multilevel ml

8/13/2019 Multilevel ML

1/28

Multi Level Minimization 1: Network Model

2-level minimization a la ESPRESSO

Manipulates (reshapes) SOP covers of functions

Heuristic: REDUCE - EXPAND - IRREDUNDANT

Multi-level minimization, where final form of logic network is not

just 2-level SOP AND-OR form

New, more general model of logic networks

New operators: forms of division

New heuristic minimization strategies to use this model + operators


2/28

Why Multi-Level Forms

Area = gates + literals, i.e., things that take space on a chip

Delay = max levels of logic gates required to compute function

2-level is minimumdelay possible (ignoring loading and fan-in effects),but usually worston area

area

delay

typical 2-level design =many gates, but only 2 levelsof logic, so fastest possible

multi-level designs =fewer gates, but > 2 levels

small,few gates+literals

big,many gates+literals

fastest, 2 levels

slower, >2 levels


3/28

Why Multi-Level?

Even smallestthings routinely done as multi-level

Example: full adder cell

sum =

ab

cin

cout=

ab

cin

1

1

11

11

11

2-level form Multilevel form

sum = abcin + abcin+ abcin+ abcin

cout= ab + acin+ bcin

sum

cout

9 gates, 25 inputs 5 gates, 10 inputs

sum

cout

a b cin

bacin


4/28

Boolean Logic Network Model

Idea: its a netlist of connected components, like a logic diagram, but

now individual components can be arbitraryBoolean functions

a

b

c

x

y

Ordinary gate netlist Same circuit as aBoolean logic network,

x, y are now Boolean functions

primary

inputs

primaryoutputs

internal

vertices

a

b

c

x

y

x=ab

y=x+c


5/28

Boolean Logic Networks

Primary inputs (usually vars)

Primary outputs (stuff network creates for other logic to consume)

Intermediate nodes that are themselves represented as single outputBoolean functions generally all in SOP form

Look at some operators that one can use to manipulate these networks

Some are fairly simple structuraloperations on graphs

Some will require entirely new operators (like division)


6/28

Boolean Logic Networks

p = ce + de

q = a + br = p + as = r + bt = ac + ad + bc + bd + e

u = qc + qc + qc

v = ad + bd + cd + aew = vx = sy = tz = u

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

Lets look at some operations on this network...

Cost = # literals = 33nodes


7/28

Reminder: Boolean Network Model

Its a graph

Has primary inputs and outputs

Internal nodes mean here is an SOP-form Boolean function Edges means here are signals going into/out of these functions

#literals = count up all literals in every SOP equation in every Booleannode

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

As gates it looks like this...

c

e

d

p

a

b

s

r

literals = 33


8/28

Network Ops: Elimination

Removes an internal vertex by replacing it (adding its SOP expression)into all the other vertices it feeds

Note: eliminate vertex for rrequires substituting (p+a) in snode

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

s=p+a+b

w

x

y

z

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

literals =

To eliminate node r, you must substitute (p + a) into node s,replacing variable r in the expression of node s.

32 ( = -1 good )


9/28

Network Ops: Decomposition

Decompose internal vertex by replacing it with 2 or more new verticesthat make a subnetwork logically equivalent to it (and simpler)

Example:

a

b

c

d

e

v=jd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

j=a+b+c

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

Decomposes into

Note that =

literals =

v=ad+bd+cd+ae

j = a+b+c v = jd + ae

32 ( = -1 good )

v=(a+b+c)d+ae


10/28

Network Ops: Extraction

Create a new vertex that represents a common subexpression for2 vertices, and add it to network

Substitute the output of the new vertex for common parts elsewhere Note that:

a

b

c

d

e

v=ad+bd+cd+ae

p=ke

t=ka+kb+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

k=c+d

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

literals =

t = ac+ad+bc+bd+e = (a+b)(c+d) + e = (a+b)k + e = ak+bk+e

p = ce+de = (c+d)e = kewhere k = c+d

29 ( = -4 good )


11/28

Network Ops: Simplification

Just run a 2-level minimizer (ESPRESSO!) at a vertex -- see if the SOPcover of the vertex gets simpler

Note -- if you dont eliminate any vars, its a localtransformation

If you actually eliminate a var, its global-- changes the network

Note:

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = q+c

r=p+a s=r+b

w

x

y

z

local change,inside vertex only

a

b

c

d

e

v=ad+bd+cd+ae

p=ce+de

t=ac+ad+bc+bd+e

q=a+b u = qc+qc+qc

r=p+a s=r+b

w

x

y

z

literals =

u = qc+qc+qc = q+c (both cover all q,c minterms but qc)

29 ( = -4 good )


12/28


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14/28

Network Ops: Scripts

Programs like MIS II, SIS, HSIS, VIS (from Berkeley)

Commercial tools from Synopsys, Ambit

Use Boolean network model

Provide collections of network operators

Users invoke scriptsthat run a sequence of these ops on their design


15/28

Scripts

A sequence of network ops...

