LR(1) Automaton

An LR(1) Automaton is a finite-state machine used in parsing, specifically in constructing LR(1) parsers, which are a type of bottom-up parser. The automaton is built from the grammar and guides the parser in determining how to analyze and reduce tokens in the input to construct the correct parse tree.

Components of an LR(1) Automaton

1. Items (States):
   The states in the LR(1) automaton represent "items." Each item is a snapshot of a possible parse at a given point in the input, typically written as:

   A -> α • β, x
   

   This means that the parser is currently in the middle of parsing the production A -> αβ, and the dot (•) indicates how much of the right-hand side has been parsed. x is the lookahead token (the next input token expected after the current derivation).

2. Lookahead (1 token):
   The "1" in LR(1) refers to the lookahead of 1 token. This means the parser makes decisions based on both the current state (items) and the next token in the input. The lookahead token helps the parser decide whether to shift (read more input) or reduce (apply a grammar rule).

3. Shift and Reduce Operations:

   • Shift: This operation moves the dot (•) in an item to the right, meaning that the parser reads one more token from the input and transitions to a new state.
   • Reduce: When the dot is at the end of a production (e.g., A -> α •), the parser applies the corresponding grammar rule and "reduces" the input by recognizing that a production is complete.

4. States and Transitions:
   The automaton consists of states connected by transitions. Each state contains a set of items (representing different parts of the parsing process), and transitions between states occur based on reading symbols (either tokens or grammar symbols).

   For example, if you're in a state where you're expecting the next symbol to be + and the input contains +, the automaton will transition to the next state.

5. Closure and GOTO Functions:

   • Closure: The closure of a set of items adds new items to a state based on grammar rules. If you're in a state where you're expecting a non-terminal symbol (like E), the closure function adds items for every possible production of that non-terminal.
   • Goto: This function moves from one state to another based on the next symbol in the input, determining the next possible parsing steps.

6. Acceptance and Conflicts:

   • The automaton reaches the accepting state when the parser has successfully parsed the entire input.
   • Shift/Reduce conflicts occur when the parser cannot decide whether to shift the input (continue reading tokens) or reduce a rule (apply a grammar production). Reduce/Reduce conflicts occur when the parser is uncertain which rule to apply. LR(1) parsers are designed to avoid most of these conflicts through the lookahead token.

How Does an LR(1) Automaton Work in Parsing?

1. Construct the Automaton:
   The automaton is built by creating states and transitions based on the grammar rules, tracking all possible parsing actions the parser might take at each step.

2. Parsing Process:
   The automaton drives the parsing process. Given an input string, the parser:

   • Shifts tokens from the input and moves to new states.
   • Reduces when it matches a rule, reducing the right-hand side of the rule to the left-hand side.
   • Continues this process until either the input is fully parsed (accepted) or an error is encountered.

Constructing the LR(1) Automaton

For more info visit the dotlr library docs