Binary Decision Diagrams BDDs A Graph Application

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Binary Decision Diagrams (BDDs)A Graph
Application
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Boolean Expressions(propositional logic)

• Boolean expressions are generated from the
  following grammar t x 0 1 not t t1
  or t2 t1 and t2 t1 --gt t2 t1 lt--gt t2where
  x is a boolean variable, and t is a boolean
  expression.
• A boolean expression t (x1,, xn) denotes a
  truth value for each assignment of truth values
  to x1,,xn, according to the standard truth
  tables defining the boolean operators.

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Truth Tables of boolean operators
0 1
0 1
0 0 1 1 1 1
or
and
not 0 1 1 0
0 0 0 1 0 1
0 1
0 1 0 1 0 1
lt--gt
0 1
0 1 1 1 0 1
--gt
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Boolean Expressions

• If we fix the ordering of the variables in the
  boolean expression, we get a function from Bn
  to B.
• f(x1, x2, , xn) 0,1
• f is satisfiable if f(x1, , xn) 1 for some
  truth assignments to x1, , x2.
• f is a tautology if f(x1, , xn) 1 for all
  truth assignments to x1, , x2.
• Ex
• ((x1 or x2) and x3) is satisfiable (ex
  x1/1, x3/1)
• ( x or not x) is a tautology.
• t1 and t2 are equivalent if f1 (x1, , xn) f2
  (x1, , xn) for all truth assignment to x1, , xn.

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Binary Decision DiagramsIf-then-else operator

• Define a new boolean operator if-then-else
• if x (is true) then y0, else (x is false) then y1
• x y0, y1 ((x and y0) or (not x
  and y1))
• In general for x - a boolean variable and t0,
  t1 - boolean expressions
• x t0, t1 is true if either x and t0 are
  true, or x is false and t1 is true.
• Always true t x t(1/x),
  t(0,x) Shannon Expansion

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Binary Decision DiagramsIf-then-else operator

• Claim
• All boolean operators can be be expressed with
  the if-then-else operator and constants 0,1, with
  the following restrictions
• tests are performed only over (positive)
  variable.
• variables appear only in the test part of the
  expression.

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If-the-else operator

• Examples Basic boolean operators
• (x or y) x --gt 1, y--gt1,0
• can be written x --gt 1, t0 t0 --gt1,
  0
• (x and y) x --gt (y --gt1,0),0
• (not x) x --gt 0,1
• (x lt--gt y) x --gt (y --gt 1,0), (y --gt 0,1)

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If-then-else operator

• An if-then-else normal form (INF) is a boolean
  expression built entirely from the if-then-else
  operator and the constants 0 and 1, such that all
  tests are performed only on (not negated)
  variables.
• Claim
• Any boolean expression is equivalent to an INF
  expression.

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If-the-else operator

• Example t ((x1 lt--gty1) and (x2 lt--gt
  y2))Choose an ordering for the variables, to
  order the INF expansion, then t x1 --gt
  t1, t0 t0 y1 --gt 0, t00
  t1 y1 --gt t11,0 t00
  x2 --gt t001, t000 t11 x2 --gt
  t111, t110 t000 y2 --gt 0,1
  t001 y2 --gt 1,0 t110 y2
  --gt 0,1 t111 y2 --gt 1,0
  I

t x1 --gt t1, t0
t0 y1 --gt 0, t00 t1 y1 --gt
t11,0 t00 x2 --gt t001, t000
t11 x2 --gt t001, t000 t000 y2
--gt 0,1 t001 y2 --gt 1,0
II
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If-the-else operator
t x1 --gt t1, t0 t0
y1 --gt 0, t00 t1 y1 --gt t00,0
t00 x2 --gt t001, t000 t000
y2 --gt 0,1 t001 y2 --gt 1,0
III
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A Decision Tree
x1
t t0 y1
t1 y1 x2 t00
t11 x2 t000
y2 t001 y2 t111 y2
t110 y2 1 0 0 1
0 0 1 0 0
1
t x1 --gt t1, t0 t0
y1 --gt 0, t00 t1 y1 --gt t11,0
t00 x2 --gt t001, t000 t11
x2 --gt t111, t110 t000
y2 --gt 0,1 t001 y2 --gt 1,0
t110 y2 --gt 0,1 t111
y2 --gt 1,0

