# Canonical LR parser

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A canonical LR parser or LR(1) parser is an LR parser whose parsing tables are constructed in a similar way as with LR(0) parsers except that the items in the item sets also contain a lookahead, i.e., a terminal that is expected by the parser after the right-hand side of the rule. For example, such an item for a rule A → B C might be

which would mean that the parser has read a string corresponding to B and expects next a string corresponding to C followed by the terminal 'a'. LR(1) parsers can deal with a very large class of grammars but their parsing tables are often very big. This can often be solved by merging item sets if they are identical except for the lookahead, which results in so-called LALR parsers.

## Contents

### Constructing LR(1) parsing tables

An LR(1) item is a production with a marker together with a terminal, e.g., [S → a A • B e, c]. Intuitively, such an item indicates how much of a certain production we have seen already (a A), what we could expect next (B e), and a lookahead that agrees with what should follow in the input if we ever reduce by the production S → a A B e. By incorporating such lookahead information into the item concept, we can make wiser reduce decisions. The lookahead of an LR(1) item is used directly only when considering reduce actions (i.e., when the • marker is at the right end).

The core of an LR(1) item [S → a A • B e, c] is the LR(0) item S → a A • B e. Different LR(1) items may share the same core. For example, if we have two LR(1) items of the form

• [A → α •, a] and
• [B → α •, b],

we take advantage of the lookahead to decide which reduction to use. (The same setting would perhaps produce a reduce/reduce conflict in the SLR approach.)

### Validity

The notion of validity changes. An item [A → β1 • β2, a] is valid for a viable prefix α β1 if there is a rightmost derivation that yields α A a w which in one step yields α β1β2 a w

### Initial item

To get the parsing started, we begin with the initial item of

Here \$ is a special character denoting the end of the string.

### Closure

Closure is more refined. If [A → α • B β, a] belongs to the set of items, and B → γ is a production of the grammar, then we add the item [B → • γ, b] for all b in FIRST(β a).

### Goto

Goto is the same. A state containing [A → α • X β, a] will move to a state containing [A → α X • β, a] with label X.

Every state has transitions according to Goto. for all

### Shift actions

The shift actions are the same. If [A → α • b β, a] is in state Ik and Ik moves to state Im with label b, then we add the action

### Reduce actions

The reduce actions are more refined than SLR . If [A→α •, a] is in state Ik, then we add the action: "Reduce A → α" to action[Ik, a]. Observe that we don’t use information from FOLLOW(A) anymore. The goto part of the table is as before.