

Transcription

1 Binary Decision Diagrams Sungho Kang Yonsei University

2 Outline Representing Logic Function Design Considerations for a BDD package Algorithms 2

3 Why BDDs BDDs are Canonical (each Boolean function has its own unique BDD, modulo ordering of variables) So two functions are identical if and only if their BDDs are identical BDDs are compact in many cases, compared to SOP Parity trees, adders, control logic (linear vs exponential) However multipliers are bad, and a few others 3

4 Why not SOP and POS The two-level representations of some functions are too large to be practical (e.g. XOR) Passing from SOP to POS and vice versa is difficult Taking the complement is difficult Taking the AND/OR of two sums of products is difficult Since SOP and POS are not canonical forms, answering the equivalence question for two functions is difficult Furthermore, deciding whether a product of sums is satisfiable is NP-complete 4

5 BDDs as MUX Circuits y 2 y 3 y 1 S = x 12 y + x 13 y = x 1 ( x 23 x + x 2 ) + x 12 x y = x + x = x y = x y + x y = x + x , ,

6 BDD Examples f (, 10) = 0 f (, 10) = 1 6

7 Effect of Variable Ordering f = abc + b d + c d 6 nodes vs. 4 nodes optimal a b c d b c a d 7

8 BDDs vs Decision Tree A decision tree is just the result of recursive Boolean expansion 2 decision tree nodes merge into 1 if they are isomorphic A decision tree node is redundant if its two children are identical 8

9 Formal Definition A BDD is a directed acyclic graph representing a multiple output switching function F The function of the terminal node is the constant function 1 The function of an edge is the function of the head node, unless the edge has the complement attribute, in which case the function of the edge is the complement of the function of the node The function of a node v in V is given by l(v)f T +l(v) f E where l(v) is label of v and f T (f E ) is the function of the T(E) edge The function of φ Φ is the function of its outgoing edge 9

10 Formal Definition An edge with (without) the attribute is called a complement (regular) edge BDDs are canonical if All internal nodes are descendents of some node in Φ There are no isomorphic subgraphs For every node, f T F E 10

11 ROBDD Reduced Ordered BDD Iteration of applying identification of isomorphic subgraphs and removal of redundant nodes Two functions are equivalent if they have the same BDD The size of the BDD is exponential in the number of variables in the worst case(e.g. multiplier) However BDDs are well-behaved for many functions that are not amenable to two-level representations The logical AND and OR of BDDs have the same complexity (polynomial in the size of the operands) Complementation is inexpensive Both satisfiability and tautology can be solved in constant time Indeed f is a tautology iff its BDD consists of the terminal node 1 11

12 ROBDD Covering problems can be solved in time linear in the size of BDD representing the constraints BDD sizes depend on the ordering Finding a good ordering is not always simple There are functions for which SOP or POS representations are mode compact than BDDs Unfortunately, many constraint functions of covering problems fall into this category In some cases SOP/POS forms are closer to the final implementation of a circuit For instance, if we want to implement a PLA, we need to generate at some point a SOP or POS form 12

13 Building (RO)BDDs 1 Order variables 2 Recursively cofactor the given SOP, in the specified order, until the cofactors become constants or single variables; 3 Merge any pair of equivalent nodes; 4 Delete any node which has identical child nodes; 5 Repeat the merge and delete steps, until neither applies to any node 13

14 Building (RO)BDDs f = ab f = a+b f a = b, f a = 0 f a = 1, f a = b f ab = 1, f a b = 0 f ab = 1, f a b = 0 14

15 Building (RO)BDDs f = abc + b d + c d f b = ac+c d, f b = d+c d f bc =a, f b c =d, f bc =d, f b c =d Each node of the BDD corresponds to one of the cofactors of the given function Each path corresponds to a minimum satisfying assignment: 15

