ECE-548 Sequential Machine Theory
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Transcript ECE-548 Sequential Machine Theory
Morphisms of State Machines
Sequential Machine Theory
Prof. K. J. Hintz
Department of Electrical and Computer
Engineering
Lecture 8
Updated and adapted by Marek
Perkowski
Notation
A relation
A function
A binary operation called multiplication
A binary operation called addition
Therefore
For all
There exists
Proper subset
Subset
+
Free 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 Word
Given
= a nonempt yset
then
a st ringis a finit esequence of element sfrom
e.g.,
1 2 , 2 4 , 1 7 , etc.
Concatenation
Given
, a nonemptyset
and
1 n , 1' m'
then
1 n 1' m'
1 n 1' m'
is theconcatenation of two strings
Partition of a Set
• Properties
A p | p P A
and,
a) p are disjoint,
b)
p A,
i
c) pi
• pi are called “pi-blocks” of a partition, (A)
Types of Relations
•
•
•
•
•
Partial, Binary, Single-Valued System
Groupoid
SemiGroup
Monoid
Group
Partial Binary Single-Valued
If
R :S S S
i.e., s1 , s2 , s3 R
such that
R is a partialfunct ion
i.e., s3 is unique single - valued
and
s1, s2 S S
i.e., propersubset partial & D R S
Groupoid
• Closed Binary Operation
• Partial, Binary, Single-Valued System with
s1, s2 S
i.e., D R S
• It is defined on all elements of S x S
• Not necessarily surjective
SemiGroup
• An Associative Groupoid
– Binary operation, e.g., multiplication
– Closure
– Associative
a bc ab c a, b, cS
• Can be defined for various operations, so
sometimes written as
S,
Closed Binary Operation
• Division Is Not a Closed Binary Operation
on the Set of Counting Numbers
6/3 = 2 = counting number
2/6 = ? = not a counting number
• Division Is Closed Over the Set of Real
Numbers.
Monoid
Semigroup With an Identity Element, e.
a A
ea a
ae a
ee e
Group
Monoid With an Inverse
e, a, b A
a a 1 e identit yelementor unit element
e must be thesame e as defined in monoid
i.e.,
a e a
‘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...
Homomorphism
“If operations such as multiplication, addition,
or multiplication by scalars are defined for
D and R, it is required that these
correspond...”
and...
Homomorphism
“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
Homomorphism
Homomorphism
• 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
Endomorphism
• A ‘morphism’ which maps back onto itself
• The range, R, is the same set as the domain,
D, e.g., the real numbers.
‘morphism’
R=D
SemiGroup Homomorphism
Given semigroups
D,
and
and a funct ion
f :D R
R, +
t hen
f is a semigroup homomorphi
sm
iff
f x f y f x y
SemiGroup Homomorphism
SmGp. HmMphsm. Example*
let D = e, g, g2 ,, g 7 with multiplication,
and R = 0,1, 2,3 with addition, +
f (e) = 0
f (g) = 1
f ( g2 ) = 2
f ( g3 ) = 3
f g 4 0
f g 5 1
f g 6 2
f g 7 3
*Larsen, Intro to Modern Algebraic Concepts, p. 53
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*
Do the results of operations correspond?
e. g., let
x g2
then
x y = g6
f g6 2
and
y g4
same
f g4 0
f g2 + f g4 = 2 + 0 = 2
f g2 2
Monoid Homomorphism
Given semigroupswith ident it yelement s
D,
and
and a function
f :D R
R, +
then
f is a monoidhomomorphi
sm
iff
f x f y f x y and f e e'
Isomorphism
• 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 Isomorphism
Injective Homomorphism
Isomorphism Example*
• Define two groupoids
– non-associative semigroups
– groups without an inverse or identity element
• SG1:
• SG2:
*Ginzberg, pg 10
A1 = { positive real numbers }
*1 = multiplication = *
A2 = { positive real numbers }
*2 = addition = +
Isomorphism Example
then
log is an isomorphism
since
xy x y
log xy log x log y
SemiGroup Isomorphism
Machine 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
, ,
where
, ,
are bijective(required for isom.)
and . . .
Machine Isomorphisms
: S1 S 2
: I1 I 2
: O2 O1 notethereverseorder of thesubscripts
such that
sS1
and
i1 i2
1 s1 , i1 2 s1 , i1
1 s1 , i1 2 s1 , i1 o1
Machine Isomorphisms
Interpret 1 s1 , i1
1 s1 , i1
x1 1 y1 z1
x2 2 y 2 z 2
therefore
1 with S I
and S is a semigroup
1 with S I and O is a semigroup
where
M1 = S1 , I1 , O1 , 1 , 1
M2 = S 2 , I 2 , O2 , 2 , 2
Machine State Isomorphism
Machine Output Isomorphism
Homo- 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- 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 Isomorphism
2
I1
I2
I1
2
M2
M1
1 1
O2
O1
O1
Block Diagram Isomorphism
Block Diagram Isomorphism
s1 s2
s1 s2
O1 O2
O1 2 s2 , i2
= 2 s1 , i1
which is the same as the preceding state
diagram and block diagram definitions
therefore M1 and M2 are Isomorphic to each
other
Machine 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 Equivalence
Let : M 1 M 2
i.e., an I/O isomorphism
then s S1
and x I1*
theoutput stringsof M 1 are equivalent to theoutput
stringsof - M 2 -
1* s1 , x * s1 , x
which is
= * 2* s2 , * x
Identity Machine Isomorphism
id : M 1 M 2
Let
: SS
: II
: O O
i.e., all elementsof theisomorphism are identityfunctions
Inverse Machine Isomorphism
Let
, , : M 1 M 2
then
1 1 , 1 , 1 : M 2 M 1
For thereto existsa 1
, , must be bijective,i.e., injectiveand surjective
Machine Equivalence
Let M 1 , M 2 , M 3 be isomorphic machines
then,
M M
M1 M 2 M 2 M1
reflexive
symmetric
M 1 M 2 and M 2 M 3 M 1 M 3 transitive
which we recognize as the properties of an equivalence
relationship, i. e., machine isomorphism is an
equivalence relationship defined on M
Machine Homomorphism
Let
, , : M1 M2
then is an I/O Homorphism of M1 intoM2
iff
: S1 S2
: I1 I 2
: O2 O1
i.e., many toone
notereverseorder of subscripts
are functions sS1 , i I1 , o O2
and
Machine Homomorphism
2 s , i 1 s, i
2 s , i 1 s, i
• If alpha is injective, then have isomorphism
– “State Behavior” assignment, M 1 M 2
– “Realization” of M1
• If alpha not injective
– “Reduction Homomorphism” M 1 < M 2
Behavioral Equivalence
T wo machines,M1 and M2 are behaviorally
equivalent
iff
I1 I 2
O1 O2
and
R S1 S 2
for which
Behavioral Equivalence
D R S1
R R S2
and if
s1 R s2
then x I1 *
*
1
s1, x s2 , x
*
2