1. Theory of Languages and Automata
Chapter 1- Regular Languages
& Finite State Automaton
Sharif University of Technology

2. Finite State Automaton
We begin with the simplest model of Computation, called finite state machine or finite automaton.
are good models for computers with an extremely limited amount of memory.
➜ Embedded Systems
Markov Chains are the probabilistic counterpart of Finite Automata
Theory of Languages and Automata
Prof. Movaghar

3. Simple Example Automatic door Front Pad Rear Pad Door
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4. Simple Example (cont.) State Diagram State Transition Table Neither
Front
Rear
Both
Closed
Open
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5. Formal Definition
A finite automaton is a 5-tuple (Q ,Σ ,δ ,q0, F), where
Q is a finite set called states,
Σ is a finite set called the alphabet,
δ : Q ×Σ ⟶ Q is the transition function,
q0 ∈ Q is the start state, and
F ⊆ Q is the set of accept states.
Theory of Languages and Automata
Prof. Movaghar

6. Example M1 = (Q, Σ, δ, q0, F) , where Q = {q1, q2, q3}, Σ = {0,1},
δ is described as
q1 is the start state, and
F = {q2}. 1 q1 q2 q3
Theory of Languages and Automata
Prof. Movaghar

7. Language of a Finite machine
If A is the set of all strings that machine M accepts, we say that A is the language of machine M and write: L(M) = A.
✓ We say that M recognizes A
or that M accepts A.
Theory of Languages and Automata
Prof. Movaghar

8. Example L(M1) = {w | w contains at least one 1 and
even number of 0s follow the last 1}.
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9. Example M4 accepts all strings that start and end with a or with b.
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Prof. Movaghar

10. Formal Definition M = (Q, Σ, δ, q0, F) w = w1w2 … wn ∀i, wi ∈ Σ
M accepts w ⇔ ∃ r0, r1, …, rn ∀i, ri ∈ Q
r0 = q0 ,
δ(ri, wi+1) = ri+1, for i = 0, … , n-1,
rn ∈ F.
Theory of Languages and Automata
Prof. Movaghar

11. Regular Language
A language is called a regular language if some finite automaton recognizes it.
Theory of Languages and Automata
Prof. Movaghar

12. Example
L (M5) = {w | the sum of the symbols in w is 0 modulo 3, except that <RESET> resets the count to 0}.
✓ As M5 recognizes this language, it is a regular language.
Theory of Languages and Automata
Prof. Movaghar

13. Designing Finite Automata
Put yourself in the place of the machine and then see how you would go about performing the machine’s task.
Design a finite automaton to recognize the regular language of all strings that contain the string 001 as a substring.
Theory of Languages and Automata
Prof. Movaghar

14. Designing Finite Automata (cont.)
There are four possibilities: You
haven’t just seen any symbols of the pattern,
have just seen a 0,
have just seen 00, or
have seen the entire pattern 001.
Theory of Languages and Automata
Prof. Movaghar

15. The Regular Operations
Let A and B be languages. We define the regular operations union, concatenation, and star as follows.
Union: A∪B = {x | x∈A or x∈B}.
Concatenation: A∘B = {xy | x∈A and y∈B }.
Star: A* = {x1x2…xk | k ≥ 0 and each xi∈A }.
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Prof. Movaghar

16. Closure Under Union THEOREM
The class of regular languages is closed under the union operation.
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Prof. Movaghar

17. Proof Let M1 = (Q1, Σ1, δ1, q1, F1) recognize A1, and
Construct M = (Q, Σ, δ, q0, F) to recognize A1∪A2.
Q = Q1 × Q2
Σ = Σ1 ∪ Σ2
δ((r1,r2),a) = (δ1(r1,a), δ2 (r2,a)).
q0 is the pair (q1, q2).
F is the set of pair in which either members in an accept state of M1 or M2.
F = (F1× Q2 ) ∪ (Q1 × F2) F ≠ F1× F2
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Prof. Movaghar

18. Closure under Concatenation
THEOREM
The class of regular languages is closed under the concatenation operation.
To prove this theorem we introduce a new technique called nondeterminism.
Theory of Languages and Automata
Prof. Movaghar

19. Nondeterminism
In a nondeterministic machine, several choices may exit for the next state at any point.
Nondeterminism is a generalization of determinism, so every deterministic finite automaton is automatically a nondeterministic finite automaton.
Theory of Languages and Automata
Prof. Movaghar

