In The Name of God

1
In The Name of God

• Binary Decision Diagrams

By Bijan Alizadeh
2
Binary Decision Diagrams

• Classical representation of logic functions
  Truth Table, Karnaugh Maps, Sum- of- Products,
  critical complexes, etc.
• Critical drawbacks
• - May not be a canonical form or is too large
  (exponential) for useful functions. Equivalence
  and tautology checking is hard
• - Operations like complementation may yield a
  representation of exponential size

3
Binary Decision Diagrams

• Reduced Ordered Binary Decision Diagrams (ROBDDs)
• A canonical form for Boolean functions
• Often substantially more compact than
  traditional normal forms
• Can be efficiently manipulated
• Introduced mainly by R. E. Bryant (1986).
• Various extensions exist that can be adapted to
  the situation at hand (e. g., the type of circuit
  to be verified)

4
Binary Decision Trees

• A Binary decision Tree (BDT) is a rooted,
  directed graph with terminal and nonterminal
  vertices
• Each nonterminal vertex v is labeled by a
  variable var( v) and has two successors
• - low( v) corresponds to the case where the
  variable v is assigned 0
• - high( v) corresponds to the case where the
  variable v is assigned 1
• Each terminal vertex v is labeled by value( v)
  . 0, 1

5
Binary Decision Trees

• Example BDT for a two- bit comparator, f( a1 ,a2
  ,b1 ,b2 )
• (a1 ? b1 ) ? (a2 ? b2 )

6
Binary Decision Trees

• We can decide if a truth assignment x (x 1 ,
  ..., x n ) satisfies a formula in BDT in linear
  time in the number of variables by traversing the
  tree from the root to a terminal vertex
• - If var( v) . x is 0, the next vertex on the
  path is low( v)
• - If var( v) . x is 1, the next vertex on the
  path is high( v)
• - If v is a terminal vertex then f( x ) f v
  (x1 , ..., xn ) value( v)
• - If v is a nonterminal vertex with var( v)
  xi , then the structure of the tree is obtained
  by Shanons expansion
• fv (x1 , ..., xn )
• ?xi ? flow(v) (x 1 , ..., x n ) ? xi ?
  fhigh(v) (x1 , ..., xn )
• For the comparator, (a1 ? 1, a2 ? 0, b1 ? 1, b2 ?
  1) leads to a terminal vertex labeled by 0, i.
  e., f( 1, 0, 1, 1) 0
• Binary decision trees are redundant
• - In the comparator, there are 6 subtrees with
  roots labeled by b2 , but not all are distinct
• Merge isomorphic subtrees
• - Results in a directed acyclic graph (DAG), a
  binary decision diagram (BDD)

7
(Canonical Form Property)Reduced Ordered BDD

• two Boolean functions are logically equivalent
  iff they have isomorphic representations
• This simplifies checking equivalence of two
  formulas and deciding if a formula is satisfiable
• Two BDDs are isomorphic if there exists a
  correspondence between the graphs such that
• - Terminals are mapped to terminals and
  nonterminals are mapped to nonterminals
• - For every terminal vertex v there exists a
  terminal vertex v , value( v) value( v) ,
  and
• - For every nonterminal vertex v there exists a
  terminal vertex v
• var( v) var( v) , low( v) low( v) , and
  high(v) high(v)
• Bryant (1986) showed that BDDs are a canonical
  representation for Boolean
• functions under two restrictions
• (1) the variables appear in the same order along
  each path from the root to a terminal
• (2) there are no isomorphic subtrees or
  redundant vertices
• Þ Reduced Ordered Binary Decision Diagrams
  (ROBDDs)

8
Canonical Form Property

• Requirement (1) Apply total order lt on the
  variables in the formula
• if vertex u has a nonterminal successor v , then
  var( u) lt var( v)
• Requirement (2) repeatedly apply three
  transformation rules
• 1. Remove duplicate terminals eliminate all but
  one terminal vertex with a given label and
  redirect all arcs to the eliminated vertices to
  the remaining one

