Sohail Aslam Compiler Construction Notes Set:2-1
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The set s changed so the repeat loop is executed again. This time, however, the item
[E → •int, $/+] does not yield any more items because the dot is followed by the
terminal int. The first set of items is
I0 = {
[S → •E, $],
[E → •E+(E), $/+],
[E → •int, $/+]
}
Let’s consider the rationale behind the Closure procedure. If [A → β•Cδ,a] ∈ s, then one
potential completion for the left context is to find a string that reduces to C, followed by
δa. This completion should cause a reduction to A, since it fills out the production’s
right-hand side (Cδ), and follows it with a valid look-ahead symbol. For a production
C → γ, closure must insert '•' before γ and add appropriate look-ahead symbols – all
terminals that can appear as the initial symbol in δa. This includes every terminal in
FIRST(δ). If ε ∈ FIRST(δ), it also includes a, thus FIRST(δa) in the algorithm.
The goto Procedure
The second critical step in the construction is to derive other parser states from I0. To
accomplish this, we compute, for each state Ii and each grammar symbol y, the state that
would arise if the parser recognized a y while in state Ii. A state s that contains
[X → α • yβ, b] has a transition (goto) labeled y to the state that contains the items
goto(s, y) where y can be terminal or a non-terminal.
goto(s, y)
m ← { }
for each item [X → α •yβ, b] ∈ s
m ← m ∪ {[X → α y•β, b]}
return closure(m)
Finite Automaton of Items
The LR(1) items are used as the states of a finite automaton (FA) that maintains
information about the parsing stack and progress of a shift-reduce parser. The FA will
start out as a nondeterministic finite automaton (NFA). A DFA can be constructed from
this NFA using the subset construction, similar to one we used for lexical analysis.
Consider the NFA of LR(0) items, i.e., no look-ahead. What are the transitions of the
NFA of LR(0) items? Consider the item A→ α•γ. Suppose γ begins with symbol X which
may be a terminal (token) or non-terminal. The item can be written as A→ α•Xη.
