Implementing a Packrat Parser in Python

This challenge asks you to implement a packrat parser in Python. Packrat parsers are a type of recursive descent parser that uses memoization to achieve linear-time parsing for a large class of context-free grammars, including those that are ambiguous. This is crucial for parsing complex languages efficiently.

Problem Description

You are to create a PackratParser class in Python. This class will be responsible for parsing input strings according to a given grammar. The parser should support memoization to avoid redundant computations, ensuring efficient parsing.

Key requirements:

1. Grammar Definition: The parser should accept a grammar definition. For simplicity, assume the grammar is defined as a dictionary where keys are non-terminal symbols (strings) and values are lists of possible productions. Each production is a list of symbols (terminals or non-terminals).
2. Memoization: Implement memoization to store the results of parsing a non-terminal at a specific input position. If a non-terminal has already been parsed at a given position, its stored result should be returned.
3. Parsing Logic: The parser should recursively try each production of a non-terminal. If a production successfully matches a part of the input, the parser should consume that part and continue parsing the rest of the production.
4. Result Format: The parser should return a structured representation of the parsed input. A simple nested list or tuple structure can represent the parse tree. If parsing fails, it should indicate failure (e.g., return None or raise an exception).
5. Error Handling: The parser should gracefully handle situations where the input does not conform to the grammar.

Example Grammar Representation:

A simple arithmetic expression grammar:

grammar = {
    "expr": [["term", "+", "expr"], ["term"]],
    "term": [["factor", "*", "term"], ["factor"]],
    "factor": [["number"]],
    "number": [["digit", "number"], ["digit"]],
    "digit": [["0"], ["1"], ["2"], ["3"], ["4"], ["5"], ["6"], ["7"], ["8"], ["9"]]
}

Important Considerations:

• Input Position Tracking: You'll need to manage the current position in the input string meticulously.
• Memoization Cache: A dictionary is a suitable data structure for the memoization cache, mapping (non_terminal, position) to parse results.
• Terminal Matching: Implement a mechanism to match terminal symbols against the input string.

Examples

Example 1: Simple Expression

Input Grammar:
{
    "expr": [["term", "+", "expr"], ["term"]],
    "term": [["factor", "*", "term"], ["factor"]],
    "factor": [["number"]],
    "number": [["digit", "number"], ["digit"]],
    "digit": [["0"], ["1"], ["2"], ["3"], ["4"], ["5"], ["6"], ["7"], ["8"], ["9"]]
}
Input String: "1+2*3"

Output: (This is a conceptual output representing a parse tree. The exact structure can vary based on your implementation.)
['expr', ['term', ['factor', ['number', ['digit', '1']]]], '+', ['expr', ['term', ['factor', ['number', ['digit', '2']]]], '*', ['expr', ['term', ['factor', ['number', ['digit', '3']]]]]]]

Explanation: The input string "1+23" is parsed according to the grammar. The outer expr rule matches term ("1"), then "+", then another expr. This inner expr rule then matches term ("23"). The term rule for "23" breaks down into factor ("2"), "", and another term ("3").

Example 2: Ambiguous Grammar (Illustrative of Packrat Benefit)

Consider a simpler ambiguous grammar for addition and multiplication:

Input Grammar:
{
    "expr": [["expr", "+", "term"], ["term"]],
    "term": [["term", "*", "factor"], ["factor"]],
    "factor": [["number"]],
    "number": [["digit"]],
    "digit": [["0"], ["1"], ["2"], ["3"], ["4"], ["5"], ["6"], ["7"], ["8"], ["9"]]
}
Input String: "1+2+3"

Output: (A packrat parser might produce one of the valid parse trees. A non-memoized parser might re-parse subexpressions unnecessarily.)
['expr', ['expr', ['term', ['factor', ['number', ['digit', '1']]]], '+', ['term', ['factor', ['number', ['digit', '2']]]]], '+', ['term', ['factor', ['number', ['digit', '3']]]]]

Explanation: For an input like "1+2+3", the grammar is ambiguous about the associativity of "+". A packrat parser, with memoization, will efficiently explore the possible parse trees and return one of the valid structures. The benefit here is not just correctness but performance, especially with deeper recursive structures.

Example 3: Parsing Failure

Input Grammar:
{
    "expr": [["term", "+", "expr"], ["term"]],
    "term": [["factor", "*", "term"], ["factor"]],
    "factor": [["number"]],
    "number": [["digit", "number"], ["digit"]],
    "digit": [["0"], ["1"], ["2"], ["3"], ["4"], ["5"], ["6"], ["7"], ["8"], ["9"]]
}
Input String: "1+a"

Output: None

Explanation: The input string contains a non-digit character 'a' where a digit is expected by the grammar, leading to a parse failure.

Constraints

• The input string will consist of alphanumeric characters and basic arithmetic operators (+, *).
• The grammar will be provided as a Python dictionary.
• The maximum length of the input string is 1000 characters.
• The number of non-terminals in the grammar will not exceed 50.
• The depth of recursion in parsing should not lead to stack overflow for typical inputs. The packrat approach with memoization should prevent excessive recursion depth for any given state.
• The parser should achieve near-linear time complexity for the given class of grammars.

Notes

• Consider creating helper methods for matching terminals and for handling the recursive calls to non-terminals.
• The structure of the returned parse tree is flexible, but it should clearly represent the hierarchical structure derived from the grammar.
• Think about how to represent the state of the parser (current position, memoization cache) effectively.
• A good starting point for parsing logic might be to implement a basic recursive descent parser first, and then add memoization.
• The grammar definition implies that the order of productions within a non-terminal matters for the first parse found.
• You will need a way to track the input position. A simple integer index will suffice.