20. Differences between DFA & NFA
First, very state of a DFA always has exactly one exiting transition arrow for each symbol in the alphabet.
In an NFA a state may have zero, one, or more exiting arrows for each alphabet symbol.
Second, in a DFA, labels on the transition arrows are symbols from the alphabet.
An NFA may have arrows labeled with members of the alphabet or ε. Zero, one, or many arrows may exit from each state with the label ε.
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Prof. Movaghar

21. Deterministic vs. Nondeterministic
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22. Example Consider the computation of N1 on input 010110.
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23. Example (cont.)
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24. Formal Definition
A nondeterministic finite automaton is a 5-tuple (Q ,Σ ,δ ,q0, F), where
Q is a finite set of states,
Σ is a finite alphabet,
δ : Q ×Σε ⟶ P(Q) is the transition function,
q0 ∈ Q is the start state, and
F ⊆ Q is the set of accept states.
Theory of Languages and Automata
Prof. Movaghar

25. Example N1 = (Q, Σ, δ, q0, F) , where Q = {q1, q2, q3, q4}, Σ = {0,1},
δ is given as
q1 is the start state, and
F = {q4}. 1 ε q1
{q1}
{q1,q2} ∅ q2
{q3}
{q4} q3 q4
Theory of Languages and Automata
Prof. Movaghar

26. Equivalence of NFAs & DFAs
THEOREM
Every nondeterministic finite automaton has an equivalent deterministic finite automaton.
PROOF IDEA convert the NFA into an equivalent DFA that simulates the NFA.
If k is the number of states of the NFA, so the DFA simulating the NFA will have 2 k states.
Theory of Languages and Automata
Prof. Movaghar

27. Proof
Let N = (Q ,Σ ,δ ,q0, F) be the NFA recognizing A. We construct a DFA M =(Q' ,Σ' ,δ' ,q0', F’) recognizing A.
let's first consider the easier case wherein N has no ε arrows.
Q' = P(Q).
q0’ = q0 .
F' = {R∈ Q’ | R contains an accept state of N}.
Theory of Languages and Automata
Prof. Movaghar

28. Proof (cont.) Now we need to consider the ε arrows. for R ⊆ Q let
E(R) = {q | q can be reached from R by traveling along 0 or more ε arrows}.
Q' = P(Q).
δ' (R,a) ={q∈ Q | q∈ E(δ(r,a)) for some r∈ R}.
q0’ = E({q0}) .
F' = {R∈ Q’ | R contains an accept state of N}.
Theory of Languages and Automata
Prof. Movaghar

29. Corollary
A language is regular if and only if some nondeterministic finite automaton recognizes it.
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Prof. Movaghar

30. Example D’s state set is {∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}.
The start state is E({1}) = {1,3}.
The accept states are {{1},{1,2},{1,3},{1,2,3}}.
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Prof. Movaghar

31. Example (cont.) After removing unnecessary states
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32. CLOSURE UNDER THE REGULAR OPERATIONS
[Using NFA]
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33. Closure Under Union
The class of regular languages is closed under the Union operation.
Let NFA1 recognize A1 and NFA2 recognize A2. Construct NFA3 to recognize A1 U A2.
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Prof. Movaghar

34. Proof (cont.)
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35. Closure Under Concatenation Operation
The class of regular languages is closed under the concatenation operation.
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Prof. Movaghar

36. Proof (cont.)
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37. Closure Under Star operation
The class of regular languages is closed under the star operation.
We represent another NFA to recognize A*.
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Prof. Movaghar

38. Proof (cont.)
𝑄= 𝑄 0 𝑈 𝑄 1 The states of N are the states of N1 plus a new start state.
The state q0 is the new start state.
𝐹= 𝑞 0 𝑈 𝐹 1
The accept states are the old accept states plus the new start state.
Define 𝛿 so that for any 𝑞∈𝑄 𝑎𝑛𝑑 𝑎∈ 𝜀 :
Theory of Languages and Automata
Prof. Movaghar

