Constructing the Canonical Collection of LR(0) Items: A Guide to LR Parsing

Learn how to construct the canonical collection of LR(0) items, a fundamental step in building LR(0) parsers for compiler design. This guide details the process of creating the deterministic finite automaton (DFA) representing the parser's states and transitions.

Constructing the Canonical Collection of LR(0) Items

Understanding LR(0) Items

LR(0) parsing is a bottom-up parsing technique used in compiler design. It's a deterministic method, meaning there's only one action the parser can take for every input symbol and state. An LR(0) item is a production rule from the grammar with a dot (•) inserted to show the parsing progress. The position of the dot indicates how much of the production has been parsed so far. The process of generating the canonical collection of LR(0) items involves determining all the possible states of the parser and the transitions between them.

Creating the Canonical Collection of LR(0) Items

Let's illustrate the process with this grammar:

S → AA
A → aA | b

1. Augment the Grammar: Add an augmented production (S' → S) with a new start symbol (S').
2. Add the Dot: Insert a dot (•) at the beginning of each production (representing the parser's current position).
3. Compute I0 (initial state): Apply the closure operation to the augmented production (S' → •S). The closure adds all productions whose non-terminal symbol directly follows the dot. This creates the initial state of the DFA (Deterministic Finite Automaton).

I0 = { S' → •S, S → •AA, A → •aA, A → •b }

State Transitions ("Go To" Operations)

From the initial state, the parser transitions to other states by consuming symbols (a or b). The dot (•) moves to indicate parsing progress. This creates the transitions within the DFA. For example, from I0, encountering an 'S' leads to state I1:

I1 = { S' → S• }

Similarly, other transitions are computed (I0 to I2 on 'A', I0 to I3 on 'a', I0 to I4 on 'b', I2 to I5 on 'A', I3 to I6 on 'A').

Creating the DFA and LR(0) Parsing Table

The DFA is created from the collection of LR(0) items. Each state in the DFA represents a set of items. Transitions between states occur based on input symbols. The LR(0) parsing table is built from the DFA. It guides the parser's actions:

• A transition on a terminal symbol indicates a shift action (moving to the next state).
• A transition on a non-terminal symbol indicates a goto action (moving to the next state).
• A state with a completed production (dot at the end) indicates a reduction action (applying a grammar rule).

The construction of the LR(0) parsing table from the DFA is based on these rules.

Conclusion

The canonical collection of LR(0) items and the resulting LR(0) parsing table are fundamental for building LR parsers. These structures guide the parser in efficiently analyzing an input string according to the defined grammar rules. This allows compilers to effectively and accurately parse complex code.