Properties of LR-DFA Construction

(1)

Properties of LR-DFA Construction

Lemma 1

(a) Each path leading to a state q with item

[A → α.β]

ends with

α

. (b) If q is a state with items

[A → α.δ]

and

[B → β.γ],

then

α

is a postfix of

β

or

β

a postfix of

α.

(c) If there is a transition marked with H from state p to state q , then p contains an item of form

[C → γ.Hδ].

Proof.

The properties hold for LR-NFA by construction and can be transferred to LR-DFA.

Ina Schaefer Context-Free Analysis 1

Properties of LR-DFA Construction (2)

Lemma 2

For any A,

ψ,α,β, it holds that, the item [A → α.β]

is contained in a state that is reached by

ψα

iff there exists a right derivation

S

⇒^∗_rm ψAu ⇒rm ψαβ

u Proof.

cf. Wilhelm, Maurer, Thm. 8.4.3, p. 363.

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LR(0) Characterization Theorem

Theorem

Let

Γ

be a reduced CFG.

Γ

is an LR(0) grammar iff LR-DFA(Γ) does not contain any conflicts.

Proof:

Left-to-Right-Direction: LR(0) property implies that LR-DFA has no conflicts.

Let q be a state of LR-DEA(Γ) with two items [A→ α.] and[B → β.γ].

We show that these items do not cause a conflict.

By Lemma 1(b), there are µ,ν with µα =νβ and with µ =εor ν = ε.

Let ψ be a path leading toq, then according to Lemma 1(a), there exists ϕ with ψ = ϕµα =ϕνβ.

LR(0) Characterization Theorem (2)

By Lemma 2, there are the following right derivations S ⇒^∗rm ϕµAu ⇒rm ϕµαu S ⇒^∗rm ϕνBv ⇒rm ϕνβγv

Case 1: Suppose, the items[A→ α.] and [B → β.γ] are different and cause a reduce/reduce-conflict, i.e. γ = ε.

We show that then it has to holds that A= B and α= β,

i.e. both items are identical which is a contradiction to the assumption.

Since γ = εand ϕµα = ϕνβ, it holds that ϕµα= ϕνβγ.

By the LR(0) property, it holds ϕµ = ϕν and A= B (and v =γv).

Fromϕν =ϕµ and ϕµα = ϕνβ, it follows thatα = β, i.e. both items are identical.

(3)

LR(0) Characterization Theorem (3)

Case 2: Suppose the items [A→ α.]and [B → β.γ]cause a

shift/reduce-conflict, i.e. γ #=εand γ starts with a terminal symbol.

We consider the cases γ ∈ T⁺ and γ =zDρ, where in the first caseγ contains no non-terminal, while in the second it contains at least one non-terminal.

If γ ∈ T⁺, forw =γ, we obtain the derivation

S ⇒^∗rm ϕνBv ⇒rm ϕνβwv = ϕµαwv The LR(0) property yields ϕν =ϕµ,A =B and v = wv.

Thus, it has to hold that w = εwhich contradicts the assumption γ #= ε.

If γ = zDρ, we can extend the above right derivation

S ⇒^∗_rm ϕνBv ⇒rm ϕνβγv =ϕµαzDρv

⇒^∗rm ϕµαzxEyv ⇒rm ϕµαzxwyv

LR(0) Characterization Theorem (4)

Then we have the right derivations

S ⇒^∗rm ϕµAu ⇒rm ϕµαu S ⇒^∗_rm ϕµαzxEyv ⇒rm ϕµαzxwyv The LR(0) property yields in particular that ϕµαzx = ϕµ.

Thus, it has to hold that z = εwhich contradicts the assumption that γ starts with a terminal.

(4)

LR(0) Characterization Theorem (5)

Right-to-Left Direction: Conflict-Freedom implies LR(0) Property.

According to the LR(0) Definition, we consider two right derivations S ⇒^∗rm ψAu ⇒rm ψαu

S ⇒^∗rm ϕBx ⇒rm ψαy In derivation 2, we assume a production B → δ such that ϕδx =ψαy.

By Lemma 2, [A→ α.] belongs to a state reached byψα and [B → δ.] to a state reached byϕδ.

Since ϕδx =ψαy, it holds ϕδ is prefix ofψα or vice versa.

LR(0) Characterization Theorem (6)

Case Distinction on relation betweenϕδ und ψα:

(a) ϕδ =ψα:

[A→ α.] and[B → δ.] belong to the same state.

Since LR-DFA has no conflicts, it holds that α= δ andA =B, thus alsoϕ = ψ andx = y. This yields the LR(0) property.

(b) ϕδ is a real prefix ofψα:

Since ϕδx =ψαy, there exists b ∈ T undz ∈T^∗ such thatx = bzy and hence ϕδbz = ψα.

By Lemma 1(c), the state reached byϕδ has to contain a transition marked with b and an item [C → µ.bν].

But this would cause a conflict with[B → δ.]which contradicts the conflict-freeness of LR-DFA such that this cases cannot occur.

(c) ψα is a real prefix ofϕδ: analog to case (b).