1. THEORY OF COMPUTATION
Lecture One:
Automata Theory
Automata Theory

2. Automaton
With every automaton, a transition function is associated which gives the next state in terms of the current state
The labels on the edges show what happens (in terms of input and output) during the transitions
Automata Theory

3. Components of an automaton
Input file : Contains strings of input symbols
Storage unit: consists of an unlimited number of cells, each capable of holding a single symbol from an alphabet
Control unit : can be in any one of a finite number of internal states and can change states in defined manner
Automata Theory

4. Some Terms used in automaton theory
Alphabets-Everything in mathematics is based on symbols. This
is also true for automata theory. Alphabets are
defined as a finite set of symbols. An example of
alphabet is a set of decimal numbers
∑={0,1,2,3,4,5,6,7,8,9}
Strings- A string is a finite sequence of symbols selected from some alphabet
If ∑ {a,b} is an alphabet then abab is string over alphabet ∑. A string is generally denoted by w. The length of string is denoted by |w|
Empty string is string with zero occurrence of symbols . This string is represented by є
Automata Theory

5. Finite Automaton
One of the powerful models of computation which are restricted model of actual computer is called finite automata. These machines are very similar to CPU of a computer .They are restricted model as they lack memory.
Finite automation is called finite because number of possible states and number of letter in alphabet are both finite and automation because the change of state is totally governed by the input.
Automata Theory

6. 2.2 Deterministic Finite Automata
graphic model for a DFA
tape
tape reader
finite
control
Automata Theory

7. Finite Automata FA
Its goal is to act as a recognizer for specific a language/pattern.
Any problem can be presented in form of decidable problem that can be answered by Yes/No.
Hence FA (machine with limited memory) can solve any problem.
Automata Theory

8. Deterministic Finite Automata DFA
FA = “a 5-tuple “ (Q, Σ, , q0, F)
Q: {q0, q1, q2, …} is set of states.
Σ: {a, b, …} set of alphabet.
(delta): represents the set of transitions that FA can take between its states.
 : Q x Σ→Q
Q x Σ to Q, this function:
Takes a state and input symbol as arguments.
Returns a single state.
q0 Q is the start state.
F Q is the set of final/accepting states.
Automata Theory

9. A DFA machine can be constructed with every input and output.
In transition, DFA cannot use n empty string, and it can be understood as one machine.
A DFA machine can be constructed with every input and output.
Automata Theory

10. Transition function  : Q x Σ→Q
Maps from domain of (states, letters) to range of states. Q
Q x Σ
(q0, a) q1
(q2, b) q2
(q1, b) q3
Automata Theory

11. How does FA work? Starts from a start state. Loop
Reads a sequence of letters
Until input string finishes
If the current state is a final state then
Input string is accepted.
Else
Input string is NOT accepted.
But how can FA be designed and represented?
Automata Theory

12. Transition System FA = “a 5-tuple “ (Q, Σ, , q0, F)
Q: {q0, q1, q2, …} is set of states.
Σ: {a, b, …} set of alphabet.
(delta): represents the set of transitions that FA can take between its states.
 : Q x Σ→Q
Q x Σ to Q, this function:
Takes a state and input symbol as arguments.
Returns a single state.
q0 Q is the start state.
F Q is the set of final/accepting states.
Automata Theory

13. Transition System Transition Diagrams Transition Tables
Automata Theory

14. Transition Diagram Notations
If any state q in Q is the starting state then it is represented by the circle with arrow as
Nodes corresponding to accepting states are marked by a double circle q
Automata Theory

15. Transition Diagram
Can be represented by directed labeled graph/Transition table
Vertex is a state
States= nodes
Starting/Initial state denoted by circle and arrow/-
Final state(s) denoted by two concentric circles/+
Transition function  =directed arrows connecting states. a
a,b S2 S1 b
Automata Theory

16. Acceptability of a string
A string is accepted by a transition system if
There exist a path from initial state to final state
Path traversed is equal to w
Automata Theory

17. Example withTransition Table
Check for * 1
A C B
B D A
C A D
D B C
Automata Theory

18. Example withTransition Table
Solution:-
(A,110101)= (B,10101)
(A,0101)
(C,101)
(D,01)
(B,1) A*
Automata Theory

19. Properties of transition function
(q,λ)=q
It comes back to same state
It requires an input symbol to change the state of a system.
(q,aw)=((q,a),w)
(q,w,a)=((q,w),a)
Automata Theory