16 BDDs are Efficient If the BDDs of functions f and g are available, then the following may be checked in constant time: f=0? (hard with SOP) f=1? (hard with SOP) f=g? (hard with SOP or POS) f=g? (hard with SOP or POS) 16

17 BDD for Linear Inequality F = abc F a = bc F a = 0 F ab =c F ab = 0 F a b = 0 F a b = 0 F = abcd 17

18 BDD for Linear Inequality... Linear growth: n variable BDD requires 2n nodes, whereas SOP grows quadratically 18

19 BDD for Linear Inequality d? Isomorphic subgraphs Linear growth: n variable BDD requires 2n nodes, 19

20 Growth of Parity Function? 20

21 Growth of Parity Function? 21

22 BDDs for FSMs BDDs can compactly represent huge sets Widely used to represent the set of reachable states of a Finite State Machine Used in every phase of synthesis and verification of sequential circuits 22

23 BDDs for Large Sets A set of m objects can be represented by a boolean function of n= log 2 (m) variables This function has a unique BDD For n=5, S(x 1,, x n ) represents 20 objects, but,... For n=6, S(x 1,, x n ) represents 40 objects Every added variable doubles the number of minterms 23

24 BDDs for Large Sets Every added variable doubles the number of minterms S = x 1 (x 2 +x 2 x 3 ) + x 1 x 2 = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 5 minterms When n = 4, 10 minterms 24

25 Design Consideration of BDD Package Shared BDD Two equivalent functions can be represented by the same BDD Unique Table Dictionary of functions represented in the DAG Best implemented as a hash table Strong Canonicity Checking for equivalence just requires checking that the pointers in the DAG associated with the functions are identical Attributed Edges Regular edge and complement edge Computed Table As a speed improvement, keep a table of recently computed functions Memory Management and Dynamic Re-Ordering Garbage collection 25

26 Shared BDDs Every node represents a Boolean function d... b c b c d 26

27 BDD Manipulation To implement operators like ITE efficiently, we need to understand to important data structures: The unique table (insures BDD is reduced) The computed table (insures that an operation like ITE(F,G,H) is never repeated) 27

28 BDD Data Structure f i=2, l=1 Node 2, variable x i = x 1 Node 3, variable x i = x 2 i=3, l=2 i=4, l=2 f = x 1 x 2 + x 1 x 2 x 3 n i=5, l=3 L T E

29 BDD Graph Data Structure Node i represents a Boolean function f f = x l(i) f xl(i) + x l(i) f xl(i) (label(i),j 0,j 1 ) (v,f v,f v ) Splitting variable v = x i l( 1) l( 2) K l( i) K l( j ) K l( j ) K 0 1 j 0 j 1 To avoid searching this (often large) data structure, we use a Hash Table 29

30 BDD Graph Data Structure BDD data structures are compact: ~22 bytes/node 10 6 nodes gives about 22MB However storage is still the main issue Spaceout on 1GB machines common Dynamic reordering necessary 30

31 Insuring Uniqueness Internally, each node is represented by a triple: (label(i),j 0,j 1 ) (v,f v,f v ) The label of the node is a variable of the function (the same at each level of the DAG) The children (j 0, j 1 ) of node i represent the cofactors of the node function Each triple is stored in memory and is never duplicated 31

32 The Unique Table i x l(i) = variable at level i Reduction rules insure that each triple is j 0 j 1 unique Hashing Function Key Node l( 1) l( 2) K l( i) K l( j ) K l( j ) K 0 1 j 0 j K i K j K j K

33 The Computed Table Similar to Unique Table, except: The hash key becomes the 4-tuple (ITE, F, G, H) The size of the table is fixed in advance, so that the computed table can fill up, and new table entries start replacing older entries This makes the Computed Table a cache, instead of a hash table 33

