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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
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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
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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
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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,
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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Lattice Example*Lattice Example*
*Hartsfield, Ringel, Pearls in Graph Theory
is a proper subset of
= S A, B, C
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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).
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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
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– 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Partition Operation PropertiesPartition Operation Properties
ii
i
ii
i
|
| |
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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
. .,
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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
. .,
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 47: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/47.jpg)
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
![Page 48: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/48.jpg)
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
![Page 49: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/49.jpg)
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
![Page 50: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/50.jpg)
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
![Page 51: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/51.jpg)
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
![Page 52: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/52.jpg)
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
![Page 53: Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d615503460f94a43b34/html5/thumbnails/53.jpg)
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