This document discusses nondeterminism in finite automata. It defines nondeterministic finite automata (NFAs) which can be in multiple states simultaneously, unlike deterministic finite automata (DFAs) which can only be in one state at a time. The document proves that NFAs and DFAs have equivalent power by describing a subset construction algorithm to transform any NFA into an equivalent DFA. It also discusses epsilon transitions in NFAs and how to remove them.

1. Nondeterminism and Epsilon Transitions
Mridul Aanjaneya
Stanford University
June 28, 2012
Mridul Aanjaneya Automata Theory 1/ 30

2. Challenge Problem
Question
Prove that any square with side length a power of 2, and one square
removed, is tileable with L’s.
Mridul Aanjaneya Automata Theory 2/ 30

3. Something to think about ...
Question
Are such tilings always possible?
Mridul Aanjaneya Automata Theory 3/ 30

4. Recap: Deterministic Finite Automata
Definition
A DFA is a 5-tuple (Q, Σ, δD, q0, F) consisting of:
• A finite set of states Q,
• A set of input alphabets Σ,
• A transition function δD : Q × Σ → Q,
• A start state q0, and
• A set of accept states F ⊆ Q.
The transition function δD:
• Takes two arguments, a state q and an alphabet a.
• δD(q,a) = the state the DFA goes to when it is in state q and
the alphabet a is received.
Mridul Aanjaneya Automata Theory 4/ 30

5. Recap: Graph representation of DFA’s
• Nodes correspond to states.
• Arcs represent transition function.
Arc from state p to state q labeled by all those input symbols
that have transitions from p to q.
• Incoming arrow from outside denotes start state.
• Accept states indicated by double circles.
q q1 2
1
0
0
q3
1
0,1
• Accepts all binary strings without two consecutive 1’s.
Mridul Aanjaneya Automata Theory 5/ 30

6. Recap: Transition table for DFA’s
0 1
→ q∗
1 q1 q2
q∗
2 q1 q3
q3 q3 q3
• A row for each state, a column for each alphabet.
• Accept states are starred.
• Arrow for the start state.
q q1 2
1
0
0
q3
1
0,1
Mridul Aanjaneya Automata Theory 6/ 30

7. Recap: Regular languages
Definition
A DFA M = (Q, Σ, δD, q0, F) accepts w if there exists a sequence
of states r0, r1, . . . , rn in Q with three conditions:
• r0 = q0
• δD(ri , wi+1) = ri+1 for i = 0, . . . , n − 1, and
• rn ∈ F
• Condition 1 says that M starts in the start state q0.
• Condition 2 says that M follows δD between two states.
• Condition 3 says that last state is an accept state.
• We say that M recognizes L if L = {w | M accepts w}.
Mridul Aanjaneya Automata Theory 7/ 30

8. Regular Languages: Examples
Example
Let L = {w | w ∈ {0,1}∗ and w, viewed as a binary integer, is
divisible by 5.}
q q0
1
2
0
1
1
0
1
0
1
0
0
q q
q
1 3
4
Mridul Aanjaneya Automata Theory 8/ 30

9. Regular Languages: Examples
Example
Show that the language of all strings over {0, 1} that do not contain
a pair of 1’s that are separated by an odd number of 0’s is regular.
1
a
0 1
1
0,10 0
e
o b
Mridul Aanjaneya Automata Theory 9/ 30

10. Regular Languages: Examples
Example
Show that the language of all strings over {0, 1} that contain an
even number of 0’s and 1’s is regular.
0 00 0
1
1
1
1
oo
ee
oe
eo
Mridul Aanjaneya Automata Theory 10/ 30

11. Nondeterminism
• Deterministic finite automata can only be in one state at any
point in time.
Recall the definition of the transition function δD.
• In contrast, nondeterministic finite automata (NFA’s) can be
in several states at once!
The transition function δN is a one-to-many function.
q1
q q q2 3 4
1
0,1
0,1 0,1
• This NFA recognizes strings in {0,1}∗ containing a 1 in the
third position from the end.
Mridul Aanjaneya Automata Theory 11/ 30

12. Transition table
0 1
→ q1 q1 q1,q2
q2 q3 q3
q3 q4 q4
q∗
4 q4 q4
• q1 is a nondeterministic state with a one-many transition on 1.
q1
q q q2 3 4
1
0,1
0,1 0,1
• Intuitively, the NFA always guesses right.
Mridul Aanjaneya Automata Theory 12/ 30

13. Nondeterministic Finite Automata
Example
An NFA that accepts all strings of the form 0k where k is a multiple
of 2 or 3.
q1
q2
q3
q4
q5
q6
0
0 0
0
0
ε
ε
Mridul Aanjaneya Automata Theory 13/ 30

14. Nondeterministic Finite Automata
Definition
An NFA is a 5-tuple (Q, Σ, δN, q0, F) consisting of:
• A finite set of states Q,
• A set of input alphabets Σ,
• A transition function δN : Q × Σε → P(Q),
• A start state q0, and
• A set of accept states F ⊆ Q.
• Here, Σε denotes the set Σ ∪ {ε}.
• P(Q) denotes the power set of Q.
Mridul Aanjaneya Automata Theory 14/ 30

15. Transition function of an NFA
• δN(q,a) is a set of states.
• Extend to strings as follows:
Basis: δN (q,ε) = q
Induction: δN (q,wa) = the union over all states of δN (p,a),
where p ∈ δN (q,w).
• A string is accepted by an NFA if δN(q0,w) contains at least
one state p ⊆ F.
• The language of an NFA is the set of strings it accepts.
Mridul Aanjaneya Automata Theory 15/ 30

