Equivalence, DFA, NDFA

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Equivalence, DFA, NDFA

• ECE-548 Sequential Machine Theory
• Prof. K. J. Hintz
• Department of Electrical and Computer Engineering
• Lecture 2

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Equivalence Relation on A

• An Equivalence Relation (Not Relationship) Is Not
  an Equality Relation
• A Relation is an Equivalence Relation if and only
  if (iff) it is
• Reflexive
• Symmetric
• Transitive

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Equivalence Relation on A
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Non-Algebraic Equivalence Relation Example

• Equivalence Relation on the Set of All Triangles
  on a Plane
• is congruent to or is similar to
• Reflexive, each triangle is similar to itself,

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Equivalence Relation Example

• Symmetric, if
• is similar to
• then
• is similar to

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Equivalence Relation Example

• Transitive, if
• is similar to
• and
• is similar to
• then
• is similar to

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Inclusion Relation
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Inclusion Relation Example
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Partition Notation

• Overbar Indicates States Which Are Elements of
  the Same ?-block.
• Single States Are Not Normally Listed

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Relations May Be Orderings

• Partial Ordering
• Total Ordering, aka Chain
• Well Ordering (not discussed here)

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Partial Ordering

• Given an Inclusion Relation, R s ? s, Defined
  on some Elements of the Set S such that s, s ?
  S, R Is a Partial Ordering If It Is
• Reflexive
• Anti-Symmetric (asymmetric)
• Transitive
• Not all orderings are specified, therefore partial

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Properties of PO

• Reflexive
• s ? s for all s ? S
• Anti-Symmetric (asymmetric)

e.g., let ? older than if Sam is older than
Bill, then Bill cannot be older than Sam
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Properties of PO

• Transitive

e.g., If the Redskins beat the Patriots and the
Patriots beat the Cowboys then the Redskins will
beat the Cowboys
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Total Ordering

• aka
• Chain, simply ordered set, totally ordered set
• A Partial Ordering for Which All Orderings Are
  Specified
• A Chain Is Connected Because

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POSET

• Partially Ordered SET
• A set on which a partial ordering is specified
• ( S, ? ) where ? is defined
• Not a chain since not all elements are connected
• We Will Revisit This Concept in the Second Half
  of the Course

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Finite Automata

• A Deterministic semi-automaton, aka Completely
  Specified Deterministic Semi-automaton Is a
  Triple
• with no Mealy machine output function, Beta (?)

Ginzburg, 1968
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FSM Set Properties
I
ib
sa
sc
S
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Language Recognizer

• aka, Rabin-Scott Automata (machine), Automaton,
  Language Recognizer
• A Recognizer Is a Quintuple of Sets
• with S, I, ? as before

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Kleene Star

• a e, a, aa, aaa, aaaa, ...
• The Kleene Star, , means NONE or more
  occurrences of something
• Star is an overloaded operator so be aware of
  context
• a ONE or more occurrences of something.
• a is Kleene Star less the null string, ?.

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Kleene Closure

• Kleene Closure Is Not Identical to Kleene Star
• Symbol is the same (overloaded)
• Kleene Closure/Star Closure
• Found in descriptions of formal language
• Language consisting of all strings over some
  alphabet

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String

• An Ordered Concatenation of Symbols From an
  Alphabet
• Used in Place of Word to Decouple From Common
  Concept of Word in Informal Language
• If ? a, 1, 0, b, then a 10b is a
  string.

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Recognizer

• If x ? I,
• i.e., a string of input symbols selected from
  the set of allowable input symbols,
• and
• the application of x to the recognizer results
  in a final state ? F,
• then
• the recognizer accepts the string.

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Strings

• A String, x, Is Accepted by a Recognizer
  Left-most Letter First, i.e.,
• if
• the input to a recognizer is a string w,
• and if
• w ? w
• then
• ? is the first letter of the string which causes
  a state transition. Subsequent letters from left
  to right do the same.

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State Transition

• Let There Be Two Configurations for a Machine

?1
q
q
S
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String Example

• Let
• w a b b a
• then
• w a w
• and
• w b b a

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Recognizer as Directed Graph

• Arbitrary State
• State Transition
• Start (initial) State
• Final State

?1
q
-
or
or

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Recognizer Examples

• Let I a, b
• Accepts no strings since no final state
• Accepts all strings
• Dead State

a, b
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Recognizer Examples

• Accepts only ?, the null string

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Recognizer Example

• This Recognizer Accepts the Language
• L ab, a (aa) b, a (aa) (aa) b, ...
• ab (bb), ab (bb) (bb), ... , etc.
• L a (aa) b ( b (aa) b )

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Rabin-Scott Example

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Rabin-Scott Example

• L (M) x ? I ? ( 1, x ) 4
• L (M) a , aa , aaa , ...

1
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Non-Deterministic FSM

• A Non-deterministic Finite Automata Is a
  Quintuple with S, I, s0, F
• as in a recognizer, but,

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Non-Deterministic FSM

• State May Change
• to two different states in response to the same
  input at the same state
• in response to a string rather than just a single
  element from the set of inputs
• in response to a null string input

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DFA-NDFA Theorem

• Every NDFA Can Be Replaced by an Equivalent DFA
• Equivalent Means Not Only Accepting All Strings
  Accepted by the NDFA, but Also NOT Accepting Any
  Others

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NDFA Example

• Non-deterministic Since
• ( ( 1, a ), 2 ) and ( ( 1, a ), 3 )

1
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NDFA Example

• Non-deterministic Since Not Completely Specified

ab
1
abb
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NDFA Example

• Non-deterministic Since State Changes in Response
  to a Null String.

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NDFA to DFA

• Theorem
• For each NDFA there is an equivalent DFA
• Constructive Proof
• 4 Difficulties to Resolve
• Missing transitions
• Single transitions due to strings gt 1
• Transitions due to ? strings
• Multiple transitions

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Problem Missing Transitions

• I a, b
• In DFA, all i ? I must be accounted for in each
  state

?
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Solution Missing Transitions

• Add a sink state which is not a final state and
  terminate all missing transitions there.

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Problem strings gt 1

• Single transition due to string of size gt 1
• Add intermediate states and sink, other
  characters in those states go to sink state

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Problem ? Strings

• Cant just combine states since

b
b
b
a
a
b
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Solution ? Strings Multiple Transitions

• Eliminate ? by defining the set of next states
  which occur in response to no input, call this
  function E( ? )
• E( ? ) is called the equivalents of ( ? )

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NDFA Example

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State Equivalents

• E( 1 ) self, explicit alternative 1, 3
• E( 2 ) 2
• E( 3 ) 3
• E( 4 ) 4
• Define a new machine based on the old using the
  E( ? ) states

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New Machine
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New Machine Transition Table
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New Machine Transition Table
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New Machine Transition Table
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DFA Equivalent of NDFA
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Reduced DFA Equivalent