What are LEADING and TRAILING operation of an operator precedence grammar?

LEADING

If production is of form A → aα or A → Ba α where B is Non-terminal, and α can be any string, then the first terminal symbol on R.H.S is

Leading(A) = {a}

If production is of form A → Bα, if a is in LEADING (B), then a will also be in LEADING (A).

TRAILING

If production is of form  A→ αa or A → αaB where B is Non-terminal, and α can be any string then,

TRAILING (A) = {a}

If production is of form  A → αB. If a is in TRAILING (B), then a will be in TRAILING (A).

Algorithm to compute LEADING

Input − Context Free Grammar G

Output − LEADING (A) = {a} iff Boolean Array L [A, a] = true

Method − Procedure Install (A, a) will make L (A, a) to true if it was not true earlier.

• begin

• For each non-terminal A and terminal a

                           L [A, a] = false ;

• For each production of form A ? aα or A → B a α

                            Install (A, a) ;

• While the stack not empty

                            Pop top pair (B, a) form Stack ;

                             For each production of form A → B α

                             Install (A, a);

• end

Procedure Install (A, a)

• begin
• If not L [A, a] then

                    L [A, a] = true

                    push (A, a) onto stack.

• end

Algorithm to compute TRAILING

Input − Context Free Grammar G

Output − TRAILING (A) = {a} iff Boolean Array T [A, a] = true

Method

• begin
• For each non-terminal A and terminal a

                 T [A, a] = false ;

• For each production of form A ? αa or A → α a B

                  Install (A, a) ;

• While the stack not empty

                Pop top pair (B, a) form Stack ;

                For each production of form A → αB

                Install (A, a);

• end

Procedure Install (A, a)

• begin
• If not T [A, a] then

                 T [A, a] = true

                 push (A, a) onto stack.

• end

Algorithm for Computing Operator Precedence Relations

Input − An Operator Grammar

Output − A Precedence Relations between terminals and symbols.

Method

• begin
• For each production A → B₁, B₂, … … … . B_n

                       for i = 1 to n – 1

              If B_i and B_i+1 are both terminals then

                      set B_i = B_i+1

             If i ≤ n − 2 and B_i and B_i+2are both terminals and B_i+1 is non-terminal then

                      set B_i = B_i+1

             If B_iis terminal & B_i+1is non-terminal then for all a in LEADING (B_i+1)

                       set B_i <. a>

            If B_iis non-terminal & B_i+1 is terminal then for all a in TRAILING (B_i)

                      set a . > B_i+1

• end