ECE%20667%20Synthesis%20and%20Verification%20of%20Digital%20Systems - PowerPoint PPT Presentation

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

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ECE 667 Synthesis and Verification of Digital Systems Binary Decision Diagrams (BDD) Outline Background Canonical representations BDD s Reduction rules Construction ... – PowerPoint PPT presentation

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


1
ECE 667Synthesis and Verificationof Digital
Systems
  • Binary Decision Diagrams
  • (BDD)

2
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

3
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 )

4
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

5
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

6
BDD Example
f abacbcd
1
0
  • Two different orderings, same function.

7
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.

8
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

9
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

10
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

11
BDD Construction naïve way
  • Reduced Ordered BDD

12
BDD Reduction Rules -1
  • Eliminate redundant nodes
  • (with both edges pointing to same node)

13
BDD Reduction Rules -2
  • Merge duplicate nodes (isomorphic subgraphs)
  • Nodes must be unique

14
BDD Construction contd
15
BDD Construction the right way
f (a b) c
16
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

17
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
18
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

19
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

20
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

21
Apply Operation - AND
F ? G x (Fx ? Gx) x(Fx ? Gx)
22
Apply Operation - OR
F G x (Fx Gx) x(Fx Gx)
23
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)

24
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

25
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
26
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

27
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)

28
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)

29
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.
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