34 ITE Operator ITE(F,G,H) = FG + F H ITE(0,0,0) = 0 0 ITE(F,G,0) = FG AND(F,G) ITE(F,G,0) = 0 F>G ITE(F,1,0) = F F ITE(F,0,G) = F G F<G ITE(1,G,0) = G G ITE(F,G,G) = F G XOR(F,G) ITE(F,1,G) = F+G OR(F,G) ITE(F,0,G ) = (F+G) NOR(F,G) ITE(F,G,G ) = (F G) XNOR(F,G) ITE(G,0,1) = G NOT(G) ITE(F,1,G ) = F+G F G ITE(F,0,1) = F NOT(F) ITE(F,G,0) = F +G F G ITE(F,G,1) = (FG) NAND(F,G) ITE(1,1,1) =

35 The ITE Operator ITE can represent all 16 functions of 2 variables ITE(F,G,H) = FG + F H ITE(F,G,0) = FG ITE(F,1,G) = F+G ITE(F,G,0) = 0 ITE(F,0,G) = F G ITE(F,G,G) = F G ITE(F,G,G ) = (F G) AND(F,G) OR(F,G) F>G F<G XOR(F,G) XNOR(F,G) 35

36 ITE Operator ITE(F,G,H) = FG + F H = v (FG + F H) v + v (FG + F H) v = v (F v G v + F v H v ) + v (F v G v + F v H v ) = (v, ITE (F v, G v, H v ), ITE(F v, G v, H v ) ) positive cofactor negative cofactor Terminal cases ITE(1,F,G) = ITE(0,G,F) = ITE(F,1,0) = ITE(G,F,F) = F 36

37 ITE Operator Key Data structure that stores an item in a location of a table identified by its key The key is the triple(v, G, H) that identifies a node in the DAG Hashing function Maps the key into the location in the table The mapping is not one-to-one Possible collision Collision Chain All colliding entries are kept in linked list 37

38 Top Variables f i=2, l=1 Top variable of f is x 1. g i=3, l=2 i=4, l=2 i=5, l=3 Top variable of g is x

39 ITE Algorithm ITE(F,G,H) { (result, terminal_case) = TERMINAL_CASE(F,G,H) if (terminal_case) { return(result) (result, in_computed_table) = COMPUTED_TABLE_HASH_ENTRY(F,G,H) } if (in_computed_table) { return(result) } v = TOP_VARIABLE(F,G,H) T = ITE(Fv, Gv, Hv) E = ITE(Fv, Gv, Hv ) if (T=E) return(t) R = FIND_ADD_UNIQUE_TABLE(v, T, E) INSERT_COMPUTED_TABLE ( (F,G,H), R) RETURN (R) } 39

40 ITE Application I = ITE(F,G,H) = (a, ITE(F a,g a,h a ), ITE(F a,g a,h a ) ) = (a, ITE(1,C,H), ITE(B,0,H) ) = (a, C, (b, ITE(B b,0 b,h b ), ITE(B b,0 b,h b ) ) ) = (a, C, (b, ITE(1,0,1), ITE(0,0,D) ) ) = (a, C, (b,0,d) ) 40

41 BDD Proliferation Larger Base Boolean Algebras,which have more constants (ADDs, MTBDDs do matrix algebra) Edge Attribution Complement dots (ROCBDDs) Numerical Values (EVBDDs) Reduction Rule Variants Zero Suppressed BDDs (ZBDDs do Large sets) Binary Moment Diagrams (BMDs do multipliers) 41

42 ROCBDDs In BDDs (ROBDDs) no two distinct nodes represent the same function In ROCBDDs (reduced, ordered, complemented) no two nodes have the same or complementary functions This magic is worked with a complement dot edge attribute and a transformation of some sub-bdds to a standard form 2X (best case) reduction possible 42

43 Shared BDDs with Complement Arcs b b a b a b? Note: Left Child is always complement of right child, so use complement pointers (2X size reduction) 43

44 Example of Complement Dots (a b) a b 1 (a b c) a b c 1 Can appear only on 0-edges (E-edges) or outputs Function evaluates to 1 on paths with even # of 44

