3-vlsicad-bdd

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VLSI CAD:Logic to Layout

Rob A. RutenbarUniversity of Illinois

Lecture 3.1

Computational BooleanAlgebra Representations:

BDD Basics, Part 1


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2013, R.A. Rutenbar

Representations, Revisited URP tautology is a great

warm up example

Shows big idea: Boolean functions asthings we manipulate with software

Data structure + operators But URP not the real way(s) we do it

Lets look at a real, important,elegantway to do this

Binary Decision Diagrams: BDDs

Slide 2

b 11 01 11

c 11 11 01

bc 11 10 10

fab fab

c 11 11 01

c 11 11 10

1 11 11 11c 11 11 01


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Background: BDDs Originally studied by many

Lee first, then Boute and Akers

Got seriously useful in 1986 Randy Bryant of CMU made

breakthrough ofROBDDs

Randy also providedmany of the nice BDD

pictures in this lecture

thanks!

Slide 3

http://www.cs.cmu.edu/~bryant/


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Binary Decision Diagrams for Truth Tables Big Idea #1: BDD

Turn a truth table for the Booleanfunction into a Decision Diagram

In simplest case, graph is just a tree By convention dont draw arrows on

the edges, we know where they go

Vertex represents a variable; edge out is a decision (0 or 1)

Follow green (dashed) line for value 0

Follow red (solid) line for value 1

Function value determined by leaf value.

Truth Table

Decision Tree

x3 x3x2

x3 x3x2

x1

Slide 4


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Binary Decision Diagrams Some terminology

A variable vertex

The lo pointer

to lo child of

vertex

The hi pointer

to hi child ofof vertex

A constant vertex

at the bottom

leaves of the tree

The variable ordering, which is the

order in which decisions about variables

are made. Here, its X1 X2 X3

0 0

x3

0 1

x3

x2

0 1

x3

0 1

x3

x2

x1

Slide 5


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Ordering Note: Different variable orders are possible

Order for this subtree is

X2then X3

Here, itsX3then X20 0

x3

0 1

x3

x2

0 1

x3

0 1

x3

x2

x1

0 0

x3

0 1

x3

x2

x2 x2

x3

x1

Slide 6


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Binary Decision Diagrams Observations

Each path from root to leaf traverses variables in a some order Each such path constitutes a row of the truth table, ie, a decision about what output

is when variables take particular values But we have not yet specified anything about the orderof decisions This decision diagram is not unique for this function

Terminology: Canonical form

Representation that does not depend on gate implementation of a Boolean function Same function of same variables always produces this exact samerepresentation Example: a truth table is canonical. We want a canonical form data structure.

Slide 7


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Binary Decision Diagrams Whats wrong with this diagram representation?

Its notcanonical, and it is way too big to be useful (its as big as truth table!)

Big idea #2: Ordering Restrictglobal ordering of variables Means:

Note Its OK to omita variable if you dont need to check it to decide which leaf node

to reach for final value of functionSlide 8


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Ordering BDD Variables Assign arbitrary orderingto vars: x1 < x2 < x3

Variables must appear in this specific order along all paths; ok to skip vars

Properties No conflicting assignments along path (see each var at most once on path).

OK No!x

1

x2x3

x1

x3

x3

x2x1

x1

x1

Slide 9


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Binary Decision Diagrams OK, now whats wrong with it?

Variable ordering simplifies things... But still too big, and not canonical

0 0

x3

0 1

x3

x2

0 1

x3

x1

Original decision diagram

Equivalent, butdifferentdiagram

0 0

x3

0 1

x3

x2

0 1

x3

0 1

x3

x2

x1

Slide 10


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VLSI CAD:

Logic to Layout


Lecture 3.2


BDD Basics, Part 2


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2013, R.A. Rutenbar

Binary Decision Diagrams Big Idea #3: Reduction

Identify redundancies in the graph that can remove unnecessary nodes and edges Removal ofX2 node and its children, replace with X3 node is an example of this

Why are we doing this? GRAPH SIZE: Want result as smallas possible CANONICAL FORM: Forsamefunction, given same variable order,

want there to be exactly one graph that represents this function

Slide 12

0 0

x30 1

x3x2

0 1

x3x1


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Reduction Rules Reduction Rule 1: Merge equivalent leaves

Just keep one copy of each constant leaf anything else is totally wasteful Redirect all edges that went into the redundant leaves into this one kept node

Apply Rule 1 to our example...

