Boolean Function Jean Vuillemin ENS, Paris Dimension d= D (f) - - PowerPoint PPT Presentation

SLIDE 1

The Ordered Dimension of a Boolean Function

• Dimension d=D(f)
• Bound d ≤ |DD(f)| on all known bit-level DDs
• Minimal multi-linear diagram d = |MLD(f)|
• Incremental operations on MLDs

Jean Vuillemin

ENS, Paris

1 Boolean Dimension

SLIDE 2

Regular Language Dimension

• MDA: minimal deterministic automaton

Boolean Dimension 2

(R) , 0 mda , 1 R

 

   (R) ( m R 0 da ) (R m a da 1 d ) m R

 

    

• Dimension:

2

(R) lo dim mda g (R), d   

• NXA: non-deterministic XOR automata

Theorem (Fliess 74)

• All NXA for R have size ≥d
• Reduced NXA of size d exist
• All reduced NXA are similar

Theorem (Ga.Vu. 09)

• Minimal NXA d=|MXA(R)|
• (R=S)(MXA(R)=MXA(S))
• Space n2 + time n3 minimization

mda s | (R)|  2d d s  

• MXA: minimal NXA w.r.t. integer word ordering

SLIDE 3

3

Binary Decision Tree

Boolean Dimension

(f) bdt

n n

f





1 2 | | 2 1 2 | |

( ) : [ ] ? :

n b n i n n n n i n

f f i f b f f x b f x f msi f

   

     

1 1 2

( ) f x x x f  (000) (100) f f

2 1

( 0) f x x f 

3 1

( 1) f x x f  ( 00) ( 10) f x f x ( 00) ( 10) f x f x (011) (111) f f

Definition

2

(f) log | bdt(f), >|    D

Example

 

2 1 1

? : ( ) 0,1, , , m x x x m x m dt x b  

1

( ), 1, , , , ( ) 4 m x x m bdt m     D

SLIDE 4

4

Reduced Ordered Decision Diagrams

Boolean Dimension

0 1 0 1 0 0 1 1

BDT

0 1 0 0 0 1 1 0

BMT

0 1

MMA BMD

1

2 1 2 2 1 2 1 2

' ? : x x x x x x x x x x x x m    

MDA MXA BDD MLD

1

[ 1]

i

f f x  

1 i

f f f x    

Invariant:

(f), = (f), = (f), = (f), (f), = (f), = (f), (f), bdt mda bdd mxa bmt mma bmd mld          

Example:

☺

i

1

(1 )

i i

f x f x f    [ 0]

i

f f x  

Shannon i

i i

f f f x x     f

Reed-Muller

SLIDE 5

Boolean Dimension 5

Examples

f mx mx~ hw spx Ő spx cfp BDD BMD MLD

 

2

( ) 2 msi 2 1

k

j k a j j k

mx k x a x k

 

    



2k ways mux

1

msi ( )

w j j k

hw k x w x k

 

  



Hidden weight

2 1

ms ( ) 2 1 i

i k i i k

spx k x x k

  

   

Scalar XOR product

mod

( )( ) ( 2) ms 2 i 1

i i i i k i k i k

cfp k z x z x z k

  

   

 

Carry-free product

☺ ☺ ☺ ☺ ☺

☠

☺

☠ ☠

☺ ☺

☠

☺ ☺

☠

☺

☠ ☠

2 1

ms ( ) 2 1 i

i k i i k

spo k x x k

  

  



Scalar OR product

spo

☠ ☠ ☠

SLIDE 6

Boolean Dimension 6

Truth Tables

' ' ' f d x f f d d x x         

for

*

f  

( ) ( tt )2 d

n

d f f n  

Disjunctive Truth-Table

dnf lip ( ) ( )

k d

f d k



 

( ) ( ) ( lip mon neg n ) ( ) g ' e

j j k

k k k k x



 

( ) mon

j j k

k x





XNF

0 1 2 0 1 2 1 2 0 1 2

2 1 2 2 1 2 1 2 ' ' ' '

? : '

x x x x x x x x x x x x

x x x x x x x x m x x x x

  