45 Complementation Rules R1 : f = xf x + x f x f = (f ) = (xf x + x f x ) R4 : f = xf x + x f x = (f ) = ( (xf x + x f x ) ) = ((x +f x )(x+f x )) = (xf x + x f x + f x f x ) = (xf x + x f x + f x f x (x + x) ) = (xf x + x f x ) R1 R2 R3 R4 Positive cofactor on left 45

46 Complementation Rules Complementation rules (for canonicity): 1-edge (T-edge, +cofactor) must be regular If complemented function needed on 1-edge, replace appropriate right subgraph with corresponding left subgraph 46

47 Conditioning of ITE Cells If we could identify all triple that give the same result, we could map all of them into the same entry of the computed table In general, finding all such triples is too expensive, but there are special cases that can be identified with very little effort Two arguments are the same function ITE(F,F,G) Two arguments are one of the complement of the other ITE(F,G,F ) One or more arguments are constants ITE(F,G,0) These checks can be performed in constant time 47

48 ITE_CONSTANT Algorithm ITE_CONSTANT (F,G,H) { (result, terminal_case) = TERMINAL_CASE(F,G,H) if (terminal_case) { return(result) } (result, in_computed_table)=computed_table_hash_entry(f,g,h) if (in_computed_table) { return(result) } v = TOP_VARIABLE(F,G,H) T = ITE_CONSTANT(Fv, Gv, Hv) if ( T 0 and T 1) return non_constant E = ITE_CONSTANT(Fv, Gv, Hv ) if (T E) return non_constant INSERT_COMPUTED_TABLE ( (F,G,H), T) return T } 48

49 Example 1 f=abd+ab d+a c+a c d =a(bd+b d)+a (c+c d) =a(b d)+a (c+d) BDD with complement edges 49

50 Example 1 - continued f=abd+ab d+a c+a c d =a(bd+b d)+a (c+c d) ROBDD with complement edges Rule 4 Rule 4 Rule 4 50

51 Example 1 -continued f=abd+ab d+a c+a c d ROBDD with complement edges Count path parity of complement dots to decide function value f f f f f f ( 11,,, 1) = 0 ( 10,,, 0) = 0 ( 10,,, 1) = 1 ( 0,, 1, ) = 1 ( 0,, 0, 1) = 1 ( 011,,, ) = 1 (1 dot: odd parity ) (3 dots: odd parity ) (2 dots: even parity ) (2 dots: even parity ) (2 dots: even parity ) (2 dots: even parity ) 51

52 Example 2 - continued F = a + b c G = b c + d H = b + c + d I = (F,G,H) =(a, ITE(F,G,H), ITE(F a, G a, H a ) ) = (a, ITE(1,G,H), ITE(p,G, H)) = (a, G, (b,ite(0,d,1),ite(c,q,r) ) ) Cofactor commutes = (a, G, (b,1,ite(c,q,r) ) ) with complementation = (a, G, (b,1,(c,ite(1,1,d),ite(0,d,1)))) = (a, G, (b,1,(c,1,1))) = (a, G, (b,1,1)) = (a,g,1) 52

53 Example 2 - continued F = G = b + cd G = b c + d H = b + c + d Note: (b,1,p) identified as F, and (b,1,c) as q by Procedure FIND_OR_ADD_UNIQUE_TABLE I = ITE(F,F,H)=ITE(F,1,H)=ITE(H,1,F) = (a,ite(h a,1,f a ),ITE( H a,1, F a )) = (a,(b,ite(1,1,1),ite(0,1,p)),(b,ite(1,1,1),ite(c,1,p))) = (a,(b,1,p),(b,1,(c,ite(1,1,d),ite(0,1,0)))) = (a,f,(b,1,(c,1,0))) = (a,f,(b,1,c)) = (a,f,q) 53