0 0

x3

0 1

x3

x2

0 1

x3

0 1

x3

x2

x1

x3 x3

x2

x3

0 1

x3

x2

x1

x3 x3

x2

x3

0 1

x3

x2

x1

Slide 13


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Reduction Rule 2: Merge isomorphic nodes

Isomorphic = 2 nodes with same variable and identicalchildren Cannot tell these nodes apart from how they contribute to decisions in graph Note: meansexact same physicalchild nodes, not just children with same label

Reduction Rules

y

x

z

x

y

x

z

Remove redundant node(extra x node).Redirect all edges that went

into the redundant node intothe one copy that you kept

(edges into right x node nowinto left as well)

Slide 14


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Reduction Rules Apply Rule 2 to our example

x3 x3

x2

x3

0 1

x3

x2

x1

x3 x3

x2

x3

0 1

x3

x2

x1

Slide 15


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Reduction Rules Reduction Rule #3: Eliminate Redundant Tests

Test: means a variable node: redundant since children go to same node... ...so we dont care what value x node takes in this diagram

y

x

Remove redundant nodeRedirect all edges intoredundant node (x) into child

(y) of the removed node

Slide 16


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Reduction Rules Apply Rule #3 to our example and were finished!

Aside: How to apply the rules? For now, iteratively: when you cant find another match, graph is reduced Is this how programs really do it? No we will talk about that later

x3

x2

0 1

x3

x2

x1

x3

x2

0 1

x3

x2

x1

x2

0 1

x3

x1

Slide 17


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2013, R.A. Rutenbar

BDDs: Big Results Recap: What did we do?

Start with a BDD, order the vars, reduce the diagram Name: Reduced Ordered BDD (ROBDD)

Big result

Same function always generates exactly same graph... forsame var ordering Two functions identical if and only if ROBDD graphs are isomorphic (ie, same)

Nice property to have: Simplest form of graph is canonicalSlide 18


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VLSI CAD:

Logic to Layout


Lecture 3.3


BDD Sharing


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BDDs: Representing Simple Things Note: can represent any function as a ROBDD

ROBDD forf(x1,x2,...xn) = 0 ROBDD forf(x1,, x,xn) = x

ROBDD forf(x1,x2,...xn) = 1NOTE: In a ROBDD, a Boolean function

is really just a pointer to the root node ofa canonical graph data structure

Slide 20


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BDD: Examples Assume variable order is X1, X2, X3, X4

Odd Parity

Linearrepresentation

Typical Function

(X1 + X2 )X4No vertex labeled X3

independent ofX3Many subgraphs shared

x2

0 1

x4

x1

x2

x3

x4

10

x4

x3

x2

x1

Slide 21


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Sharing in BDDs Very important technical point

Every BDD node (notjustroot) represents someBoolean function in a canonicalway BDDs good at extracting & representing sharing of subfunctions in subgraphs

(X1 + X2 )X4

x2

0 1

x4

x1

x2

x3

x4

10

x4

x3

x2

x1

( x3!!x4 )

X1X2

X3

X4

x3!!x4

x4

Slide 22


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BDDs Sharing: Consider a 4bit Adder Look at sum S3 msb & carry out Cout

4 bit binary adder

a3 b3 a0 b0a1 b1a2 b2

S3

b3

b2

b1

0

b0

a0

b1

a1

b2

a2

1

b3

a3

Cout

b3

b2

b1

b0

a0

b1

a1

b2

a2

b2

b1

0

b0

a0

b1

a1

1

b2

a2

b3

a3

S3

Xor

Negative

LogicCarryChain

PositiveLogicCarryChain

Basically sameshared

subfunction

Cout

Slide 23


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BDD Sharing: Multi-rooted Graph Dont represent it twice!

BDD can have multipleentry points, or roots

Called a multi-rooted BDD

Recall (again!) Everynode in a BDD represents

some Boolean function

This multi-rooting idea just explicitlyexploits this to better share stuff

b3 b3

a3

Cout

b3

b2

b1

b0

a0

b1

a1

b2

a2

b2

b1

0

b0

a0

b1

a1

1

b2

a2

b3

a3

S3

PositiveLogicCarryChain

2 roots

Slide 24


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Sharing: Multi-rooted BDD Graphs Why stop at 2 roots?