   

2 8

202 ( ) (01010 dt 011) 1 2 t m   

2

( ) 010001 xt 8 t 1 9 m   DNF

2 2 2

( 1) ( ) ( ) (0 1 ) ) t ( 1 dt 1

i i

i

x i x i x x i      

2

2 ( ) x 2

j j

k k j k

j



  

( ) x

k x j k

j

 

 

( ) 2 xtt

k k f

x f



  

Exclusive Truth-Table

xnf mon ( ) ( )

k x

f x k



 

2

( ) ( ) t 2 x t

j

j

x x j   (0) (0) (2) (1) (2) 98 x x x x x   

x j

k x j k  



Binary Moebius Transform

( ) ( ) O O

n d

• g n

f d g f f g



   

' ' x x f f   

1 1 1 1 1

1 ' ' ' x x x x x x x x x x  

( ) ( ) ( ) ( ) BDT g BMT f BDT f BMT g  

SLIDE 7

Multi-Linear Diagram

Boolean Dimension 7

f

1

f

x 

f



f  ' x f

1

xf f

 x

f

1

f f

x

MLD(m) 1

2

x

1 2

x x

2 '

x x

2 1

? : x x x 1 x

1

x

2 1

( ) x x x 

1 2

( ) x x x x  

2 1 2 2 1 2 1 2

' ? : x x x x x x x x x x x x m    

f f x f   

x

f f 

Reed-Muller

1

f f f   

f f  f

x

1

' f x f xf  

x

f

1

f

Shannon

1

f f

x x’

f

Davio

1

' f f x f   

x

f 

1

f

1

f f  f

x

MXA(m)

SLIDE 8

Dimension Properties

Boolean Dimension 8

Theorem Let fBiB have dimension d=D(f)N

1.  DD DD: d≤|DD(f)| linear decomposition 2. d = D(f~) mirror 3. d=|MLD(f)| minimal base 4. f=gMLD(f)=MLD(g) strong normal form 5. d = D(f’)  f≠0 not 6. D(fg) ≤ D(f,g)< D(f)+ D(g) xor 7. D(fg) < D(f) x D(g) and 8. f: D(f) < 2i/2+1 worst/average dim

   

BDD BMD FDD KDD HDD MXA IDD BDD MLD    DD

deterministic non-deterministic f: |BDD(f)| < 2i+1/i worst/average BDD

SLIDE 9

Boolean Dimension 9

Minimal Base

[b1 … bd ] is a Reduced Basis for f if d= (f) and bdt(f)<b1 … bd >. All RB are linearly similar. Theorem: Unique minimal basis B=base(f)=LC[b1 … bd ]: All RB A=[a1 … ad ] a.s.t. k: bk ak and k: bk <ak iff BA.

2 1 2 20100011

98 {0100011,01,011,0,1} {0,1,2,6,98} {1,0 xn 1,001,0000011} {1,2, f xtt i 8, dd b s } e 6 a 9 x x x x x         

m

Equally defined by:

• Xor Sorted i<j: bi <bj <bibj
• Reduced Echelon Form i  j: 0=bi2l(bj)

SLIDE 10

10

Minimal Multi-linear Diagram

Boolean Dimension

1

2

1

BDD

2 1

? : x x x

1

x

x

2 1

( ? : ) 4 m x x x   D

1 2 1

( ) 1 ( ) base m x x x x x  

2 1

( ) m x x x x   

MLD

2 1

( ) x x x 

x

1

x

1

m

1 1 2 1 2 3 1 2 3 2 1 2 1 3 1 3

(0) 2 (1) 4 (2)( ) ( ) 96 98 b x t x b x t x t x t t b x b b x t t m b b            

SLIDE 11

Boolean Dimension 11

MLD Adder

SLIDE 12

Incremental Operations on MLDs

Boolean Dimension 12

Broad Word Computer 1 bwc= d bops Mirror(f) l2(d) bwc dl2(d) bop Xor(f,g) 1 bwc d bop Minimize(f) n2 bwc dn2 bop And(f,g) d2 bwc d3 bop