x 0 x1
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Binary Decision Diagram (BDD)
If we identify all identical expressions (or
subtrees), we get a directed acyclic graph
(DAG), which is called a Binary Decision
Diagram, or BDD.
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Binary Decision Diagram
A BDD for t ((x1 lt--gty1) and (x2 lt--gt
y2))with variable ordering x1ltx2lty1lty2
x1
t t0 y1
t1 y1 x2 t00
t11 x2 t000
y2 t001 y2
1 0 0 1 0
0
x1
t t0 y1
t1 y1 x2 t00
t000 y2 t001
y2 1
0

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Reduced, Ordered Binary Decision Diagrams(ROBDDs)

• A reduced, ordered binary decision diagram, is a
  directed acyclic graph G satisfying
• (maximum of) two terminal nodes, representing
  values 1 and 0.
• Nonterminal nodes are labeled with a variable v,
  and pntrs to low(v) and high(v) (maximal out
  degree 2)
• In all paths variables have the same ordering.
  (O)
• For all nodes u,v in G
• var(v) var(u), low(v) low(u), high(v)
  low(u) implies v u
• low(u) not high(u) (R)

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Reducedness
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Canonicity of ROBDDs

• ClaimFor any function f Bn ---gt B and
  ordering x1lt x2ltltxn on the variables of f,
  there is exactly one ROBDD b such thatbv1/x1,
  , vn/xn t(v1, , vn)for all (v1, , vn) in
  Bn
• Consequence
• b is a tautology if and only if b 1
• b is satisfiable if and only if b 0

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ROBDDs

• ROBDD give a compact representation of boolean
  expressions (functions).
• Fast for evaluating equivalence of boolean
  expressions.
• Efficient algorithm for boolean expressions over
  the ROBDD representations.
• Variable ordering has a great influence on the
  size of an ROBDD associated with a boolean
  expression.

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Constructing ROBDDs

• The challenge is to build an OBDD which is
  already reduced, or reducing during construction.
• Representation
• Nodes - represented as numbers 0,1,2, where 0,1
  are reserved for the terminal nodes.
• Variables, ordered x1lt x2lt .. lt xn are
  represented by their indices 1,2,,n.
• An ROBDD is stored in an array T, where each
  entry represents a node u, with vu, lowu and
  highu all indices, namely
• T u --gt (v, low, high)

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Array Implementation
(var ) 1
node var low high
T
0 7 1 7 2 5
1 0 3 6 0
1 4 4 2
3 5 2 4 0 6
3 0 4 7 1
5 6
x1 7
x2 5
x3 6 x4 4
x5 2 3
x6 1 0

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Using a Hash table to ensure Reducedness

• To construct a reduced OBDD b, need to know for
  any given (v,low, high) whether a node with
  these 3-values is already in b?
• To get an efficient reply, maintain a hash table
  H (v, l, h) u mapping triples to
  nodes.
• Note
• H is the inverse of the node table T u (v,
  l, h) (not hashed. Direct access by node ).

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Makenode and Build ROBDD

• T u (i, l, h)
• init(T)
  initialize node table
• u add (T, i, l, h)
  allocate a new node in table
• H (i, l, h) u
• init (H)
  initialize hash table
• b member(H, i, l, h) check if
  (i,l,h) already in T
• u lookup (H, i, l, h) find u
  with (i,l,h)
• insert (H, i, l, h, u)
  make (i,l,h) map to u in
  

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Make a New Node
mk checks whether a node with (i,l,h) already
exists, constructs one if not, and returns the
node (u) O(1)
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
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Build an ROBDD
Building an ROBDD from a boolean expression t,
using the ordering x1
lt x2 lt lt xn
build T,H (t, i) if i gt n then if
t false return 0 else return 1
else v0 build (t0/xi, i1) v1
build (t1/xi, i1) return mk(i,v0,v1)
Initiation buildT,H (t,1)