ECE%20667%20Synthesis%20and%20Verification%20of%20Digital%20Systems

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ECE 667Synthesis 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 terminal nodes 0, 1 (sinks)
• Each node has exactly two children, associated
  with 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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BDD Example
f abacbcd
1
0

• 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 the same order.

not ordered
ordered a,c,b

• Reduced Ordered BDD (ROBDD) - reduction rules
• if the two children of a node are the same, the
  node is eliminated
• f v f v f
• 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 along each path
• 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 BDD encodes 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
  pairwise 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
  measured by the number of the nodes, and not
  paths.
• A DAG can represent an exponential number of
  paths with a linear size (number of nodes) in
  terms of its variables.
• 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.

Reduction (redundant node) fa b, f?a 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

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BDD Construction naïve way

• 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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BDD Construction the right way
f (a b) c
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Logic Manipulation using BDDs

• Useful operators

• Complement F F
• (switch the terminal nodes)

• Restrict Fxb F(xb) where b const

• Restrict Fxb F(xb) where b const

• To restrict variable x to 1, reconnect all
  incoming edges to nodes x to their 1-nodes
• To restrict variable x to 0, reconnect all
  incoming edges to nodes x to their 0-nodes

• To restrict variable x to 1, reconnect all
  incoming edges to nodes x to their 1-nodes
• To restrict variable x to 0, reconnect all
  incoming edges to nodes x to their 0-nodes

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Restrict Operator ( f (c0, d1) )
f (ad)(bc)adbc
fc (ad)b
fcd (a1)b b
fcd b
Set c 0
Set d 1
Restricted BDD
Original BDD
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Useful BDD Operators Apply Operation

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

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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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Apply Operation (contd)

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

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

?

• Any logic operation can be expressed using only
  Restrict and Apply
• Efficient algorithms, work directly on BDDs
• Apply can be used to construct a BDD bottom-up
• From primary inputs, through internal logic
  gates, to output

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Apply Operation - AND
F ? G x (Fx ? Gx) x(Fx ? Gx)
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Apply Operation - OR
F G x (Fx Gx) x(Fx Gx)
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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 key (v,G,H), where F
  ITE(v,G,H).
• avoids duplication of existing nodes
• Hash-Table hash-function(key) value
• identical to the use of a hash-table in
  AND/INVERTER circuits

hash value of key
collision chain

• Computed Table key (F,G,H)
• avoids re-computation of existing results

hash value of key
No collision chain
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Unique Table - Hash Table
hash index of key
collision chain

• Before a node (v, g, h ) is added to BDD data
  base, it is looked up in the unique-table. If
  it is there, then existing pointer to node is
  used to represent the logic function. Otherwise,
  a new node is added to the unique-table and the
  new pointer returned.
• Thus a strong canonical form is maintained. The
  node for f (v, g, h ) exists iff(v, g, h ) is
  in the unique-table. There is only one pointer
  for (v, g, h ) and that is the address to the
  unique-table entry.
• Unique-table allows single multi-rooted DAG to
  represent all users functions

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Computed Table

• Keep a record of (F, G, H ) triplets already
  computed by the ITE operator
• software cache ( cache table)
• simply hash-table without collision chain (lossy
  cache)

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Extension - Complement Edges

• Combine inverted functions by using complemented
  edge
• similar to circuit case
• reduces memory requirements
• BUT MORE IMPORTANT
• makes some operations more efficient (NOT, ITE)

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Extension - Complement Edges

• To maintain strong canonical form, need to
  resolve 4 equivalences

Solution Always choose one on left, i.e. the
then leg must have no complement edge.