LR-Grammars

LR(0), LR(1), and LR(K)

Deterministic Context-Free

Languages

DCFL

A family of languages that are accepted by a Deterministic Pushdown Automaton (DPDA)

Many programming languages can be described by means of DCFLs

Prefix and Proper Prefix

Prefix (of a string) Any number of leading symbols of that

string

Example: abc Prefixes: , a, ab, abc

Proper Prefix (of a string) A prefix of a string, but not the string itself

Example: abc Proper prefixes: , a, ab

Prefix Property

Context-Free Language (CFL) L is said to have the prefix property whenever w is in L and no proper prefix of w is in L

Not considered a serve restriction

Why?

Because we can easily convert a DCFL to a

DCFL with the prefix property by introducing an

endmarker

Suffix and Proper Suffix

Suffix (of a string)

Any number of trailing symbols

Proper Suffix

A suffix of a string, but not the string itself

Example Grammar

This is the grammar that will be used in many of the examples:

S’ Sc

S SA | A

A aSb | ab

LR-Grammar

Left-to-right scan of the input producing a rightmost derivation

Simply:

L stands for Left-to-right

R stands for rightmost derivation

LR-Items

An item (for a given CFG)

A production with a dot anywhere in the

right side (including the beginning and end)

In the event of an -production: B B · is an item

Example: Items

Given our example grammar:

S’ Sc, S SA|A, A aSb|ab

The items for the grammar are:

S’·Sc, S’S·c, S’Sc·

S·SA, SS·A, SSA·, S·A, SA·

A·aSb, Aa·Sb, AaS·b, AaSb·, A·ab, Aa·b, Aab·

Some Notation

* = 1 or more steps in a derivation

*rm = rightmost derivation

rm = single step in rightmost derivation