1. Lecture # 8 (Transition Graphs)

2. Example Consider the language L of strings, defined over Σ={a, b}, having (containing) triple a or triple b. Consider the language L of strings, defined over Σ={a, b}, having (containing) triple a or triple b. The language L may be expressed by RE The language L may be expressed by RE (a+b)* (aaa + bbb) (a+b)* This language may be accepted by the following TG This language may be accepted by the following TG

3. Example Continued …

4. OR

5. OR

6. Example Consider the language L of strings, defined over Σ = {a, b}, beginning and ending in different letters. Consider the language L of strings, defined over Σ = {a, b}, beginning and ending in different letters. The language L may be expressed by RE The language L may be expressed by RE a(a + b)*b + b(a + b)*a The language L may be accepted by the following TG The language L may be accepted by the following TG

7. Example continued …

8. Example Consider the Language L of strings of length two or more, defined over Σ = {a, b}, beginning with and ending in same letters. Consider the Language L of strings of length two or more, defined over Σ = {a, b}, beginning with and ending in same letters. The language L may be expressed by the following regular expression The language L may be expressed by the following regular expression a(a + b)*a + b(a + b)*b This language may be accepted by the following TG This language may be accepted by the following TG

9. Example Continued …

10. Example Consider the EVEN-EVEN language, defined over Σ={a, b}. Consider the EVEN-EVEN language, defined over Σ={a, b}. As discussed earlier that EVEN-EVEN language can be expressed by a regular expression As discussed earlier that EVEN-EVEN language can be expressed by a regular expression ( aa + bb + (ab+ba)(aa+bb)*(ab+ba) )* ( aa + bb + (ab+ba)(aa+bb)*(ab+ba) )* The language EVEN-EVEN may be accepted by the following TG The language EVEN-EVEN may be accepted by the following TG

11. Example continued …

12. Example Consider the language L, defined over Σ={a, b}, in which a’s occur only in even clumps (bundles) and that ends in three or more b’s. Consider the language L, defined over Σ={a, b}, in which a’s occur only in even clumps (bundles) and that ends in three or more b’s. The language L can be expressed by its regular expression The language L can be expressed by its regular expression (aa)*b (b*+(aa(aa)*b)*) bb OR OR (aa)*b (b*+( (aa) + b)*) bb The language L may be accepted by the following TG The language L may be accepted by the following TG

13. Example Continued …

14. Example Consider the following TG Consider the following TG

15. Example Continued … Consider the string abbbabbbabba. It may be observed that the above string traces the following three paths, (using the states) Consider the string abbbabbbabba. It may be observed that the above string traces the following three paths, (using the states) (a)(b) (b) (b) (ab) (bb) (a) (bb) (a) (a)(b) (b) (b) (ab) (bb) (a) (bb) (a) (-)(4)(4)(+)(+)(3)(2)(2)(1)(+) (-)(4)(4)(+)(+)(3)(2)(2)(1)(+) (a)(b) ((b)(b)) (ab) (bb) (a) (bb) (a) (a)(b) ((b)(b)) (ab) (bb) (a) (bb) (a) (-)(4)(+)(+)(+)(3)(2)(2)(1)(+) (-)(4)(+)(+)(+)(3)(2)(2)(1)(+) (a) ((b) (b)) (b) (ab) (bb) (a) (bb) (a) (a) ((b) (b)) (b) (ab) (bb) (a) (bb) (a) (-) (4)(4)(4)(+) (3)(2)(2)(1)(+) (-) (4)(4)(4)(+) (3)(2)(2)(1)(+)

16. Example Continued … Which shows that all these paths are successful, (i.e. the path starting from an initial state and ending in a final state). Which shows that all these paths are successful, (i.e. the path starting from an initial state and ending in a final state). Hence the string abbbabbbabba is accepted by the given TG. Hence the string abbbabbbabba is accepted by the given TG.

17. Generalized Transition Graphs A generalized transition graph (GTG) is a collection of three things: A generalized transition graph (GTG) is a collection of three things: 1. Finite number of states, at least one of which is start state and some (maybe none) final states. 2. Finite set of input letters (Σ) from which input strings are formed. 3. Directed edges connecting some pair of states labeled with regular expression It may be noted that in GTG, the labels of transition edges are corresponding regular expressions

18. Example Consider the language L of strings, defined over Σ={a,b}, containing double a or double b. Consider the language L of strings, defined over Σ={a,b}, containing double a or double b. The language L can be expressed by the following regular expression The language L can be expressed by the following regular expression (a+b)* (aa + bb) (a+b)* The language L may be accepted by the following GTG. The language L may be accepted by the following GTG.

19. Example continued … Or Or

20. Example Consider the Language L of strings, defined over Σ = {a, b}, beginning with and ending in same letters excluding Λ. Consider the Language L of strings, defined over Σ = {a, b}, beginning with and ending in same letters excluding Λ. The language L may be expressed by the following regular expression The language L may be expressed by the following regular expression (a + b) + a(a + b)*a + b(a + b)*b This language may be accepted by the following GTG This language may be accepted by the following GTG

21. Example

22. Example Consider the language L of strings of, defined over Σ = {a, b}, beginning and ending in different letters. Consider the language L of strings of, defined over Σ = {a, b}, beginning and ending in different letters. The language L may be expressed by RE a(a + b)*b + b(a + b)*a The language L may be expressed by RE a(a + b)*b + b(a + b)*a The language L may be accepted by the following GTG The language L may be accepted by the following GTG

23. Example Continued … Another option of GTG for above language L Another option of GTG for above language L

24. Example Continued …

25. Example Consider the language L of strings, defined over Σ={a, b}, having triple a or triple b. Consider the language L of strings, defined over Σ={a, b}, having triple a or triple b. The language L may be expressed by RE (a+b)* (aaa + bbb) (a+b)* The language L may be expressed by RE (a+b)* (aaa + bbb) (a+b)* This language may be accepted by the following GTG This language may be accepted by the following GTG

26. Example Continued … OR

27. NonDeterminism TGs and GTGs provide certain relaxations i.e. there may exist more than one path for a certain string or there may not be any path for a certain string, this property creates nondeterminism and it can also help in differentiating TGs or GTGs from FAs. TGs and GTGs provide certain relaxations i.e. there may exist more than one path for a certain string or there may not be any path for a certain string, this property creates nondeterminism and it can also help in differentiating TGs or GTGs from FAs. Hence an FA is also called a Deterministic Finite Automaton (DFA). Hence an FA is also called a Deterministic Finite Automaton (DFA).

28. Kleene’s Theorem If a language can be expressed by 1. FA or 2. TG or 3. RE then it can also be expressed by other two as well. It may be noted that the theorem is proved, proving the following three parts It may be noted that the theorem is proved, proving the following three parts

29. Kleene’s Theorem continued … Kleene’s Theorem Part I Kleene’s Theorem Part I If a language can be accepted by an FA then it can be accepted by a TG as well. Kleene’s Theorem Part II Kleene’s Theorem Part II If a language can be accepted by a TG then it can be expressed by an RE as well. Kleene’s Theorem Part III Kleene’s Theorem Part III If a language can be expressed by a RE then it can be accepted by an FA as well

30. Kleene’s Theorem continued … Proof(Kleene’s Theorem Part I) Proof(Kleene’s Theorem Part I) Since every FA can be considered to be a TG as well, therefore there is nothing to prove.

31. Thank You…