Orderings and BoundsOrderings and Bounds

Parallel FSM Decomposition

Prof. K. J. HintzDepartment of Electrical and Computer

Engineering

Lecture 10

Update and modified by Marek Perkowski

Orderings on Inclusion RelationsOrderings on Inclusion RelationsOrderings on Inclusion RelationsOrderings on Inclusion Relations

• Inclusion Relation– Reflexive– Anti-symmetric– Transitive

• Partial Ordering– An inclusion relationship for which some

orderings are not specified

Orderings on Inclusion RelationsOrderings on Inclusion Relations

Total Ordering

• Chain, simply ordered set

• An inclusion relationship for which all orderings are specified

• A chain is “connected” because S kjjkkj ssssss ,or

Partially Ordered Set (POSET)Partially Ordered Set (POSET)

• POSET: a set on which a partial ordering is specified,

• One way to show a partial ordering is with a Hasse Diagram– AKA directed graph

S,

POSET Example*POSET Example*

*Lee, 1.1.4

Let

, where

is defined as is an

integral divisor of

A 1 2 4 5 10 20 25 50 100, , , , , , , ,

:

" "

R x y

x

y

Least ElementLeast Element

one) thanmore be could (thereelement least is

aexist not does there

,

a POSET is where

Given

a

B

ΦB

A

AB

j

ji

ji

b

then

bb

if

bb

Lower BoundLower Bound

Given

= POSET =

an inclusion relation

is lower bound of

P A

A

a

,

, , ,

,

x y z and

if

z x

and

z y

then

z x y

Lower Bound ExampleLower Bound Example

From the Hasse Diagram Example, e.g.,

under the integer divide relation,

, the lower bounds

let x y

then

z

z

and

z z

i

j

i j

50 20

50 50 25

20 20 4

10 5 2 1

, ,

, , , , ,

, , , , ,

, , ,

10 5 2 1

10 5 2 1

Greatest Lower BoundGreatest Lower Bound

Infimum of a Set, Greatest Lower BoundIf z x m z x y

and z y

and m x

and m y

and z m z

then

m

, , ,

A

A

the is greatest lower bound

Greatest Lower Bound ExampleGreatest Lower Bound Example

From the Hasse Diagram Example, e.g.,

under the integer divide relation,

, the lower bounds, glb = 10

let x y

then

z

z

and

z z

i

j

i j

50 20

50 50 25

20 20 4

10 5 2 1

, ,

, , , , ,

, , , , ,

, , ,

10 5 2 1

10 5 2 1

Greatest Lower Bound ExampleGreatest Lower Bound Example

• 10 Is the glb of { 50, 20 } Under the Inclusion Relation Integer Division Given the Set A.

• The glb of { x, y } Is Written As

And Called the Meet of x and y

• The Meet Is Not the Boolean AND since we are only seeking the greatest value

x y

Upper BoundUpper Bound

Given

= POSET =

an inclusion relation

is upper bound of

P A

A

an

,

, , ,

,

x y z and

if

x z

and

y z

then

z x y

Upper Bound ExampleUpper Bound Example

From the Hasse Diagram Example, e.g.,

under the integer divide relation,

, the upper bounds

let x y

then

z

z

and

z z

i

j

i j

4 10

4 4

10 10 50

20 100

, ,

, ,

, , ,

,

20 100

20 100

Least Upper BoundLeast Upper Bound

Supremum of a Set, Least Upper Bound

If x l l z x y

and y l

and l z

z x z y z

then

l x y

, , ,

,

A

the

such that and

is least upper bound of

Least Upper Bound ExampleLeast Upper Bound Example

From the Hasse Diagram Example, e.g.,

under the integer divide relation,

, the upper bounds, with lub =

let x y

then

z

z

and

z z

i

j

i j

4 10

4 4

10 10 50

20 100 20

, ,

, ,

, , ,

,

20 100

20 100

Least upper bound = lub

Least Upper Bound ExampleLeast Upper Bound Example

• 20 Is the lublub of { 4, 10 } Under the Inclusion Relation Integer Division Given the Set A.

• The lublub of { x, y } Is Written As

And Called the JoinJoin of x and y

• The Join Is NotIs Not the Boolean OR since we are only seeking the greatest value

x y

BoundsBounds

• A Set May Have No Upper or Lower Bound or It May Have Many

• If the meet and join exist, they are unique

• Not every POSET has the property that each pair of elements possesses a glb or lub.

• If a POSET has a glb and lub for every pair of elements, then they form a lattice.lattice.

