1. Finite State Automata
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2. Formal Specification of Languages
Generators
Context-free Grammars
Regular Expressions
Recognizers
Parsers, Push-down Automata
Finite State Automata
FSA is a mechanism to recognize a set of valid inputs before carrying out an action.
FSA is a notation for describing a family of language recognition algorithms.
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3. Parity Problem 1 EP OP 1
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4. Basic Features Consumes the entire input string.
Remembers the parity of the bit string by abstracting from the number of 1s in the string.
Finite amount of memory required for this purpose.
Observe that counting requires unbounded memory, while computing the parity requires very small and fixed amount of memory.
Accepts/Rejects the input in a deterministic fashion.
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5. Final / Accepting state State Transition
Indicates the status of the machine after consuming some portion of the input.
Summarizes the history of the computation that is relevant to the future course of action.
Initial / Start State
Final / Accepting state
State Transition 1
Even Parity
Odd Parity
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6. Deterministic Finite State Automaton (DFA)
Q: Finite set of states
S: Finite Alphabet
d: Transition function
total function from QxS to Q
: Initial/Start State
F : Set of final/accepting state
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7. Operation of the machine1
Input Tape
Finite Control
Read the current letter of input under the tape head.
Transit to a new state depending on the current input and the current state, as dictated by the transition function.
Halt after consuming the entire input.
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8. Associating Language with the DFA
Machine configuration:
Yields relation:
Language:
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9. Examples
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10. Set of strings over {a,b} that contain bb
Design states by parititioning S*.
Strings containing bb q2
Strings not containing bb
Strings that end in b q1
Strings that do not end in b q0
Initial state: q0
Final state: q2
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11. State Diagram and Tableq0 q1 q2 b a b d a b q0 q1 q2
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12. Strings over {a,b} that do not contain bbq0 q1 q2 b a b d a b q0 q1 q2
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13. DFA for the complement of L given DFA for L
Let M = (Q,S,d,q0,F) be a DFA. Then,
M’ = (Q,S,d,q0,Q-F) is a DFA with
L(M’) = S* - L(M).
Implication: Languages associated with DFAs are closed under complementation.
(Recall that languages associated with regular expressions are closed under union, concatenation, and Kleene Star operations, by definition.)
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14. Strings over {a,b} containing even number of a’s and odd number of b’s.Ea Oa Eb Ob Eb Ob b
[Ea,Eb]
[Ea,Ob] b a a a a b
[Oa,Ob]
[Oa,Eb] b
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15. (ab)*c S* valid prefix invalid prefix end_a end_b end_c a a Ea Eb b c
Incompletely specified DFA if Err deleted Bold multiple arcs. c
a,c l b c Ec
Err
a,b,c
a,b,c b
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16. (ab)*c S* valid prefix invalid prefix end_a end_b end_c a Ea Eb b c
Incompletely specified DFA if Err deleted Bold multiple arcs.
Eb is where /\ leads us. If necessary, we can either relabel the state because /\ is not ending in b
Or create a new start state with three out arrows yielding a non-minimal DFA. c
a,c b Ec
Err
a,b,c
a,b,c
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17. Nondeterministic Finite Automataqi qi qj qj a
DFA a qk q
NFA
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18. How do we associate a language with an NFA?
Every DFA is an NFA. However, does non-determinism make NFAs strictly more expressive (powerful) than DFAs?
DFA: Unique computation for a given string
NFA: Accept if there exists an accepting computation
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19. NFA State Diagram (Strings over {a,b} ending in bb)q0 q1 q2 b b d a b q0
{q0}
{q0,q1} q1 f
{q2} q2
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20. Halts in non-accepting state after consuming the input.
Halts in accepting state
after consuming the input.
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21. NFA State Diagram (a U b)* bb (a U b)*q2 q0 q1 a b
DFA a b a q0 q1 q2 b b a
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22. NFA for (a U b)* (aa U bb) (a U b)*q2 q0 q1 a b a b
q11
q22 b b
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23. Introducing l-transitions into NFA
A l-transition causes the machine to change its state non-deterministically, without consuming any input.
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24. Closure Properties of NFA-lsM1 l l M M2 l l
L(M)*
L(M1) U L(M2)
These results do not hold for ordinary NFAs.
Compositionality; Modularity. l l l M1 M2
L(M1) L(M2)
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25. … … a a a a a0 a1 b1 a2 b2 a3 b3 a3 b3 an b b b b b b b bn
This construction cannot be generalized to recognize
because the machine will have infinite number of states.
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