context free grammars_simplification
- 1. Costas Busch - LSU 1 Simplifications of Context-Free Grammars
- 2. Costas Busch - LSU 2 A Substitution Rule bB aAB abBcA aaAA aBS Substitute Equivalent grammar aAB abbcabBcA aaAA abaBS | | bB
- 3. Costas Busch - LSU 3 Equivalent grammarabaAcabbcabBcA aaAA aaAabaBS || || aAB abbcabBcA aaAA abaBS | | Substitute aAB
- 4. Costas Busch - LSU 4 In general: 1yB xBzA Substitute zxyxBzA 1| equivalent grammar 1yB
- 5. Costas Busch - LSU 5 Nullable Variables :production X Nullable Variable: Y Example: M aMbM aMbS Nullable variable production
- 6. Costas Busch - LSU 6 M M aMbM aMbS Substitute abaMbM abaMbS | | Removing sproduction After we remove all the all the nullable variables disappear (except for the start variable) sproduction
- 7. Costas Busch - LSU 7 Unit-Productions YX Unit Production: (a single variable in both sides) bbB AB BA aA aAS Example: Unit Productions
- 8. Costas Busch - LSU 8 bbB AB BA aA aAS Substitute BA bbB BAB aA aBaAS | | Removal of unit productions:
- 9. Costas Busch - LSU 9 Remove bbB BAB aA aBaAS | | bbB AB aA aBaAS | BB XX can be removed immediately Unit productions of form
- 10. Costas Busch - LSU 10 Substitute AB bbB aA aAaBaAS || bbB AB aA aBaAS |
- 11. Costas Busch - LSU 11 Remove repeated productions bbB aA aBaAS | bbB aA aAaBaAS || Final grammar
- 12. Costas Busch - LSU 12 Useless Productions aAA AS S aSbS aAaaaaAaAAS Some derivations never terminate... Useless Production
- 13. Costas Busch - LSU 13 bAB A aAA AS Another grammar: Not reachable from S Useless Production
- 14. Costas Busch - LSU 14 In general: If there is a derivation )(GLwxAyS Then variable is usefulA Otherwise, variable is uselessA consists of terminals
- 15. Costas Busch - LSU 15 A production is useless if any of its variables is useless xA DC CB aAA AS S aSbS Productions useless useless useless useless Variables useless useless useless
- 16. Costas Busch - LSU 16 Example Grammar: aCbC aaB aA CAaSS || Removing Useless Variables and Productions
- 17. Costas Busch - LSU 17 First: find all variables that can produce strings with only terminals or aCbC aaB aA CAaSS || },{ BA },,{ SBA Round 1: Round 2: (possible useful variables) (the right hand side of production that has only terminals) (the right hand side of a production has terminals and variables of previous round) This process can be generalized
- 18. Costas Busch - LSU 18 Then, remove productions that use variables other than aCbC aaB aA CAaSS || },,{ SBA aaB aA AaSS |
- 19. Costas Busch - LSU 19 Second: Find all variables reachable from aaB aA AaSS | S A B Use a Dependency Graph where nodes are variables unreachable S
- 20. Costas Busch - LSU 20 Keep only the variables reachable from S aaB aA AaSS | aA AaSS | Final Grammar Contains only useful variables
- 21. Costas Busch - LSU 21 Removing All Step 1: Remove Nullable Variables Step 2: Remove Unit-Productions Step 3: Remove Useless Variables This sequence guarantees that unwanted variables and productions are removed