Morphisms of State Machines

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Morphisms of State Machines

• Sequential Machine Theory
• Prof. K. J. Hintz
• Department of Electrical and Computer Engineering
• Lecture 7

Updated and adapted by Marek Perkowski
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Notation

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Free SemiGroup

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String or Word

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Concatenation

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Partition of a Set

• Properties
• pi are called pi-blocks of a partition, ?(A)

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Types of Relations

• 1. Partial, Binary, Single-Valued System
• 2. Groupoid
• 3. SemiGroup
• 4. Monoid
• 5. Group

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Partial Binary Single-Valued

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Groupoid

• Closed Binary Operation
• Partial, Binary, Single-Valued System with
• It is defined on all elements of S x S
• Not necessarily surjective

arguments
Surjective each y in the R has at least one x
in the D
Also
a b c
a(ba) ac a (ab) a ba c
a a b a
b c a b
value
c c a c
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SemiGroup

• An Associative Groupoid
• Binary operation, e.g., multiplication
• Closure
• Associative
• Can be defined for various operations, so
  sometimes written as

a b c
a a b c
b b c a
c c a b
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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.

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Monoid

• Semigroup With an Identity Element, e.

a b c
a a b c
b b c a
c c a b
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Group

• Monoid With an Inverse

0 1 2
Operation is modulo addition. Check that this is
a group
a b c
a b c
a a b c
a a b c
b b c a
b b c a
c c a b
c c a b
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Morphisms

• Homomorphism
• 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...

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Homomorphism continued

• If operations such as multiplication, addition,
  or multiplication by scalars are defined for D
  and R, it is required that these correspond...
• and...

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Example Homomorphism of groups

• 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
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Homomorphism

Note that homomorphism can map many elements to
one.But homomorphic properties must be preserved
in the range
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Homomorphism preserves correspondence

• 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

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Endomorphism

• Question What is endomorphism?
• Answer An endomorphism is a morphism which
  maps back onto itself
• The range, R, is the same set as the domain, D,
  e.g., the real numbers.

morphism
RD
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SemiGroup Homomorphism

Operation in range
Operation in domain
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Graphical Explanation of Homomorphism of
Semi-Groups

Operation in range
Operation in domain
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Homomorphism of Semi-Groups. Example

Larsen, Intro to Modern Algebraic Concepts, p. 53
Ask a student to draw operations in domain and
range and then show this homomorphism graphically
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Homomorphism of Semi-Groups. 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

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Homomorphism of Semi-Groups. Example

• Do the results of operations correspond?

same
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\Homomorphism of Monoids
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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

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Graphical illustration of Isomorphism of
Semi-Groups

Injective Homomorphism
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Example of function Log being Isomorphism of two
semi-groups

• 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
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Isomorphism Example
Example of function Log being Isomorphism of two
semi-groups (continued)

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Graphical illustration of this SemiGroup
Isomorphism

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Machine Isomorphisms

• Formally, it should be called Machine
  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

alpha
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Machine Isomorphisms (cont)
alpha

iota
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Machine Isomorphisms (cont)
delta

• Interpret

Machine state isomorphism
Machine output isomorphism
Two machine isomorphisms should be introduced,
for states and for outputs
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Machine State Isomorphism

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Machine Output Isomorphism

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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

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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

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Block Diagram Isomorphism

I1
I2
O2
O1
M2
O1
M1
I1
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Block Diagram Isomorphism

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Block Diagram Isomorphism

• which is the same as the preceding state diagram
  and block diagram definitions therefore M1 and M2
  are Isomorphic to each other

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Information in Isomorphic Machines

• 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

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Output Equivalence

Output strings of one machine are equivalent to
output strings of other machine
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Identity Machine Isomorphism

Al three are identity functions
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Inverse Machine Isomorphism

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Machine Equivalence

Remember machine isomorphism is an equivalence
relation defined on M
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Machine Homomorphism

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Machine Homomorphism

• If alpha is injective, then have isomorphism
• State Behavior assignment,
• Realization of M1
• If alpha not injective
• Reduction Homomorphism

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Behavioral Equivalence of two State Machines

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Behavioral Equivalence

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Homework Problem

• Take an arbitrary machine M and minimize it to
  machine M2 which has less states.
• Next specify the homomorphism between Machine M
  and Machine M2 that corresponds to the relation
  of combining compatible states.
• To specify this homomorphism use the formalisms
  and notations from this lecture.