morphisms of state machines sequential machine theory prof. k. j. hintz department of electrical and...
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Morphisms of State MachinesMorphisms of State Machines
Sequential Machine Theory
Prof. K. J. HintzDepartment of Electrical and Computer
Engineering
Lecture 8
Updated and adapted by Marek Perkowski
NotationNotation
A relation
A function
A binary operation called multiplication
A binary operation called addition +
Therefore
For all
There exists
Proper subset
Subset
Free SemiGroupFree SemiGroup
The free semigroup generated by the set
is the set of all strings (words) from
where
= symbols
This is equivalent to
+
I*
String or WordString or Word
.,,,
.,
from elements of sequence finitea is stringa
setnonempty a =
Given
714221 etc
e.g
then
ConcatenationConcatenation
strings twoof ionconcatenat theis
,
setnonempty a ,
Given
''11
''11
''11
mn
mn
mn
then
and
Partition of a SetPartition of a Set
• Properties
• pi are called “pi-blocks” of a partition, (A)
i
i
P
p
Ap
p
Ap|pA
c)
, b)
disjoint, are a)
and,
Types of RelationsTypes of Relations
• Partial, Binary, Single-Valued System
• Groupoid
• SemiGroup
• Monoid
• Group
Partial Binary Single-ValuedPartial Binary Single-Valued
S
SS
SSS:
R
R
R
R
D & partial subsetproper .,.
,
valued-single unique is.,.
function partiala is
,,.,.
21
3
321
ei
ss
and
sei
such that
sssei
If
GroupoidGroupoid
• Closed Binary Operation
• Partial, Binary, Single-Valued System with
• It is defined on all elements of S x S
• Not necessarily surjective
S
S
RD .,.
, 21
ei
ss
SemiGroupSemiGroup
• An Associative Groupoid– Binary operation, e.g., multiplication– Closure– Associative
• Can be defined for various operations, so sometimes written as
,S
S cbacbacba ,,
Closed Binary OperationClosed Binary Operation
• Division Is Not a Closed Binary Operation on the Set of Counting Numbers6/3 = 2 = counting number
2/6 = ? = not a counting number
• Division Is Closed Over the Set of Real Numbers.
MonoidMonoid
Semigroup With an Identity Element, e.
eee
aea
aae
a
A
GroupGroup
Monoid With an Inverse
aea
ei
ee
eaa
bae
.,.
monoid in defined as same thebemust
elementunit or element identity
,,1
A
‘Morphisms’‘Morphisms’
Homomorphism (J&J)“A correspondence of a set D (the domain)
with a set R (the range) such that each element of D determines a unique element of R [single-valued] and each element of R is the correspondent of at least one element of D.“
and...
HomomorphismHomomorphism
“If operations such as multiplication, addition, or multiplication by scalars are defined for D and R, it is required that these correspond...”
and...
HomomorphismHomomorphism
“If D and R are groups (or semigroups) with the operation denoted by * and
x corresponds to x’ and
y corresponds to y’
then
x * y must correspond to x’ * y’ “
Product of Correspondence = Correspondence of product
HomomorphismHomomorphism
HomomorphismHomomorphism
• Correspondence must be– Single-valued: therefore at least a partial
function– Surjective: each y in the R has at least one x in
the D– Non-Injective: not one-to-one else
isomorphism
EndomorphismEndomorphism
• A ‘morphism’ which maps back onto itself
• The range, R, is the same set as the domain, D, e.g., the real numbers.
R=D
‘morphism’
SemiGroup HomomorphismSemiGroup Homomorphism
yxfyfxf
iff
f
f
smhomomorphi semigroupa is
then
:
functiona and
+, and ,
semigroups Given
RD
RD
SemiGroup HomomorphismSemiGroup Homomorphism
SmGp. HmMphsm. Example*SmGp. HmMphsm. Example*
3 3 = )(
2 2 = )(
1 1 = )(
0 0 = )(
+ addition, with 32,1,0,=
tion,multiplica with,,gg,e,=
73
62
5
4
72
gfgf
gfgf
gfgf
gfef
and
glet
R
D
*Larsen, Intro to Modern Algebraic Concepts, p. 53
SmGp. HmMphsm. Example*SmGp. HmMphsm. Example*
Is the relation • single-valued?