39. Regular Expression Say that R is a regular expression if R is:
a for some a in the alphabet Σ 𝜀 ∅
𝑅 1 𝑈 𝑅 2 , 𝑤ℎ𝑒𝑟𝑒 𝑅 1 𝑎𝑛𝑑 𝑅 2 𝑎𝑟𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑒𝑥𝑝.
𝑅 1 𝑜 𝑅 2 , 𝑤ℎ𝑒𝑟𝑒 𝑅 1 𝑎𝑛𝑑 𝑅 2 𝑎𝑟𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑒𝑥𝑝
𝑅 1 ∗ , 𝑤ℎ𝑒𝑟𝑒 𝑅 1 𝑎𝑛𝑑 𝑅 2 𝑎𝑟𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑒𝑥𝑝
Circular Definition?
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Prof. Movaghar

40. Regular Expression Language
Let R be a regular expression. L( R ) is the language that is defined by R:
𝑖𝑓 𝑅=𝑎 𝑓𝑜𝑟 𝑎∈Σ then L( R )={a}
𝑖𝑓 𝑅=𝜀 𝑡ℎ𝑒𝑛 𝐿 𝑅 ={𝜀}
𝑖𝑓 𝑅=∅ 𝑡ℎ𝑒𝑛 𝐿 𝑅 =∅
𝑖𝑓 𝑅= 𝑅 1 𝑈 𝑅 2 𝑡ℎ𝑒𝑛 𝐿 𝑅 =𝐿 𝑅 1 𝑈𝐿 𝑅 2
𝑖𝑓 𝑅= 𝑅 1 ∩ 𝑅 2 𝑡ℎ𝑒𝑛 𝐿 𝑅 =𝐿 𝑅 1 ∩𝐿 𝑅 2
𝑖𝑓 𝑅= 𝑅 1 ∗ 𝑡ℎ𝑒𝑛 𝐿 𝑅 = 𝐿 𝑅 1 ∗
Theory of Languages and Automata
Prof. Movaghar

41. Examples(cont.)
1. 0∪𝜀 1 ∗ = 01 ∗ ∪ 1 ∗ 2. Σ ∗ 1 Σ ∗ ={𝑤|𝑤 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 1} 3. 0 ∗ 10 ∗ ={𝑤|𝑤 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 1} 4. Σ ∗ 001 Σ ∗ ={𝑤|𝑤 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 001 𝑎𝑠 𝑎 𝑠𝑢𝑏𝑠𝑡𝑟𝑖𝑛𝑔} 5. 01𝑈10={01,10} 6. (ΣΣ) ∗ ={𝑤|𝑤 𝑖𝑠 𝑎 𝑠𝑡𝑟𝑖𝑛𝑔 𝑜𝑓 𝑒𝑣𝑒𝑛 𝑙𝑒𝑛𝑔𝑡ℎ} 7. (ΣΣΣ) ∗ ={𝑤|𝑡ℎ𝑒 𝑙𝑒𝑛𝑡𝑔ℎ 𝑜𝑓 𝑤 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3} 8 (ΣΣΣΣ) ∗ ={𝑤|𝑡ℎ𝑒 𝑙𝑒𝑛𝑡𝑔ℎ 𝑜𝑓 𝑤 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4} 9. 0∪𝜀 1 ∗ = 01 ∗ ∪ 1 ∗ 10. 0∪𝜀 (1∪𝜀) ={𝜀,0,1,01} ∗ ∅=∅ 12. ∅ ∗ ={𝜀}
Theory of Languages and Automata
Prof. Movaghar

42. Equivalence of DFA and Regular Expression
A language is regular if and only if some regular expression describes it.
Lemma:
If a language is described by a regular expression, then it is regular.
If a language is regular, then it is described by a regular expression.
Theory of Languages and Automata
Prof. Movaghar

43. Building an NFA from the Regular Expression
We consider the six cases in the formal definition of regular expressions
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44. Examples
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45. Other direction of the proof
We need to show that, if a language A is regular, a regular expression describes it!
First we show how to convert DFAs into GNFAs, and then GNFAs into regular expressions.
We can easily convert a DFA into a GNFA in the special form.
3-state DFA
5-state GNFA
4-state GNFA
Regular Expression
2-state GNFA
3-state GNFA
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Prof. Movaghar

