ECE 697B (667) Spring 2006 Synthesis and Verification of Digital Systems

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ECE 697B (667)Spring 2006Synthesis and
Verificationof Digital Systems

• Binary Decision Diagrams
• (BDD)

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Outline

• Background
• Canonical representations
• BDDs
• Reduction rules
• Construction of BDDs
• Logic manipulation of BDDs
• Application to verification and SAT
• Reading
• read one of the BDD tutorials available on class
  web site
• Anderson, or
• Somenzi

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Common Representations

• Boolean functions ( f B ? B )
• Truth table, Karnaugh map
• SoP, PoS, ESoP
• Reed-Muller expansions (XOR-based)
• Decision diagrams (BDD, ZDD, etc.)
• Each minimal, canonical representation is
  characterized by
• Decomposition type
• Shannon, Davio, moment decomposition, Taylor
  exp., etc.
• Reduction rules
• Redundant nodes, isomorphic sub-graphs, etc.
• Composition method (Apply, compose rule)
• What they represent
• Boolean functions (f B ? B)
• Arithmetic functions (f B ? Int )
• Algebraic expressions (f Int ? Int )

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Binary Decision Diagrams (BDD)

• Based on recursive Shannon expansion
• f x fx x fx
• 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

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ROBDDs

• Directed acyclic graph (DAG)
• one root node, two terminals 0, 1
• each node, two children, and a variable
• Shannon co-factoring tree, except reduced and
  ordered (ROBDD)
• Reduced
• any node with two identical children is removed
• two nodes with isomorphic BDDs are merged
• Ordered
• Co-factoring variables (splitting variables)
  always follow the same order along all paths
• xi1 lt xi2 lt xi3 lt lt xin

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Example
root node
a
f abacbcd
a
1
cbd
b
c
cbd
b
b
0
cd
d
db
c
c
c
d
b
b
d
0
1
0
1

• Two different orderings, same function.

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ROBDD

• Ordered BDD (OBDD) Input variables are ordered -
  each path from root to sink visits nodes with
  labels (variables) in ascending order.

not ordered
ordered order a,c,b
a
a
c
c
c
b
b
b
c
1
1
0
0

• Reduced Ordered BDD (ROBDD) - reduction rules
• if the two children of a node are the same, the
  node is eliminated f vf vf
• if two nodes have isomorphic graphs, they are
  replaced by one of them
• These two rules make it so that each node
  represents a distinct logic function.

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Efficient Implementation of BDDs

• BDDs is a compressed Shannon co-factoring tree
• f v fv v fv
• leafs are constants 0 and 1
• Three components make ROBDDs canonical (Proof
  Bryant 1986)
• unique nodes for constant 0 and 1
• identical order of case splitting variables along
  each paths
• hash table that ensures
• (node(fv) node(gv)) Ù (node(fv) node(gv)) Þ
  node(f) node(g)
• provides recursive argument that node(f) is
  unique when using the unique hash-table

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Onset is Given by all Paths to 1
F bac abacbac all paths to the 1
node
f
a
0
1
fa cbc
c
1
fa b
b
0
0
1
0
1

• Notes
• By tracing paths to the 1 node, we get a cover of
  pair wise disjoint cubes.
• The power of the BDD representation is that it
  does not explicitly enumerate all paths rather
  it represents paths by a graph whose size is
  measures by its nodes and not paths.
• A DAG can represent an exponential number of
  paths with a linear number of nodes.
• BDDs can be used to efficiently represent sets
• interpret elements of the onset as elements of
  the set
• f is called the characteristic function of that
  set

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Implementation

• Variables are totally ordered If v lt w then v
  occurs higher up in the ROBDD
• Top variable of a function f is a variable
  associated with its root node.
• Example f ab abc abc. Order is (a lt b
  lt c).
• fa b, f?a b

b is top variable of f
f
a
f
b
reduced
b
f does not depend on a, since fa f?a . Each
node is written as a triple f (v,g,h) where g
fv and h f?v . We read this triple
as f if v then g else h ite (v,g,h) vgv
h
0
1
0
1
v is top variable of f
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BDD Construction

• Reduced Ordered BDD

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BDD Reduction Rules -1

• Eliminate redundant nodes
• (with both edges pointing to same node)

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BDD Reduction Rules -2

• Merge duplicate nodes (isomorphic subgraphs)
• Nodes must be unique

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BDD Construction contd
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APPLY Operator

• Apply F G, any Boolean operation
• (AND, OR, XOR, ?)

• Useful in constructing BDD for arbitrary Boolean
  logic
• Any logic operation can be expressed using Apply
  (ITE)
• Efficient algorithms, work directly on BDD graphs

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Logic Manipulation using BDDs

• Useful operators

• Complement F F
• (switch the terminal nodes)

• Restrict Fxb F(xb) where b const

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Useful BDD Operators Apply Operation

• Basic operator for efficient BDD manipulation
  (structural)
• Based on recursive Shannon expansion
• F OP G x (Fx OP Gx) x(Fx OP Gx)
• where OP OR, AND, XOR, etc

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Apply Operation (contd)

• Apply F G
• where stands for any Boolean operator (AND,
  OR, XOR, ?)

?

• Any logic operation can be expressed using only
  Restrict and Apply
• Efficient algorithms, work directly on BDDs

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Apply Operation - AND
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Apply Operation - OR
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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)

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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)

• Problem size explosion

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Efficient Implementation of BDDs

• Unique Table
• avoids duplication of existing nodes
• Hash-Table hash-function(key) value
• identical to the use of a hash-table in
  AND/INVERTER circuits
• Computed Table
• avoids re-computation of existing results

hash value of key
collision chain
hash value of key
No collision chain