Lattice Example*Lattice Example*

*Hartsfield, Ringel, Pearls in Graph Theory

is a proper subset of

= S A, B, C

LatticeLatticeLatticeLattice• Lattice Properties

– Idémpotent– Commutative– Associative

x x x x x x ,

x y y x x y y x ,

x y z x y z x y z x y z ,

•Lattice: a Lattice: a POSET, L, in POSET, L, in Which Which Any Any two Elements, two Elements, XX and and YY, Have , Have Both a Meet Both a Meet (glb) and a (glb) and a Join (lub).Join (lub).

Properties of LatticesProperties of Lattices

– Absorptive– Isotone

x x y x x x y x ,

If

x y

then

x z y z

and

x z y z

Check these

properties on the example at the right

– Modular Inequality

– Distributive Inequality

If

x z

then

x y z x y z

x y z x y x z

x y z x y x z

Check these properties on the example at the right

Properties of LatticesProperties of Lattices

Partial Orderings on PartitionsPartial Orderings on Partitions

2121

21

j

212

211

21

then, if

subsets of setsbut blocks-y necessarilnot are ,

states of sets are ,,

,,

,,

,,Let

jii

i

Y

then

where

P

XX

YXS

YY

XX

S

Check these properties on the example at the right

Partial Orderings on PartitionsPartial Orderings on Partitions

2121

21

j

212

211

21

then, if

subsets of setsbut blocks-y necessarilnot are ,

states of sets are ,,

,,

,,

,,Let

jii

i

Y

then

where

P

XX

YXS

YY

XX

S

Partial Orderings on PartitionsPartial Orderings on Partitions

a POSET is

,

then

, states, ofset theof

partitions SP)y necessaril(not ofset thedenotes

If

S

S

S

not necessarily

Trivial PartitionsTrivial Partitions

TopTop• The partition consisting of a single pi-block

containing all states

n,,3,2,1 |

nsss ,,, |

is TOP, partition,

trivialThe set.empty -nona be Let

21 S

S

Trivial PartitionsTrivial Partitions

BottomBottom• The partition consisting of one state per pi-

block n,,3,2,1

nsssss ,,

is BOTTOM,partition,

trivialThe set.empty -nona be Let

21 S

S

Machine Lattice Example*Machine Lattice Example*Machine Lattice Example*Machine Lattice Example*

*Lee, p. 287

0 1 0 1

1 3 1 1 0

2 3 1 1 0

3 2 1 0 1

4 1 3 0 0

5 6 4 1 0

6 5 4 0 1

6,5,2,1

6,5,4,3,2,1 3,2,1

6,5,3,2,16,5

4,3,2,13,2

6,5,3,22,1

Partitions

5

94

83

72

61

Closed partitions

Machine Lattice ExampleMachine Lattice Example

6,5,2,1

6,5,4,3,2,1 3,2,1

6,5,3,2,16,5

4,3,2,13,2

6,5,3,22,1

Partitions

5

94

83

72

61

Operations on PartitionsOperations on Partitions

• Product, or meet, of 2 Partitions– the intersection of all blocks of two partitions

• Sum, or join, of 2 Partitions– merging all blocks of two partitions which have

one or more states in common

Meet of Two PartitionsMeet of Two Partitions

The Intersection of All Blocks of Two Partitions, e.g.,

Which Just Happens to Be the First Partition, but This Is Not Always the Case nor Is It Required.

5,4,3,2,1

5,4,3,2,1

5,4,3,2,1

21

2

1

Join of Two PartitionsJoin of Two Partitions

Merging All Blocks of Two Partitions Which Have One or More States in Common, e.g.,

which just happens to be the 2nd partition, but this isn’t always the case nor is it required

5,4,3,2,1

5,4,3,2,1

5,4,3,2,1

21

2

1

Partition Operation PropertiesPartition Operation Properties

ii

i

ii

i

|

| |

Parallel DecompositionParallel DecompositionParallel DecompositionParallel Decomposition

A machine, , has a parallel decomposition

machines and such that

is contained in

is behaviorally equivalent to

is a realization of

1 2

1 2

||

||

M

M M

M M M

M M

M M

M M||

iff

i e

. .,

Parallel DecompositionParallel Decomposition

The parallel decomposition is non - trivial

The size of and (number of states) is strictly

smaller than the size of 1

iff

and

i e

S S

S S

S S

S

1

2

2

. .,

Parallel DecompositionParallel Decomposition

Necessary and Sufficient Conditions

If has a parallel decomposition, then there is a

state - behavior assignment,

= , ,

where is injective, and , not necessarily identity.