Big savings forsets of functions Minimize size overseveralseparate

BDDs by maximum sharing

Example: Adders Separately

51nodes for4-bit adder 12,481 for64-bit adder

Shared 31 nodes for4-bit adder 571 nodes for64-bit adder

b3 b3

a3

Cout

b3

b2 b2

a2

b2 b2

a2

b3

a3

S3

b2

b1 b1

a1

b1 b1

a1

b2

a2

S2

b1

a0 a0

b1

a1

S1

b0

10

b0

a0

S0

Slide 25


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VLSI CAD:

Logic to Layout


Lecture 3.4


BDD Ordering


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How Are BDDs Really Implemented Recursive methods, like URP

Shannon cofactor divide & conquer is key

As a set ofoperators (ops) onthe BDDs

AND, OR, NOT, EXOR, CoFactor Quant,Quant, Satisfy, etc

Operate on a universe ofBoolean data as input/output Constants 0,1; vars; Bool functions

represented as shared BDD graphs

Big trick: Implement each op so ifinputs shared, reduced, ordered,

outputs are too

Slide 27

H = (bdd)OP(bdd F, bdd G)

bdd nodes


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BDDs: Build Up Incrementally

For example: by walking a gate-level network

Each inputis a BDD, each gatebecomesan operatorthat produces a new outputBDD

Build BDD for Out as a scriptof callsto basic BDD operators

A

B

C

T1

T2

Out

A B C

T1 T2

Out

0 1

a

0 1

c

0 1

b

0 1

b

a

0 1

c

b

c

b

0 1

b

a

BDD operator script

1. A = CreateVar(A)2. B = CreateVar(B)3. C = CreateVar(C)4. T1 = And(A,B)5. T2 = And(B,C)6. Out = Or(T1,T2)

Slide 28


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Apps: Comparing Logic Implementations Are these two Boolean

functions F, G the same?

Build BDD forF. Build BDD forG

Compare pointers to roots ofF, G If pointers are same(!!), F==G

What inputs make functionsF, G give different answers?

Build BDD forF. Build BDD forG.

Build the BDD forH = FG

Ask satisfiable forH (!!)

Slide 29

bdd nodes bdd nodes


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More Very Useful BDD Applications Tautology checking

Was complex with cubelist URP,subtle algorithm, lots of work

With BDDs, its trivial. Just build theBDD, check if BDD graph ==

Satisfiability Find values (0,1) for vars so F == 1? No idea how to do it with cubelists Any pathfrom root to 1 leaf is soln!

x2

0 1

x4

x1 Satisfiability: X1 X2 X3 X4 =

Slide 30


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BDDs: Seem Too Good To Be True?! Problem : Variable ordering matters. Ex: a1b1 + a2b2 + a3b3

Good Ordering Bad Ordering

Linear Growth Exponential Growth

0

b3

a3

b2

a2

1

b1

a1

a3 a3

a2

b1 b1

a3

b2

b1

0

b3

b2

1

b1

a3

a2

a1

Slide 31


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Variable Ordering: Consequences Interesting problem

Some problems known to be exponentially hard are very easydone with BDDs(!) Trouble is, they are easy only if size of the BDD for the problem is reasonable Some problems make nice (small) BDDs, others make pathological (large) BDDs No universal solution (or could always to solve exponentially hard problems easily)

How to handle? Variable ordering heuristics:make nice BDDs for reasonable probs Characterization: know which problems nevermake nice BDDs (eg, multipliers) Dynamic ordering: let the BDD software package pick the order on the fly

Slide 32


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Variable Ordering: Intuition Rules of thumb for BDD ordering

Related inputsshould be neareach other in order; groupsof inputs that candetermine function by themselves should be (i) close together, and (ii) neartop of BDD

0

b3

a3

b2

a2

1

b1

a1

a3 a3

a2

b1 b1

a3

b2

b1

0

b3

b2

1

b1

a3

a2

a1

Slide 33

Ex:(a1b1 + a2b2 + a3b3)


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Aside: Variable Ordering Arithmetic circuits are important logic; how are their BDDs?

Many carry chain circuits have easy linear sized ROBDD orderings:Adders, Subtractors, Comparators, Priority encoders

Rule is alternatevariables in the BDD order: a0 b0 a1 b1 a2 b2 an bn So are all arithmetic circuits easy then? No, sorry

General experience with BDDs Many tasks have reasonable OBDD sizes; algorithms practical to ~100M nodes People spend a lot of effort to find orderings that work

Function Class Best Order Worst Order

Addition linear exponential

Multiplication exponential exponential

Slide 34


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BDD Summary Reduced, Ordered, Binary Decision Diagrams, ROBDDs

Canonical form a data structure for Boolean functions Two boolean functions the same if and only if they have identical BDD A Boolean function is just a pointer to the root node of the BDD graph Every node in a (shared) BDD represents some function With shared BDD, test (F==G) check pointers for equality, point to same root Basis for much of todays general manipulation or Boolean stuff

Problems Variable ordering matters; sometimes BDD is just too big Often, we just want to know SAT dont need to build the whole function

Slide 35

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