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