9
Canonical Form Property

• 2. Remove duplicate nonterminals if nonterminals
  u and v have var( u) var( v) , low( u) low(
  v) and high( u) high( v) , eliminate one of the
  two vertices and redirect all incoming arcs to
  the other vertex

3. Remove redundant tests if nonterminal vertex
v has low( v) high( v) , eliminate v and
redirect all incoming arcs to low( v)
10
Canonical Form Property

• A canonical form is obtained by applying the
  transformation rules until no further application
  is possible
• Applications
• -checking equivalence verify isomorphism
  between ROBDDs
• -non- satisfiability verify if ROBDD has
  only one terminal node, labeled by 0
• -tautology verify if ROBDD has only one
  terminal node, labeled by 1
• Example
• ROBDD of 2- bit Comparator with variable order
  a1 lt b1 lt a2 lt b2

11
Variable Ordering Problem

• The size of an ROBDD depends critically on the
  variable order
• For order a 1 lt a 2 lt b 1 lt b 2 , the comparator
  ROBDD becomes

• For an n- bit comparator
• a1 lt b1 lt ... lt an lt bn gives 3n 2 vertices
  (linear complexity)
• a1 lt ... lt an lt b1 ... lt bn , gives 32n - 1
  vertices (exponential complexity!)

12
Variable Ordering Problem

• The problem of finding the optimal variable order
  is NP- complete
• Some Boolean functions have exponential size
  ROBDDs for any order (e. g., multiplier)
• Heuristics for Variable Ordering
• Heuristics developed for finding a good variable
  order (if it exists)
• ROBDDs tend to be smaller when related variables
  are close together in the order (e. g., ripple-
  carry adder)
• Variables appearing in a subcircuit are related
  they determine the subcircuits output Þ
  should usually be close together in the order
• Dynamic Variable Ordering
• Useful if no obvious static ordering heuristic
  applies
• During verification operations (e. g.,
  reachability analysis) functions change, hence
  initial order is not good later
• Good ROBDD packages periodically internally
  reorder variables to reduce ROBDD size
• Basic approach based on neighboring variable
  exchange ... lt a lt b lt ... Þ ...lt b lt a lt ...

13
Logic Operations on ROBDDs

• Residual function (cofactor) b Î 0, 1
• f (x1,..., xn ) xi b f( x1 , ..., xi-1 , b,
  xi1 , ..., xn )
• ROBDD of f computed by a depth- first traversal
  of the ROBDD of f
• (1) For any vertex v which has a pointer to a
  vertex w such that var( w) xi , replace the
  pointer by low(w) if b is 0 and by high(w) if b
  is 1.
• (2) If not in canonical form, apply Reduce to
  obtain ROBDD of
• f xi b.
• All 16 two- argument logic operations on Boolean
  function implemented efficiently on ROBDDs in
  linear time in the size of the argument ROBDDs.

14
Logic Operations on ROBDDs

• Based on Shannons expansion
• f Ø x Ù f x 0 Ú x Ù f x 1
• Bryant (1986) gave a uniform algorithm, Apply,
  for computing all 16 operations
• f f an arbitrary logic operation on Boolean
  functions f and f
• v and v the roots of the ROBDDs for f and f ,
  x var( v) and x var( v)
• Consider several cases depending on v and v
• (1) v and v are both terminal vertices f f
  value( v) value( v)
• (2) x x use Shannons expansion
• f f Ø x Ù (f x 0 f x 0)
  Ú x Ù (f x 1 f x 1)
• to break the problem into two subproblems, each
  is solved recursively
• The root is v with var( v) x
• Low( v) is (f x 0 f x 0 )
• High( v) is (f x 1 f x 1 )