46. Formal Definition
A generalized nondeterministic finite automaton is a 5-tuple, a 5-tuple (Q ,Σ ,δ ,qstart , qaccept), where
Q is a finite set called states,
Σ is a the input alphabet,
δ : (𝑄−{ 𝑞 𝑎𝑐𝑐𝑒𝑝𝑡 }) ×(Q−{ 𝑄 𝑠𝑡𝑎𝑟𝑡 } ⟶𝑅 is the transition function,
qstart is the start state, and
qaccept is the accept state.
Theory of Languages and Automata
Prof. Movaghar

47. Assumptions
For convenience we require that GNFAs always have a special form that meets the following conditions:
The start state has transition arrows going to every other state but no arrows coming in from any other state.
There is only a single accept state, and it has arrows coming in from every other state but no arrows going to any other state. Furthermore, the accept state is not the same as the start state.
Except for the start and accept states, one arrow goes from every state to every other state and also from each state to itself.
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Prof. Movaghar

48. How to Eliminate a State?
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49. Example
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50. Example
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51. Grammar A grammar G is a 4-tuple G = (V, Σ, R, S) where:
V is a finite set of variables,
Σ is a finite, disjoint from V, of terminals,
R is a finite set of rules,
S is the start variable.
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Prof. Movaghar

52. Rule
A rule is of the form 𝑥→𝑦 where 𝑥∈(𝑉∪Σ)+ and 𝑦∈(𝑉∪Σ)∗ The rules are applied in the following manner: given a string w of the form w = uxv, We say that the rule x → y is applicable to this string, and we may use it to replace x with y, thereby obtaining a new string z = uyv, This is written as 𝑤⇒𝑧.
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53. Derivation If 𝑊 1 ⇒ 𝑊 2 ⇒ …⇒ 𝑊 𝑛 we say that W1 derives Wn and write
𝑊 1 ⇒ 𝑊 2 ⇒ …⇒ 𝑊 𝑛
we say that W1 derives Wn and write
𝑊 1 ⇒ ∗ 𝑊 𝑛
Thus, we always have
𝑊 ⇒ ∗ 𝑊
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54. Language of a Grammar
Let G = (V, Σ, R, S) be a grammar. Then, the set 𝑳 𝑮 ={𝑾∈ 𝑻 ∗ :𝑺 ⇒ ∗ 𝑾} is the language generated by G.
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Prof. Movaghar

55. Example
Consider the grammar G = ({S}, {a,b}, P, S} with P given by S → aSb S → ε Then 𝑆→𝑎𝑆𝑏→𝑎𝑎𝑆𝑏𝑏→𝑎𝑎𝑏𝑏 So we can write 𝑆 ⇒ ∗ 𝑎𝑎𝑏𝑏 Then, L(G) = {an bn: n ≥ 0}
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56. A Notation for Grammars
Consider the grammar
G = ({S}, {a,b}, P, S}
with P given by
S → aSb
S → ε
The above grammar is usually written as:
G: S → aSb | ε
Theory of Languages and Automata
Prof. Movaghar

57. Regular Grammar
A grammar G = (V, Σ, R, S) is said to be right-linear if all rules are of the form A → xB A → x Where A, B V, and X Σ*. A grammar is said to be left-linear if all rules are of the form A → Bx A regular grammar is one that is either right-linear or left-linear.
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Prof. Movaghar

58. L(G) is a regular language.
Theorem
Let G = (V, Σ, R, S) be a right-linear grammar. Then:
L(G) is a regular language.
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59. Example
Construct a NFA that accepts the language generated by the grammar V0 → aV1 V1 → abV0 | b a b V0 V1 Vf a b V2
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60. Theorem
Let L be a regular language on the alphabet Σ. Then: There exists a right-linear grammar G = (V, Σ, R, S) Such that L = L(G).
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61. Theorem
Theorem A language is regular if and only if there exists a left-linear grammar G such that L = L(G).
Outline of the proof:
Given any left-linear grammar with rules of the form
A → Bx
A → x
We can construct a right-linear Ĝ by replacing every such rule of G with
A → xRB
A → xR
We have L(G) = L(Ĝ)R .
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62. Theorem
A language L is regular if and only if there exists a regular grammar G such that L = L(G). 𝐿 𝑖𝑠 𝒓𝒆𝒈𝒖𝒍𝒂𝒓 ↔∃ 𝐺:𝐿=𝐿 𝐺 ;𝐺 𝑖𝑠 𝒓𝒆𝒈𝒖𝒍𝒂𝒓
Theory of Languages and Automata
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