Monday, 8 August 2016

How To Construct LR(0) Parsing Table

                    In this post, we will see how to construct LR(0) parsing table for given grammar.
                    LR(0) is Bottom Up Parsing algorithm.

What is meaning of name LR(0)???
L: Left To Right Scanning of String
R: Reverse to Right Most Derivation
(0): Size of Look ahead is 0. In LR(0), the reduce action is not dependent on next symbol that is why it is 0.


Example: Find the LR(0) parsing table for the following grammar.


S->AA

A->aA/b

Note:
1. Remember A->BCD is called a production while A->.BCD or A->B.CD or A->BC.D or A->BCD. are called as "items". The difference between production and item is the placement of new symbol '.' (Dot). Now, what is the significance of Dot? Dot indicates LHS of Dot is visited while RHS of Dot is still remained to get visited.

2. We have to find items set called as Closure. Closure indicates a state. We move from one closure to another closure after reading a Terminal or a Non-Terminal.

3. We will name these closures as I0, I1,I2..... Here I indicates "Items set" or simply a "closure".

Finding Closure and goto:
Method:
1. Now we will see how to find a closure:
                   Let us find first closure i.e. I0.
In I0, add item (S'->.S). This indicates we have not read S till now. As Dot is before S, we need productions to evaluate S. Add the items for S i.e. (S->.AA). 
                   In item (S->.AA), Dot is followed by A. Hence again we have to add items for A i.e. (A->.aA) and (A->.b).
                   In item (A->.aA/.b), Dot is followed by terminals a and b. So no need to add any other item.

2. Now let us see how to find goto:
                  For each symbol (terminal or non-terminal) which follows Dot, we have to add goto (i.e. edge) and we have to find the closures in the similar way.

                  Check following Diagram of Closures and Goto.


Dia. Closure and Goto
                 In above example, I0,I1,I2,I3 are closures and edges are the "goto" moves.

Note: In above diagram, from I3, for 'a', we get the same state/closure. Hence loop is shown there for goto move 'a'.

Constructing LR(0) parsing table:
                 We have to use above Diagram of Closures and Goto.
Note:
1. Rows will have Closure numbers(e.g. 0 for I0, 1 for I1 etc.) while column have Action (for terminals) and Goto (for Non-Terminals).

2. Action part involves two actions Shift and Reduce while Goto involves just a transition from one Closure to another Closure.

3. Check into following parsing table:
                In s3, s4.... 's' stands for Shift action while '3' or '4' stands for Closure number.
                In r3, r2, r1... 'r' stands for Reduce action while '3' or '2' or '1' stands for production number. In this example. we have numbered productions as follows:
                1. S->AA
                2. A->aA
                3. A->b

Method:
1. For a closure, 
                 For each transition(edge) with terminal, add Si into the table where 'S' represents Shift action and 'i' represents Destination Closure number.
                 While for each transition(edge) with Non-Terminal, add just a Destination Closure number into Goto part of table.
    
2. For the closures, which includes items like (A->BCD.) i.e. where Dot is at the end of item, we have to add entry ri where 'r' stands for Reduce action and i stands for production number.
                 For LR(0) parsing table, add ri into all entries of action part of table.

3. There is a special entry for row 1 and column $ i.e. "Accept".

LR(0) Parsing Table:


Closures Action Goto
a b $ A S
0 s3 s4 2 1
1 Accept
2 s3 s4 5
3 s3 s4 6
4 r3 r3 r3
5 r1 r1 r1
6 r2 r2 r2

Note: If LR(0) parsing table contains two entries in any table cell like (S,R) or (R,R), then the grammar is not LR(0).




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