sweep; eliminate -1simplify -m nocomp

eliminate - 1

sweep; eliminate 5

simplify -m nocompresub -a

fxresub -a; sweep

eliminate -1; sweepfull_simplify -m nocomp


16/28

Rugged Ops: Sweep

Eliminates all single-inputvertices

Eliminates vertices with aconstant function (i.e., ==0, ==1always)

Sort of a basic clean up op


eliminate - 1

sweep; eliminate 5simplify -m nocompresub -a

fxresub -a; sweep

eliminate -1; sweep

full_simplify -m nocomp


17/28

Sweep Examples

Sweep examples

a

G

F=a

G=F

HH=F

a

G

H


18/28

Network Ops: Eliminate

Eliminates node by collapsing it into itsfanout nodes

Eliminates all nodes with value less than

or equal to threshold. Value of node is number of literals added

by eliminating the node:

Reduce literals by number of placesnode variable is used.

Increase literals by size of node timesnumber of extra times (>1) its used

-1 means eliminate nodes only usedonce elsewhere in network

sweep; eliminate -1

simplify -m nocompeliminate -1

sweep; eliminate 5

simplify -m nocompresub -a

fxresub -a; sweep


threshold > 0 means we allow some increase in literal count

Sets up opportunities for other simplifications (kind of likeREDUCE in ESPRESSO)


19/28

Value of Elimination

We have a vertex that has L literals; Its used N times in other vertices

What happens if we eliminate it? What is value of this?

Answer is: change in total number of literals in design

F = L literals

G1 = F + ...

G2 = F + ...

GN = F + ...

Total literals in thissub-piece =

Now, eliminate vertex F

Total literals aftereliminate=

Change = value =

G1 = L lits + ...

G2 = L lits + ...

GN = L lits + ...

L + N

N*L

N*L - L - N


20/28

Eliminate Examples

Eliminate -1

Eliminate 5

F = ab

G = F + x G = ab + x

F = abc

G4 = Fd + e

G1 = F + d

G2 = F + ef

G3 = F + gh

G4 = abcd + e

G1 = abc + d

G2 = abc + ef

G3 = abc + gh

literals =19 - 14 = +5

literals =3 - 4 = -1


21/28

Network Ops: Simplify

Run ESPRESSO on each node

Minimize SOP 2-level form of each

-m nocomp says dont try tocompute the full offset for eachnode-- makes it run faster

Run time of expand dependson size of blocking matrix

Same as simplify, but uses a

larger set of dont cares...

...works harder to try to get a

better (smaller SOP) answer


eliminate -1

sweep; eliminate 5simplify -m nocomp

resub -a

fxresub -a; sweep



22/28

Simplify Examples

Simplify

F = a + ab + c

G = a + a

F = a + c

G = 1


23/28

Network Ops: Resub

Re-substitute each node in thenetwork into each other node inthe network

In other words, for each pair ofnodes S, T, checks if S is a factorof T, or if T is a factor of S

Tries to use both the true andcomplemented form of the outputof each node it tries to substitute

Loops until network stops gettingbetter, ie, literal count stopsdecreasing

-a means that algebraic divisionis used to see if one node cansubstitute (divide) into another

(We talk aboutalgebraic divisionnext )

sweep; eliminate -1simplify -m nocompeliminate -1

sweep; eliminate 5simplify -m nocomp

resub -a

fxresub -a; sweep



24/28

Resub Example

Resub example 1

Resub example 2

G = ab+c

F = ab

H = ab+e

G = F+c

F = ab

H = F+e

G = ab+c

F = ab

H = a+b+cd

G = F+c

F = ab H = F+cd


25/28

Network Ops: Fx

Extracts common subexpressionsthat are either

A single cube (eg, bcd)

A double cube (eg, ab + bcd)

Result is a whole bunch of newnodes in the network thatrepresent these common factors

removed Note that after you get these

factors, you run resub to seewhich ones you can use in otherplaces

sweep; eliminate -1simplify -m nocompeliminate -1

sweep; eliminate 5simplify -m nocompresub -a

fxresub -a; sweep



26/28

fx Example

fx example

G = abx+cx+d

F = ab+c+x

H = ab+dExtract

ab+c

G = Nx+d

F = N+x

H = ab+d

N=ab+c


27/28

resub != fx

It addsnodes to the network to do this

It substitutes these new nodes in where it helps overall literal count

It CANNOT go find or extract new factors

It just looks at what nodes are alreadyaround in network

It tries to use these to substitute one node into another to save literals

Cost is opposite as well

Extract N instances of a factor with L literals

Literal savings is N + L - N*L


28/28

Rugged Script

sweep; eliminate -1

simplify -m nocompeliminate -1

sweep; eliminate 5

simplify -m nocomp

resub -a

fxresub -a; sweep

eliminate -1; sweep

full_simplify -m nocomp

Initial

houskeeping

Prepare for

factoring

Aggressivefactoring

Optimize

2-level nodes

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