CSC312 Automata Theory Lecture # 28 Chapter # 13 by Cohen Grammatical Format

1. CSC312 Automata Theory Lecture # 28 Chapter # 13 by Cohen Grammatical Format

2. Semiword: A semiword is a string of terminal (maybenone) concaenated with exactly one nonterminal (on the right). i.e. of the shape (terminal) (terminal) … (terminal)(Nonterminal) Theorem 22: If all the productions in a given CFG fit one of the two forms: Nonterminal  semiword or Nonterminal  word ( where the word may be ), then the language generated by this CFG is regular.

3. Regular Grammar: A CFG is called a regular grammar if each of its productions is of one of the two forms Nonterminal  semiword or Nonterminal  word Example: Consider the CFG S  aaS | bbS |  Example: Consider the regular CFG S  aaS | bbS | abX | baX |  S  aaS | bbS | abS | baS

4. NULL-Productions: -Productions The production of the form Non-terminal   is said to be NULL production. Note: If a CFG has a Null production, then it is possible to construct another CFG without Null production accepting the same language with the exception of the word . Removing Null-Production: Delete all the Null productions and add new productions by substituting the  in the old productions.

5. Example: Consider the CFG S  aSbSa |  Now remove the  and add new productions as Old production New productions S  aSbSa S  aSba S  abSa S  aba So the new CFG without  is S  aSbSa |aSba | abSa | aba

6. Nullable-Productions: A production is called nullable if is of the form N   or there is derivation that starts at N and leads to  i.e. N1  N2  N3  … Nn   where N1,N2,…,Nn are non-terminals Example: S  AA | bB A  aa | B B  aS |  .

7. Example: S  AA | bB A  aa | B B  aS |  Here S  AA and A  B are nullable productions while B   is null production Removing Nullable-Production: Remove all -productions by substituting the  in the old productions and also substitute  for the nullable productions.

8. Example: S  XY X  Zb Y  bW Z  AB W  Z A  aA | bA |  B  Ba | Bb |  Here A   and B   are null productions, whereas Z  AB, W  Z are nullable productions Now removing all the null productions we get the following new productions

9. Old productions New Productions S  XY A  a | b X  Zb B  a | b Y  bW X  b Z  AB Y  b W  Z Z  A (by substituting  for B) A  aA | bA Z  B B  Ba | Bb So the final CFG is S  XY A  aA | bA | a | b X  Zb | b B  Ba | Bb | a | b Y  bW | b Z  AB | A | B W  Z

10. Unit Production: The production of the form nonterminal  one terminal, is called unit production. Removing Unit Production: Examples: Consider the following CFG S  A | bb A  B | b B  S | a Separate the unit production from non-unit productions. Now delete unit productions and add the new productions that can be generated by using the deleted unit productions.

11. Chomsky Normal Form (CNF): A CFG is said to be in CNF if it has all its productions of the form nonterminal  string of two nonterminals nonterminal  on terminal Note: Any CFG can be converted into CNF, if the null productions and the unit productions are remobed. Also if a CFG contains nullable productions as well then the corresponding new productions are also to be added.

12. Examples: Consider the following CFG S  aSa | bSb | a | b | aa | bb Convert it into equivalent CNF. Steps for converting any CFG into CNF • Remove all the null and nullable productions • Kill all the unit productions • Convert all the productions of multiple letters into the form nonterminal  string of two nonterminal by converting all terminal in strings of multiple letters into nonterminals and then introduce new productions in the form of pairs of nonterminals.