Canonical collection of LR(0) items -2

Closure Function of LR(0)

Let G be the grammar and I be the set of items of it then closure (I) is defined as the set of items constructed from I as follows.

1. Initially, closure (I) contains the item in I.
2. If closure (I) contains an item A →α• β ץ and there is a production B then item B → ץ is added to I if it does not exist already.
3. Repeat step 2 until there are no new items that can be added to the closure

If α β1 is a viable prefix then X → β1, β2 forms a valid item for αβ1. However, an item can be viable for the number of viable prefixes.

Now, with a valid item for a viable prefix, it is easy to decide whether to perform the shift or reduce action which is as follows,

• If β2 ≠ ∊, then it represents that handle is not shifted onto the stack and thus shift action has to be performed.
• If  β2 = ∊, then it represents X → β1as the handle and thus reduce action has to be performed. 

Thus, for a single viable, prefix there is a possibility of two valid items directing to perform different actions thereby leading to conflicts. However, these conflicts are solvable by cooking at the next input symbol or by other methods.

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Canonical collection of LR(0) items -2