1. Lecture # 5 Theory Of Automata By Dr. MM Alam
2. Lecture#4 Recap • Recursive way of defining languages • Recursive way examples • Regular expressions • Regular expression examples
3. • Construct a regular expression for all words that contain exactly two b’s or exactly three b’s, not more. a*ba*ba* + a*ba*ba*ba* or a*(b + Λ)a*ba*ba*
4. Construct a regular expression for: (i) all strings that end in a double letter. (ii) all strings that do not end in a double letter (i) (a + b)*(aa + bb) (ii)
5. • Construct a regular expression for all strings that have exactly one double letter in them • (b + Λ)(ab)*aa(ba)*(b + Λ) + (a + Λ) (ba)*bb(ab)*(a + Λ)
6. Construct a regular expression for all strings in which the letter b is never tripled. This means that no word contains the substring bbb (Λ + b + bb)(a + ab + abb)* Words can be empty and start and end with a or b. A compulsory ‘a’ is inserted between all repetitions of b’s.
7. • Construct a regular expression for all words in which a is tripled or b is tripled, but not both. This means each word contains the substring aaa or the substring bbb but not both. (Λ + b + bb)(a + ab + abb)*aaa(Λ + b + bb)(a + ab + abb)* + (Λ + a + aa)(b + ba + baa)*bbb(Λ + a + aa)(b + ba + baa)*
8. • Let r1, r2, and r3 be three regular expressions. Show that the language associated with (r1 + r2)r3 is the same as the language associated with r1r3 + r2r3. Show that r1(r2 + r3) is equivalent to r1r2 + r1r3. This will be the same as providing a ‘distributive law’ for regular expressions.
9. (r1+ r2)r3: The first expression can be either r1 or r2. The second expression is always r3. There are two possibilities for this language: r1r3 or r2r3. r1(r2 + r3): The first expression is always r1. It is followed by either r2 or r3.
10. Question • Can a language be expressed by more than one regular expressions, while given that a unique language is generated by that regular expression?
11. Deterministic Finite Automata • Also known as Finite state machine, Finite state automata • It represents an abstract machine which is used to represent a regular language • A regular expression can also be represented using Finite Automata • There are two ways to specify an FA: Transition Tables and Directed Graphs.
12. Graph Representation – Each node (or vertex) represents a state, and the edges (or arcs) connecting the nodes represent the corresponding transitions. – Each state can be labeled.
13. Finite Automata state a, a, a, 0 1 2 3 ∑ = { a, b } b b b final state start state transition Input • Table State a b Representation (continued) 0 1 1 – An FSA may also be 1 2 2 represented with a 2 3 3 state-transition table. The table for the
14. Finite Automata Definition • An FA is defined as follows:- – Finite no of states in which one state must be initial state and more than one or may be none can be the final states. – Sigma Σ provides the input letters from which input strings can be formed. – FA Distinguishing Rule: For each state, there must be an out going transition for each input letter in Sigma Σ.
15. • Σ = {a,b} and states = 0,1,2 where 0 is an initial state and 1 is the final state. • Transitions: 1. At state 0 reading a,b go to state 1, 2. At state 1 reading a, b go to state 2 3. At state 2 reading a, b go to state 2
16. Transition Table Representation Old state Input Next State 0 a 1 0 B 1 1 a 2 1 b 2 2 a 2 2 b 2
17. Another way of representation… Old States New States Reading a Reading b 0 ­ 1 1 1 2 2  2 + 2 2
18. FA directed graph a,b 0- a,b 1 a,b 2+
19. Example • Σ = {a,b} and states = 0,1,2 where 0 is an initial state and 1 is the final state. • Transitions: 1. At state 0 reading a go to state 1, 2. At state 0 reading b go to state 2, 3. At state 1 reading a,b go to state 2 4. At state 2 reading a, b go to state 2
20. Transition Table Representation Old state Input Next State 0 a 1 0 b 2 1 a 2 1 b 2 2 a 2 2 b 2