Machine Composition

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Machine Composition

• ECE-548 Sequential Machine Theory
• Prof. K. J. Hintz
• Department of Electrical and Computer Engineering
• Lecture 9

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Behavioral Equivalence

• A behavioral Equivalence is
• Reflexive
• M M
• Symmetric
• M1 M2 implies M2 M1
• Transitive
• M1 M2 M3 implies M1 M3

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Morphisms

• Reduction Homomorphism
• Shows behavioral equivalence of machines of
  different sizes
• Can limit our analysis to minimized machines
• Isomorphism
• Shows equivalence of same, but not necessarily
  minimal size
• Shows equivalence between machines with different
  labels for inputs, states, and outputs

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Reduction Homomorphism

• Two machines, M1 and M2, along with a
  homomorphism, ? (?, ?, ?), is a reduction if
• ? is surjective (onto)
• I1 I2 and ? is the identity function
• O1 O2 and ? is the identity function

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Reduction Homomorphism Ex.

Shields, p. 65
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M1 State Diagram

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Reduction Homomorphism Ex.

Shields, p. 65
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M2 State Diagram

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State Relation

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Red. Homo. Proof, States

• and.

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Red. Homo. Proof, Output
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Partitions from RH

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Partition from e.g. RH

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Partition from e.g. RH
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Inverse Mapping

• This Particular Surjective Mapping Is of Interest
  Because
• This mapping leads to an inverse, ?-1(s2) which
  determines a partition of S1 since many elements
  of S1 map to fewer elements of S2.
• Although there are other partitions of S1 this
  particular one is of interest because it has the
  substitution property

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Needed Definitions

• The next state of a transition function resulting
  from any of a set of states, ?, in response to a
  single input is itself a set of all states which
  could be reached from any of the initial states
  element of ?.

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Needed Definitions

• The output caused by an input when the machine is
  in any of a set of states is equal to the set of
  outputs generated by the response of the
  individual states to the single input.

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Behavior Equivalence

• B. E. as defined is awkward to work with, need a
  property which infers B.E.
• For a single machine, M1, one can define a new
  state transition function, ? and output
  function, ?

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Substitution Property

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Substitution Property

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Substitution Property Example

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SP Counter Example

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Substitution Property Example

• Given a
• Group

Ginzburg, p. 20
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Substitution Property Example

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SP Counter Example

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Output Consistent SP (OCSP)

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Output Consistent SP (OCSP)

• The Output Produced by All States That Are
  Elements of a Particular Pi-Block Is the Same
  When Under the Influence of the Same Input.
• The Output Does Not Have to Be the Same for Each
  Different Input, Only the Same for Each
  Individual Input.

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OCSP Example
Lee, p. 254
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OCSP Example

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OCSP Example

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Kernel(?)

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Kernel(?)

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Quotient Machine

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Machine Composition

• Parallel Composition
• Two or more machines operating synchronously to
  produce outputs
• Inputs/outputs of machine composition are vectors
  constructed by adjoining the inputs/outputs of
  the individual machines
• Represented as M M1 M2

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Machine Composition

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Parallel Composition

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Parallel Composition

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Parallel Composition Example

Lee, p. 291
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Parallel Composition Example

Lee, p. 291
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Parallel Composition Example

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Parallel Composition Example

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Machine Composition

• Serial Composition
• Output of first machine fed to input of second
  machine through mapping function, ?.
• Machines do not need to operate synchronously
• Input of first machine is input to serial
  composed machine
• Output of last machine is output of serial
  composed machine

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Machine Composition

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Serial Composition

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Serial Composition

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Serial Composition

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Serial Composition Example

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Serial Composition Example

Lee, p. 291
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Serial Composition Example

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Serial Composition Example