. ., a single state of is equal to a state vector

of the parallel machine and there is a 1:1

correspondence between them1

M

M

i e

s s, 2

Parallel Decomposition TheoremParallel Decomposition Theorem

21

21

thatsuch

on , and , ,partitions SP trivial-non 2

of iondecomposit parallel trivial-nona exists There

M

M

iff

Parallel Decomp. Example*Parallel Decomp. Example*Parallel Decomp. Example*Parallel Decomp. Example*

*Lee, p.291

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

1

2

123 456

16 25 34

,

,

, ,

, ,

A B

C D E

not OC

Output compatible

Parallel Decomposition ExampleParallel Decomposition Example

Does a parallel decomposition exist?• Are the two partitions non-trivial?

S S

S S

1

2

2 6

3 6

?

?

yes

yes

the decomposition is non - trivial

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

• Are the partitions SP?1

Input PS NS PI-block0 1 4 B0 2 6 B0 3 5 B

1 1 3 A1 2 3 A1 3 2 A

Substitution property

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

• Are the partitions SP?1 cont' d

Input PS NS PI-block0 4 2 A0 5 1 A0 6 3 A

1 4 5 B1 5 4 B1 6 4 B

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. Example

Parallel Decomp. Example

• Are the partitions SP?2

Input PS NS PI-block0 1 4 E0 6 3 E

1 1 3 E1 6 4 E

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

• Are the partitions SP?2 (cont' d)

Input PS NS PI-block0 2 6 C0 5 1 C

1 2 3 E1 5 4 E

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. Example

Parallel Decomp. Example

• Are the partitions SP?2 (cont' d)

Input PS NS PI-block0 3 5 D0 4 2 D

1 3 2 D1 4 5 D

both partitions are SP

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

• Are the partitions orthogonal?

defined. are partitions these whichon machine original

theof iondecomposit parallela exist does there

|

4 ,5 ,6 ,3 ,2 ,1

,,,,,

21

21

?

21

SS

EBDBCBEADACA

ii s

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

1

2

123 456

16 25 34

,

,

, ,

, ,

A B

C D E

not OC

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

1,,0,,1,,0,

1,0

,,,,

1

1

111 111

BBAA

BA

IO

I

,

II MMstates

Output states

Parallel Decomp. ExampleParallel Decomp. Example

Construct an Image Machine (cont’d)1

Input

S1 0 1

A B A

B A B

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

EDC

BA

,,

34,25,16

,

456,123

2

1

Parallel Decomp. ExampleParallel Decomp. Example

Construct an Image Machine (cont’d) EDC

BA

,,

34,25,16

,

456,123

2

1

6,5,41,1,

3,2,10,0,

3,2,11,1,

6,5,40,0,

1

1

1

1

BBBB

ABAB

AAAA

BABA

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Input

S1 0 1

A ( A, 0 ) ( A, 1 )

B ( B, 0 ) ( B, 1 )

Copy inputs to columns

Parallel Decomp. ExampleParallel Decomp. Example

Construct an Image Machine (cont’d)

1,1,

0,0,

1,1,

0,0,

1

1

1

1

BB

BB

AA

AA

Input

S1 0 1

A ( A, 0 ) ( A, 1 )

B ( B, 0 ) ( B, 1 )

outputs

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

Construct an Image Machine

1

2

123 456

16 25 34

,

,

, ,

, ,

A B

C D E

not OC

1,,0,

,1,,0,

,1,,0,

1,0

,

,,,,

2

2

222 222

EE

DD

CC

EDC

IO

I

,

II MM

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

We create We create cartesian cartesian productsproducts

Parallel Decomp. ExampleParallel Decomp. Example

Construct an Image Machine (cont’d)

1

2

123 456

16 25 34

,

,

, ,

, ,

A B

C D E

not OC

S1 0 1 0 1

C E E ( C, 0 ) ( C, 1 )

D C E ( D, 0 ) ( D, 1 )

E D D ( E, 0 ) ( E, 1 )

2

2

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

State Behavior Assignment of Parallel Decomposition

, ,

CBEA

DBDA

EBCA

siffsswhere

,6,3

,5,2

,4,1

,

:

2121

21

SSS

S

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

State Behavior Assignment of Parallel Decomposition

, ,

: , |

,

I I I

I

i i i I

where

i i1 2

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0

Parallel Decomp. ExampleParallel Decomp. Example

X i Y js i i j

i j

A C

A D

A E

B E

B D

B C

, , ,,

, , ,

, , ,

, , ,

, , ,

, , ,

, , ,

if X Y = s and

if X Y = or 0

1 0

2 0

3 1

4 0

5 0

6 0

Input Output

0 1

1 4 3 0

2 6 3 0

3 5 2 1

4 2 5 0

5 1 4 0

6 3 4 0