Conflict Removal. Less Than, Equals ( <= ) Conflict

Transcription

1 Conflict Removal As you have observed in a recent example, not all context free grammars are simple precedence grammars. You have also seen that a context free grammar that is not a simple precedence grammar can be transformed to achieve a simple precedence grammar. The goal in this transformation is to change the grammar so that it is simple precedence while keeping the language unchanged. Hence, the new context free, simple precedence grammar is called similar to the original context free, non-simple precedence grammar. We say that symbol V i has a simple precedence conflict with symbol V j if more than one simple precedence relation holds between V i and V j. There are four different types of simple precedence conflicts:. V i <= V j 2. V i >= V j 3. V i <> V j 4. V i <=> V j We will consider each type of simple precedence conflict below. Our goal is to keep the same language while changing the grammar so as to remove the simple precedence conflict. Less Than, Equals ( <= ) Conflict In the less than, equals conflict, we have V i <= V j. Remember that an equals relation is defined as: V i E V j holds if a right hand side of a production of the form V i V j where V i,v j V and, V *. and that a less than relation is defined as: V i L V j holds if a right hand side of a production of the form V i V k and V k + V j where V k V n, V i, V j V, and,, V *. Thus, if we have A = B and A < B, what we have is a right hand side of some production of the form α A B β, and another production whose right hand side is γ A C δ where C + B μ. The general strategy for a less than equal conflict resolution is to introduce a new non-terminal symbol so that the equals relation is turned into a less than relation. Thus, if we have the productions: and α A B β γ A C δ we would change the first production to α A B-

2 and add the production B- B β Thus, A is equal to B-, and A is less than B. Since B- only appears in the one production, there cannot be any other simple precedence conflicts between A and B-, and there will be no other production whose right hand side is α A B-. You do need to consider that there may be another production whose right hand side is B β. At this point you might try several other approaches to combine or substitute symbols while keeping the grammars similar. Greater Than, Equals ( >= ) Conflict In the greater than, equals conflict, we have V i >= V j. Remember that an equals relation is defined as: V i E V j holds if a right hand side of a production of the form V i V j where V i, V j V and, V *. and that a greater than relation is defined as: V i G V j holds if a right hand side of a production of the form V k V j and V k + V i or if a right hand side of a production of the form V k V l and V k + V i and V l + V j where V k, V l V n, V i, V j V, and,,, V *. Thus, if we have A = B and A > B, what we have is a right hand side of some production of the form α A B β, and either another production whose right hand side is γ C B δ where C + μ A or another production whose right hand side is γ C D δ where C + μ A and D + B σ. The general strategy for a greater than equal conflict resolution is to introduce a new non-terminal so that the equals relation is turned into a greater than relation. Thus, if we have the productions: and α A B β γ C B δ we would change the first production to A- B β and add the production A- α A Thus, A- is equal to B, and A is greater than B. Since A- only appears in the one production, there cannot be any other simple precedence conflicts between A- and B and there will be no other production whose right hand side is A- B β. You do need to check and make sure that there is no production whose right hand side is α A. If this is the case, than another attempt at splitting the α A B β right hand side so that A is not equal to B needs to be considered.

3 Less Than, Greater Than ( <> ) Conflict In the less than, greater than conflict, we have V i <> V j. Remember that a less than relation is defined as: V i L V j holds if a right hand side of a production of the form V i V k and V k + V j where V k V n, V i, V j V, and,, V *. and that a greater than relation is defined as: V i G V j holds if a right hand side of a production of the form V k V j and V k + V i or if a right hand side of a production of the form V k V l and V k + V i and V l + V j where V k, V l V n, V i, V j V, and,,, V *. Thus, if we have A < B and A > B, what we have is a right hand side of some production where α A C β where C + B and either another production whose right hand side is γ E B δ where E + μ A or another production whose right hand side is γ E D δ where E + μ A and D + B σ. The general strategy for a less than greater than conflict resolution is to introduce a new non-terminal symbol so that the less than relation is turned into a greater than relation. Thus, if we have the productions: and α A C β γ E B δ we would change the first production to A- C β and add the production A- α A Thus, A- is less than to B, and A is greater than B. Since A- only appears in the one production, there cannot be any other simple precedence conflicts between A- and B or A- and C. We also need to consider identical right hand sides as we did with the other conflict resolutions. Less than, Equals, Greater Than ( <=> ) Conflict In the less than, equals, greater than conflict, we have V i <=> V j. The general strategy is to first reduce the less than equals conflict so that you have only a less than greater than conflict. Then reduce the less than greater than conflict as described above. When I have been faced with this challenge in the past, I have found that adding one or two new terminal symbols makes the conflict resolution much easier. However, by adding any new terminal symbols, you have changed the language. Example Suppose that we have the following grammar. E E + T 2. T

4 3. T T * F 4. F 5. F P ^ F 6. P 7. P R & P 8. R 9. R V 0. ( E ) When we run the simple precedence analyzer, we get the simple precedence matrix for the above grammar below: E T F P R + * ^ v ( ) E..... =..... = 2 T..... G =.... G 3 F..... G G.... G 4 P..... G G 5... G 5 R..... G G G =.. G L L L.... L L. 7 *.. = L L.... L L. 8 ^.. = L L.... L L = L.... L L. 0 v..... G G G G.. G ( 4 L L L L.... L L. 2 )..... G G G G.. G Notice that there is a less than/equals, represented by 4, conflict between ( and E, and between + and T. There is also a greater than/equals, represented by 5, conflict between P and ^. Our standard approach is to remove the less than/equals conflict and then remove the greater than/equals conflict. Eliminating the less than/equals conflict we want to introduce a new non-terminal system and turn the equals relation into a less than relation. Hence, we will make the following changes to the grammar by introducing a new non-terminal E- and T- while keeping the language the same. Our new set of productions is. E E + T- 2. T 3. T T * F 4. F 5. F P ^ F 6. P

5 7. P R & P 8. R 9. R V 0. ( E- ). E- E 2. T- T where we have changed productions and 0 to eliminate the less than/equals conflicts and added productions and 2. We note that there are now two productions that have the same symbol on the RHS, namely productions 2 and 2. We see that we can change production 2 so that we have 2. E T- The resulting simple precedence matrix is E T E T F P R * ^ v ( ) E =..... G 2 T G =.... G 3 F G G.... G 4 P G G 5... G 5 R G G G =.. G 6 E = 7 T G..... G 8 +. L L L L. =.... L L. 9 *.. = L L L L. 0 ^.. = L L L L.... = L L L. 2 v G G G G.. G 3 ( L L L L L =..... L L. 4 ) G G G G.. G Notice that we only have the greater than/equals conflict between P and ^ in our grammar. Applying the idea that we want to change the equals relation into a greater than by introducing a new non-terminal symbol we might have the following grammar:. E E + T- 2. T- 3. T T * F 4. F 5. F P- ^ F 6. P

6 7. P R & P 8. R 9. R V 0. ( E- ). E- E 2. T- T 3. P- P where we have changed production 5 and added production 3. Again we see that we have two productions that have the same RHS, namely productions 6 and 3. Again, we change production 6 to 6. F P- The new simple precedence matrix is E T P E T F P R * ^ v ( ) E =..... G 2 T G =.... G 3 F G G.... G 4 P G G G... G 5 R G G G =.. G 6 E = 7 T G..... G 8 P = L L L L. = L.... L L. 0 *.. = L L.. L.... L L. ^.. = L L.. L.... L L = L L L. 3 v G G G G.. G 4 ( L L L L L =. L.... L L. 5 ) G G G G.. G We now have a simple precedence grammar whose language is the same as the grammar that we started with.