15
Logic Operations on ROBDDs

• (3) x lt x f x 0 f x 1 f since f
  does not depend on x
• In this case the Shannons expansion simplifies
  to
• f f Ø x Ù (f x 0 f) Ú x Ù (f x
  1 f), similar to (2)
• and compute subproblems recursively,
• (4) x lt x similar to the case above
• Improvement using the if- then- else (ITE)
  operator
• ITE( F, G, H) F . G F . H where F, G and
  H are functions
• Recursive algorithm based on the following, v is
  the top variable (lowest index)
• ITE( F, G, H) v.( F. G F. H)v v.( F. G
  F. H)v
• v.( Fv .Gv Fv .Hv ) v.( Fv .Gv Fv
  .Hv )
• (v, ITE( Fv , Gv , Hv ), ITE( Fv , Gv , Hv
  ))
• With terminal cases being F ITE( 1, F, G)
  ITE( 0, G, F) ITE( F, 1, 0) ITE( G, F, F)
• we define NOT( F) ITE( F, 0, 1) AND( F, G)
  ITE( F, G, 0)
• OR( F, G) ITE( F, 1, G) XOR( F, G)
  ITE( F, Ø G, G)
• etc.

16
Logic Operations on ROBDDs

• By using dynamic programming, it is possible to
  make the ITE algorithm polynomial
• (1) The result must be reduced to ensure that it
  is in canonical form
• - record constructed nodes ( unique table )
• - before creating a new node, check if it
  already exists in this unique hash table

17
Logic Operations on ROBDDs

• (2) Record all previously computed functions in
  a hash table ( computed table )
• - must be implemented efficiently as it may grow
  very quickly in size
• - before computing any function, check table for
  solution already obtained

18
Logic Operations on ROBDDs

• Return Build (t,1)
•  function Build (t,i)
• if i ? n then
• if t is false then
• return 0
• else return 1
• else
• ? Build (t0/ , i1)
• ? Build (t1/ , i1)
• return MK(i , , )
• end Build
•  

MKT, H(i, l, h) if l h then return l
else if member(H, i, l, h) then return
lookup(H, i, l, h) else u ?
add(T,i,l,h) insert(H, i, l, h, u) return u
unique Table
19
Example

• F (x1 x2) x3

20
Example
T u ? (i,l,h)

• Return Build (t,1)
•  function Build (t,i)
• if i ? n then
• if t is false then
• return 0
• else return 1
• else
• ? Build (t0/ , i1)
• ? Build (t1/ , i1)
• return MK(i , , )
• end Build

MKT, H(i, l, h) if l h then return l
else if member(H, i, l, h) then return
lookup(H, i, l, h) else u ?
add(T,i,l,h) insert(H, i, l, h, u) return u
H (i,l,h) ? u
21
Example
T u ? (i,l,h)
22
Operations on ROBDD

• Construct the ROBDD resulting from applying op on
  u1 and u2
• ApplyT,H(op,u1,u2)
• init(G) //Computed Table
• return APP(u1,u2)
• function APP(u1,u2)
• if G(u1,u2) ? empty then return G(u1,u2)
• else if u1 ? 0,1 and u2 ? 0,1 then u ?
  op(u1,u2)
• else if var(u1) var(u2) then
• u ? MK( var(u1) , APP( low(u1) , low(u2) ) ,
  APP( high(u1) , high(u2) ) )
• else if var(u1) ? var(u2) then
• u ? MK( var(u1) , APP( low(u1) , u2 ) , APP(
  high(u1) , u2 ) )
• else //var(u1) ? var(u2)
• u ? MK( var(u2) , APP( u1 , low(u2) ) , APP(
  u1 , high(u2) ) )
• G(u1,u2) ? u
• return u
• end APP

23
Example of Apply

• Compute AND of two ROBDDs

24
Example of Apply
25
Example of Apply
T u ? (i,l,h)
G(u1,u2) ? APP(u1,u2)
H (i,l,h) ? u
26
Example of Apply
T u ? (i,l,h)
27
Operations on ROBDD

• Restrict the ROBDD u according to the truth
  assignment b/xj
• RESTRICTT,H(u,j,b)
• return res(u)
• function res(u)
• if var(u) ? j then return u
• else if var(u) ? j then return
• MK( var(u), res( low(u) ), res( high(u) ) )
• else ( var(u) j ) if b 0 then
  return
• res( low(u) )
• else ( var(u) j, b 1 ) return
• res( high(u) )
• end res

