MVSIS Front End 219b Project Presentation Spring 2000 Minxi Gao, EECS, UC Berkeley.

MVSIS Front End

219b Project PresentationSpring 2000

Minxi Gao, EECS, UC Berkeley

Outline

Motivation

Experiment results

Implementation

Theory

Problem definition

Motivation

MVSIS Front End

– multi level minimization• decomposition, substitution, collapsing, elimination, factorization – was not done

– 2 level minimization:• espresso-mv, node don’t care minimization – was done

– visualization of network-node information• node factored form, value, kernels… - need first to finish

multi-level minimization – was not done

Problem definition How to factor a multi-valued logic

function?

How to divide a multi-valued logic function given a divisor?

))(( }2,1,0{}0{}3{}3,2,1{}3,1{}2,1,0{

}2,1,0{}3,2,1{}0{}3{}3,2,1{}1{}0{}3{}2,1,0{

cacbca

cbacbcaca

Example:

Theory

Satisfiability-matrix based kernel searching (factorization)– Can do both kernel searching (factorization)

and division

Direct method for multi-valued division– Only applies to division – Similar to the weak-division in the binary

setting

Satisfiability-matrix

Given a matrix M and its entries– Value condition

• Value val of a variable var satisfies the value condition if it appears in all combinations of the rows and columns that it appears in M

– Satisfiability• M is satisfiable if all values of all variables

of the entries in M satisfy the value condition

An example

1 21 2 3 3

1 1 X1{0,1}X2

{1,2}

22 X1{1}X2

{1,4} X1{0,1}X2

{4}

– All values of X1 satisfy the value condition. Value 4 of X2 satisfies the value condition

X1{0,1}X2

{2,3}

X1{1}X2

{1,3} X1

{0,1}X2{2,3}

Obtain factorization

, ,

, ,

( )( ) ( )( )

( )( )

ij r i c j r cij ij

ij r i c j

M e e e e

M e e

Then if M is satisfiable ,

Place as many cubes of F as possible in M so that M is satisfiable:

– F= (er)(ec) + R– R is the remainder and contains the cubes

that will make M not satisfiable if put in M

er,i=supercube(all cubes in row i) ec,j=supercube(all cubes in column j)

– supercube({ci}): smallest cube containing all ci

Techniques

Pre-select number of rows Fill M in column order, check satisfiability at

position

Branch and bound search Exponential but good heuristics have

been used to speed up the search

An example

1 21 2

11

22 b{1,2,3} c{3}

a{0}b{1,2,3}

c{0}

F = a{0,1,2}c{3} + b{1,2,3} c{3} + a{0}b{1,2,3} c{0} +a{0}c{1}

= a{0,1,2} b{0,1,2,3}c{3} + b{1,2,3} c{3}

+ a{0}b{1,2,3} c{0} +a{0} b{0,1,2,3}c{1}

a{0,1,2} b{0,1,2,3}c{3}

a{0} b{0,1,2,3} c{1}

An example

a{0,1,2} b{0,1,2,3}c{3} a{0} b{0,1,2,3}

c{1}

b{1,2,3} c{3} a{0}b{1,2,3}

c{0}

F = a{0,1,2} c{3} + b{1,2,3} c{3} + a{0}b{1,2,3} c{0} +a{0} c{1}

= (c{3} +a{0}c{0,1})(a{0,1,2} c{1,3} +b{1,2,3}

c{0,3})

c{3} a{0}c{0,1}

a{0,1,2} c{1,3}

b{1,2,3} c{0,3}

Direct method for exact division

• Similar to WEAK_DIV in SIS • Given:

– F =a{0,1,2}c{3}+b{1,2,3} c{3} +a{0}c{1}+ a{0}b{1,2,3} c{0}

– d =c{3}+a{0}c{0,1,2}

• Candidate cubes: contained in divisor cube– c{3}: – a{0}c{0,1,2}:

a{0,1,2}c{3}

, a{0}c{1}, a{0}b{1,2,3}

c{0}

b{1,2,3} c{3}

Direct method for exact division

Candidate quotient cubes:– c{3}: a{0,1,2}c S1, b{1,2,3} c S2

– a{0}c{0,1,2}: a S3c S4, a S5b{1,2,3} c S6

– {3}S1{0,1,2,3}, {3}S2{0,1,2,3}, {0}S3 {0,1,2,3}, {1} S4 {1,3}, {0} S5{0,1,2,3}, {0}S6 {0,3}

Take quotient cubes in common (compatible)– c{3}: – a{0}c{0,1,2}:

Result:– F =(c{3} +a{0}c{0,1,2})(a{0,1,2} c{1,3} +b{1,2,3} c{0,3})

a{0,1,2}c{1,3}, a{0,1,2}c{1,3},

b{1,2,3} c{0,3}

b{1,2,3} c{0,3}

Direct method for exact division

Run time for direct method:– 2-cube divisor:

• Can be reduced to maximum matching problem: maximum number of edges in a graph none of which share a vertex

• n3, have been implemented, result show it is fast

– >2-cube divisor• can be reduced to covering problem• Have not been implemented yet.

Implementation MV-SIS multi-level minimization

packages related to algebraic factorization and division– kernel extraction– factorization– decomposition– collapsing– substitution– elimination

Implementation

Kernelling and factorization– Satisfiability-matrix based method

Substitution– Direct method

Collapsing Elimination

– Compute the value of a node: number of cubes before and after elimination

– Collapse if value is less than thresh hold given by the user

Decomposition Best-kernel based method

– Divide these kernels into all other nodes to compute their value

– Choose the best one, extract it into a new node

– Find all 2-cube kernels in all nodes by making each pair of cubes cube-free

– Divide the new node into all other nodes

Visualization commands mv_print n1

– print SOP form of the function at node n1, print the whole network if n1 not specified

mv_print_io n1– print fanins and fanouts of n1. Print PI and PO if n1

not specified mv_print_factor n1

– print factored form of n1. print the whole network if n1 not specified

mv_print_value n1– print the value of n1, print the whole network if n1

not specified

Experimental results Multi-valued PLA examples:

– All 2-cube kernel method decomposition

PLA Run time/sec Num cubes saved

C1908_40_20.club 2 613->523, 15%

Apex6_40_25.club 1.8 505->429, 15%

Frg2_60_25.club 6 699->507, 27%

Pair_40_15.club 10 1418->1279, 10%

Toolarge_25_20.club 0.35 403->312, 23%

Experimental results• Hand-made examples

• Test examples from last year’s project and an adder_mod4 example

• All 2-cube kernel method decomposition

Example methodNum cubes

saved

test3Eliminate 0->dcomp-

>eliminate 0286->107,

63%

test3 Collapse->decomp 488->71, 85%

test2 Eliminate 0->decomp 73->42, 42%

Adder_mod4 Decomp 32->24, 25%

Adder_mod5 Decomp 40->34, 15%

Experimental results Machine_learning examples:

– All 2-cube kernel method decomposition

Name Num cubes saved

balance.mv 153->121, 21%

car.mv 58->45, 22%

employ2.mv 60->33, 45%

nursery.mv 144->53, 63%

pal3.mv 36->18, 50%

Conclusion

Multi-level minimization in MVSIS is efficient and fast

We have built a SIS version with MV layer with a menu of optimization commands