– Each symbol of D maps to only one symbol of R
• surjective?– Each symbol of R has a corresponding element in D
• not-injective?– e and g4 correspond to the same symbol, 0
SmGp. HmMphsm. Example*SmGp. HmMphsm. Example*
Do the results of operations correspond?
e g
x g y g
x y g f g
f g f g
f g f g
. ., let
and
then
=
+ = 2 + 0 = 2
2 4
6 2
6 4
2 4
2
2 0same
Monoid HomomorphismMonoid Homomorphism
' and
smhomomorphi monoida is
:
+, ,
elementsidentity withsemigroups Given
eefyxfyfxf
iff
f
then
f
tionand a func
and
RD
RD
IsomorphismIsomorphism
• An Isomorphism Is a Homomorphism Which Is Injective
• Injective: One-to-One Correspondence– A relation between two sets such that pairs can
be removed, one member from each set until both sets have been simultaneously exhausted
SemiGroup IsomorphismSemiGroup Isomorphism
Injective Homomorphism
Isomorphism Example*Isomorphism Example*
• Define two groupoids– non-associative semigroups– groups without an inverse or identity element
• SG1: A1 = { positive real numbers }
*1 = multiplication = *
• SG2: A2 = { positive real numbers }
*2 = addition = +*Ginzberg, pg 10
Isomorphism ExampleIsomorphism Example
yxxy
yxxy
since
then
logloglog
misomorphis an is log
SemiGroup IsomorphismSemiGroup Isomorphism
Machine IsomorphismsMachine Isomorphisms
• Input-output isomorphism, but usually abbreviated to just isomorphism
• An I/O isomorphism exists between two machines, M1 and M2 if there exists a triple
and . . .
where
isom.)for (required bijective are ,,
,,
Machine IsomorphismsMachine Isomorphisms
1112111
112111
211
12
21
21
,,
,,
and s
thatsuch
subscripts theoforder reverse thenote :
:
:
oisis
isis
ii
S
OO
II
SS
Machine IsomorphismsMachine Isomorphisms
Interpret
, , ,, =
, , ,, =
semigroupa is and with
semigroupa is and with
,
,
222222
11111
1
1
2222111
1111111
OI S
OI S
O IS
SIS
M
M1
where
therefore
zyxis
zyxis
Machine State IsomorphismMachine State Isomorphism
Machine Output IsomorphismMachine Output Isomorphism
Homo- vice Iso- MorphismHomo- vice Iso- Morphism
Reduction Homomorphism• Shows behavioral equivalence between
machines of different sizes• Allows us to only concern ourselves with
minimized machines (not yet decomposed, but fewest states in single machine)
• If we can find one, we can make a minimum state machine
Homo- vice Iso- MorphismHomo- vice Iso- Morphism
Isomorphism
• Shows equivalence of machines of identical, but not necessarily minimal, size
• Shows equivalence between machines with different labels for the inputs, states, and/or outputs
Block Diagram IsomorphismBlock Diagram Isomorphism
I1 I2 O2 O1M2
M1
2 2
1 1
I1O1
Block Diagram IsomorphismBlock Diagram Isomorphism
Block Diagram IsomorphismBlock Diagram Isomorphism
which is the same as the preceding state diagram and block diagram definitions therefore M1 and M2 are Isomorphic to each other
112
222121
2121
,=
,
is
isOss
OOss
Machine InformationMachine Information
• Since the Inputs and Outputs Can Be Mapped Through Isomorphisms Which Are Independent of the State Transitions, All of the State Change Information Is Maintained in the Isomorphic Machine
• Isomorphic Machines Produce Identical Outputs
Output EquivalenceOutput Equivalence
, =
is which
, ,
- - of strings
output the toequivalent are of stringsoutput the
x and
s then
misomorphis I/O an .,. Let
*2
*2
*
1*
1*1
2
1
*1
1
21
xs
xsxs
M
ei
M
MM
I
S
:
Identity Machine IsomorphismIdentity Machine Isomorphism
functionsidentity are misomorphis theof elements all .,.
Let
21id
ei
OO
II
SS
:
:
:
:
MM
Inverse Machine IsomorphismInverse Machine Isomorphism
surjective and injective .,. bijective, bemust ,,
a exists toFor there
,,
then
,,
Let
1
21111
21
ei
1MM
MM
:
:
Machine EquivalenceMachine Equivalence
Let be isomorphic machines
then,
reflexive
symmetric
and transitive
which we recognize as the properties of an equivalence
relationship, . ., machine isomorphism is an
equivalence relationship defined on
M M M
M M
M M M M
M M M M M M
M
1 2 3
1 2 2 1
1 2 2 3 1 3
, ,
i e
Machine HomomorphismMachine Homomorphism
,,s functions are
subscripts oforder reverse note
one many to .,.
into of m HomorphisI/O an is
,,
Let
211
12
21
21
21
21
and
oi
OO
II
eiSS
iff
then
OIS
:
:
:
:
MM
MM
Machine HomomorphismMachine Homomorphism
• If alpha is injective, then have isomorphism– “State Behavior” assignment,
– “Realization” of M1
• If alpha not injective– “Reduction Homomorphism”
isis
isis
,,
,,
12
12
M M1 2
M <M1 2
Behavioral EquivalenceBehavioral Equivalence
equivalent
ly behavioral are and machines, Two
21
21
21
21
for which
and
iff
SS
OO
II
R
MM
Behavioral EquivalenceBehavioral Equivalence
xsxs
xthen
ss
and if
,,
*
2*21
*1
1
21
2
1
I
SR
SD
R
R
R