28
Example of RESTRICT

• F (x1 ? x2) x3 x20 gt j2 b0

T u ? (i,l,h)
H (i,l,h) ? u
29
Example of RESTRICT

• After applying RESTRICT algorithm

30
Example of RESTRICT

• Reconstruction phase

31
Example of RESTRICT
T u ? (i,l,h)
H (i,l,h) ? u
32
Operations on ROBDD

• SatCountT(u) Return number of valid truth
  assignments of u
• SatCountT(u)
• return 2var(u)-1 count(u)
• function count(u)
• if u0 then res ? 0
• else if u1 then res ? 1
• else
• res ? 2var(low(u))-var(u)-1 count(low(u))
• 2var(high(u))-var(u)-1 count(high(u))
• Return res
• end count

33
Operations on ROBDD

• AnySat(u) Return a satisfying truth assignment
  for u
• AnySat(u)
• if u0 then Error
• else if u1 then
• else if low(u)0 then return xvar(u) ?1,
  AnySat(high(u))
• else return xvar(u) ?0, AnySat(low(u))

34
Operations on ROBDD

• AllSat(u) Return all satisfying truth assignment
  for u
• AllSat(u)
• if u0 then ltgt
• else if u1 then ltgt
• else return
• ltadd xvar(u) ?0 in front of all
  truth-assignment in
• AllSat(low(u)),
• add xvar(u) ?1 in front of all
  truth-assignment in
• AllSat(high(u))gt

35
Derived Operations

• Compose Given F(x) and G(y), return F(G(y))

• Compose is reduced to two operations Restrict and
  three operations Apply

36
Derived Operations

• Quantifications
• Given a function F(x1, x2, x3)
• Existential Quantification
• ?x1F(x1, x2, x3) F(0, x2, x3) OR F(1, x2, x3)
• Universal Quantification
• ?x1F(x1, x2, x3) F(0, x2, x3) AND F(1, x2,
  x3)
• Unique Quantification
• !x1F(x1, x2, x3) F(0, x2, x3) XOR F(1, x2,
  x3)

37
Logic Operations on ROBDDs

• Complement edges can reduce the size of an ROBDD
  by a factor of 2
• - Only one terminal node is labeled 1
• - Edges have an attribute (dot) to indicate if
  they are inverting or not
• - To maintain canonicity, a dot can appear only
  on low( v) edges
• - Complementation achieved in O( 1) time by
  placing a dot on the function edge
• - F and F can share entry in computed table
• - Adaptation of ITE easy

38
References

• H.R. Andersen, An Introduction to Binary
  Decision Diagrams, Lecture notes 1997. Web
  http//www.it.dtu.dk/hra
• T. Kropf, Introduction To Formal Verification,
  Springer 1999.
• K.S. Brace, R.L. Rudell and R.E. Bryant,
  Efficient Implementation of a BDD Package, 27th
  ACM/IEEE Design Automation Conference 1990, pp.
  40.
• E. M. Clarke, O. Grumberg and D.A. Peled, Model
  Checking, 1999 E.M. Clarke, O. Grumberg and
  Lucent Technologies.

39
Variable Ordering

• Window Permutation Algorithm
• Example window size k 3 Strating point x2
• k!-1 total variable exchange k(k-1)/2
  exchanging to go to best position

40
Variable Ordering

• Sifting Algorithm
• Starting point x4

41
Variable Ordering

• Group Sifting Algorithm
• is an extension of sifting. It moves a group of
  variables at the time, instead of a single
  variable.
• Symmetric variables
• A boolean function f(x1,x2,,xn) is symmetric in
  xi and xj if the interchange of xi and xj leaves
  the function identically the same.
• Once two variables are identified as symmetric,
  they are locked together. This effectively
  leads to an algorithm that sifts groups of
  variables of varying size.

42
Variable Ordering

• Relative Absolute Position (RAP)
• In each iteration do Sifting and Symmetry
  checking (Window Permutation)
• 1. At each iteration, a variable that has not
  been sifted yet is chosen
• 2. In each Sifting, Check it against its
  neighbor for symmetry
• 3. If symmetry is found, a group is formed and
  the relative position is fixed
• References
• 1. Who Are the Variables in Your
  Neighborhood, Fabio Somenzi
• 2. Dynamic Variable Ordering for Ordered
  Binary Decision Diagrams, Richard Rudell