16. Equivalence of DFA’s and NFA’s
• Every DFA is also an NFA by definition.
There is simply no nondeterminism.
• Surprisingly, for every NFA, there is also an equivalent DFA!
Two equivalent machines recognize the same language.
Nonintuitive, as we’d expect NFA’s to be more powerful.
Useful, as describing an NFA is much simpler.
• Proof is the subset construction.
• The number of states of the DFA can be exponential in the
number of states of the NFA.
• Thus, NFA’s accept exactly the regular languages.
Mridul Aanjaneya Automata Theory 16/ 30

17. Equivalence of DFA’s and NFA’s
• DFA for recognizing strings with a 1 in the third last position.
q q q q6 4
1
0
1 1
1
00
1
0 0
11 0
0
1
1 0
q q q1
q
2
35 7
8
Mridul Aanjaneya Automata Theory 17/ 30

18. Subset construction
• Given an NFA (Q,Σ,δN,q0,F), construct equivalent DFA with:
States P(Q) (set of subsets of Q).
Inputs Σ.
Start state {q0}.
Final states = all those with a member of F.
Note:
• The DFA states have names that are sets of NFA states.
• But as a DFA state, an expression like {p,q} must be read as
a single symbol.
Mridul Aanjaneya Automata Theory 18/ 30

19. Subset construction
• Example: We’ll construct the DFA for the following NFA
which recognizes strings in {0,1}∗ containing a 1 in the
second position from the end.
1
0,1
0,1
a b c
Mridul Aanjaneya Automata Theory 19/ 30

20. Subset construction
1
0,1
0,1
a b c
1
a
0
a,b a,b,c
a,c
1
00
0 1
1
Mridul Aanjaneya Automata Theory 20/ 30

21. Proof of equivalence: subset construction
• Show by induction on length of w that
δN(q0,w) = δD({q0},w)
• Basis: w = ε. δN(q0,ε) = δD({q0},ε) = {q0}.
• Inductive step: Assume IH is true for all strings shorter than
w. Let w = xa, then IH is true for x.
Let δN (q0,x) = δD({q0},x) = S.
Let T = the union over all states p in S of δN (p,a).
Then δN (q0,w) = δD({q0},w) = T (by definition).
Mridul Aanjaneya Automata Theory 21/ 30

22. NFA’s with ε-transitions
• State-to-state transitions on ε input.
• These transitions are spontaneous, and do not consider the
input string.
0
0
1
1 1
ε
A
DCB
E F
0
ε
ε
Mridul Aanjaneya Automata Theory 22/ 30

23. Closure of States
• CL(q) = set of states that can be reached from state q
following only arcs labeled ε.
0
0
1
1 1
ε
A
DCB
E F
0
ε
ε
• CL(A) = {A}, CL(E) = {B, C, D, E}.
• Closure of set of states = union of closure of each state.
Mridul Aanjaneya Automata Theory 23/ 30

24. Extended transition function
• Basis: δE (q,ε) = CL(q).
• Induction: δE (q,xa) is computed as follows:
1 Start with δE (q,x) = S.
2 Take the union of CL(δ(p,a)) for all p in S.
• Intuition: δE (q,w) is the set of states you can reach from q
following a path labeled w with ε’s in between.
Mridul Aanjaneya Automata Theory 24/ 30

25. Example: Extended transition function
0
0
1
1 1
ε
A
DCB
E F
0
ε
ε
• δE (A,ε) = CL(A) = {A}.
• δE (A,0) = CL(E) = {B, C, D, E}.
• δE (A,01) = CL(C,D) = {C, D}.
Mridul Aanjaneya Automata Theory 25/ 30

26. Example: Extended transition function
0
0
1
1 1
ε
A
DCB
E F
0
ε
ε
• Language of an ε-NFA is the set of strings w such that
δE (q0,w) contains a final state.
Mridul Aanjaneya Automata Theory 26/ 30

27. Equivalence of NFA, ε-NFA
• Every NFA is an ε-NFA.
It just has no ε-transitions.
• Converse requires us to take an ε-NFA and construct an NFA
that accepts the same language.
• This is done by combining ε-transitions with the next
transition on a real input.
• Start with an ε-NFA (Q,Σ,q0,F,δE ) and construct an ordinary
NFA (Q,Σ,q0,F’,δN).
Mridul Aanjaneya Automata Theory 27/ 30

28. Equivalence of NFA, ε-NFA
• Compute δN(q,a) as follows:
Let S = CL(q).
δN (q,a) is the union over all p in S of δE (p,a).
• F’ = set of states q such that CL(q) contains a state of F.
• Intuition: δN incorporates ε-transitions before using a.
• Proof of equivalence is by induction on |w| that
CL(δN(q0,w)) = δE (q0,w).
• Basis: CL(δN(q0,ε)) = CL(q0) = δE (q0,ε).
• Inductive step: Assume IH is true for all x shorter than w.
Let w = xa.
Then CL(δN (q0,xa)) = CL(δE (CL(δN (q0,x)),a)) (by definition).
But from IH, CL(δN (q0,x)) = δE (q0,x).
Hence, CL(δN (q0,w)) = CL(δE (δE (q0,x),a)) = δE (q0,w).
Mridul Aanjaneya Automata Theory 28/ 30

29. Example
0
0
1
1 1
ε
A
DCB
E F
0
ε
ε
0
0
1
1 1
A
DCB
E F
0
11
1
Mridul Aanjaneya Automata Theory 29/ 30

30. Summary
• DFA’s, NFA’s and ε-NFA’s all accept exactly the same set of
languages: the regular languages.
• NFA types are easier to design and may have exponentially
fewer states than a DFA.
• But only a DFA can be implemented!
Mridul Aanjaneya Automata Theory 30/ 30