1. Equivalence, DFA, NDFA Sequential Machine Theory Prof. K. J. Hintz
Department of Electrical and Computer Engineering
Lecture 2
Updated and modified by Marek Perkowski

2. 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

3. Equivalence Relation on A

4. 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,

5. Equivalence Relation Example
Symmetric, if
is similar to
then

6. Equivalence Relation Example
Transitive, if
is similar to
and
then

7. Inclusion Relation

8. Inclusion Relation Example

9. Partition Notation
Overbar Indicates States Which Are Elements of the Same -block.
Single States Are Not Normally Listed

10. Relations May Be Orderings
Partial Ordering
Total Ordering, aka Chain
Well Ordering (not discussed here)

11. 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

12. Properties of PO Reflexive Anti-Symmetric (asymmetric)
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

13. 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

14. Total Ordering
aka
Chain, simply ordered set, totally ordered set
A Partial Ordering for Which All Orderings Are Specified
A Chain Is “Connected” Because

15. 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 a later part of the Course

16. Finite Automata
A Deterministic semi-automaton*, aka Completely Specified Deterministic Semi-automaton Is a Triple
with no Mealy machine output function, Beta ()
* Ginzburg, 1968

17. FSM Set Properties I ib sa sc S

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

19. 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, .

20. 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

21. 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 “1%0b” is a string.

22. 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.

23. 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.

24. State Transition Let There Be Two Configurations for a Machine 1 q q’

25. String Example
Let
w = a b b a
then
w = a w’
and
w’ = b b a

26. Recognizer as Directed Graph
Arbitrary State
State Transition
Start (initial) State
Final State q 1 q q’ - or or +

27. Recognizer Examples Let I = { a, b }
Accepts no strings since no final state
Accepts all strings
Dead State
a, b -
a, b
a, b

28. Recognizer Examples
Accepts only , the null string
a,b
+/-

29. Recognizer Example This Recognizer Accepts the Language
L= { ab, a (aa) b, a (aa) (aa) b, ...
ab (bb), ab (bb) (bb), ... }
L = a (aa)* b ( b (aa)* b )* - a b

30. Rabin-Scott Example

31. Rabin-Scott Example L (M) = { x  I* | * ( 1, x ) = 4 }
L (M) = { a; , a+a; , a+a+a; , ... } a 2 3 4 +
+ ; ;
a + ; 1

32. Non-Deterministic FSM
A Non-deterministic Finite Automata Is a Quintuple with S, I, s0, F
as in a recognizer, but,

33. 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

34. 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

35. NDFA Example Non-deterministic Since
( ( 1, a ), 2 ) and ( ( 1, a ), 3 ) a b 2 3 4 1 1

36. NDFA Example Non-deterministic Since Not Completely Specified ab 1 abb4
abb

37. NDFA Example
Non-deterministic Since State Changes in Response to a Null String. a b 2 3 4 1 bb

38. NDFA to DFA Theorem Constructive Proof 4 Difficulties to Resolve
For each NDFA there is an equivalent DFA
Constructive Proof
4 Difficulties to Resolve
Missing transitions
Single transitions due to | strings | > 1
Transitions due to  strings
Multiple transitions

39. Problem: Missing Transitions
I = { a, b }
In DFA, all i  I must be accounted for in each state a b ?

40. Solution: Missing Transitions
Add a “sink” state which is not a final state and terminate all missing transitions there. a b
a, b

41. Problem: | strings | > 1
Single transition due to string of size > 1
Add intermediate states and “sink”, other characters in those states go to “sink” state a ab
a, b b

42. Problem:  Strings
Can’t just combine states since a b a b b a a b

43. 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 (  )

44. NDFA Example
> 1 2 3 4  a b

45. State Equivalents E( 1 ) = {self, explicit alternative} = { 1, 3 }
Define a new machine based on the old using the E(  ) states

46. New Machine

47. New Machine Transition Table

48. New Machine Transition Table

49. New Machine Transition Table

50. DFA Equivalent of NDFA

51. Reduced DFA Equivalent