The blog post details methods for eliminating left and mutual recursion in context-free grammars, crucial for parser construction. Left recursion, where a non-terminal derives itself as the leftmost symbol, is problematic for top-down parsers. The post demonstrates how to remove direct left recursion using factorization and substitution. It then explains how to handle indirect left recursion by ordering non-terminals and systematically applying the direct recursion removal technique. Finally, it addresses mutual recursion, where two or more non-terminals derive each other, converting it into direct left recursion, which can then be eliminated using the previously described methods. The post uses concrete examples to illustrate these transformations, making it easier to understand the process of converting a grammar into a parser-friendly form.
This blog post, titled "Fixing left and mutual recursions in grammars," addresses the challenges posed by left and mutual recursion in context-free grammars, particularly during the process of top-down parsing. These types of recursion can cause infinite loops in recursive descent parsers, which try to expand a non-terminal by recursively calling the production rules. The post meticulously explains why these issues arise and provides solutions for resolving them.
Left recursion occurs when a non-terminal immediately expands into a derivation that starts with itself. This creates a problem because the parser will endlessly attempt to expand the same non-terminal without consuming any input, leading to an infinite loop. The post illustrates this concept with a clear example of a grammar for arithmetic expressions. It then demonstrates a systematic method for eliminating left recursion by introducing new non-terminals and restructuring the grammar rules. This transformation effectively converts left-recursive productions into right-recursive ones. The resulting grammar is functionally equivalent to the original but is amenable to top-down parsing. The post carefully explains each step of this transformation, providing a general formula that can be applied to any left-recursive grammar. It emphasizes the importance of factoring out common prefixes to avoid unnecessary duplication in the rewritten grammar.
Further, the post delves into mutual recursion, which arises when two or more non-terminals refer to each other in a cyclical manner. Similar to left recursion, this can cause infinite loops in recursive descent parsing. The post presents a comprehensive strategy for eliminating mutual recursion. This strategy involves selecting one of the mutually recursive non-terminals and substituting its productions into the other non-terminal's rules. This process effectively removes the direct mutual dependency, potentially creating left recursion in the process. The previously described method for eliminating left recursion is then applied to resolve any newly introduced left-recursive productions. The post uses a concrete example to demonstrate the steps involved in eliminating mutual recursion, again providing a clear and generalizable approach.
Finally, the post briefly touches upon the role of tools like ANTLR and Yacc in handling left and mutual recursion. While these parser generators can handle direct left recursion, they generally do not handle indirect left recursion, underscoring the importance of understanding these concepts for grammar design. The post concludes by reiterating the benefits of understanding these techniques, particularly for building efficient and correct parsers.
Summary of Comments ( 20 )
https://news.ycombinator.com/item?id=42907139
Hacker News users discussed the potential inefficiency of the presented left-recursion elimination algorithm, particularly its reliance on repeated string concatenation. They suggested alternative approaches using stacks or accumulating results in a list for better performance. Some commenters questioned the necessity of fully eliminating left recursion in all cases, pointing out that modern parsing techniques, like packrat parsing, can handle left-recursive grammars directly. The lack of formal proofs or performance comparisons with established methods was also noted. A few users discussed the benefits and drawbacks of different parsing libraries and techniques, including ANTLR and various parser combinator libraries.
The Hacker News post titled "Fixing left and mutual recursions in grammars" sparked a brief but insightful discussion with a few key comments.
One commenter questioned the practicality of the presented transformations, particularly for parsing, expressing concern that they seemed to prioritize generating strings from a grammar rather than the more common task of parsing a string into an abstract syntax tree. They pointed out that the transformations might complicate parsing by obscuring the original structure of the grammar. This commenter also hinted at a potential connection to the Pumping Lemma for context-free languages, suggesting it might be relevant to understanding the limitations of such transformations.
Another comment offered an alternative approach to handling left recursion, suggesting the use of parsing techniques like packrat parsing or operator precedence parsing, which can handle left-recursive grammars directly without requiring transformations. This commenter argued these techniques offer a more practical solution for parsing in real-world scenarios. They further pointed out that the transformations presented in the article, while theoretically interesting, might not be the most efficient or straightforward way to deal with left recursion in practical parser implementations.
A subsequent reply acknowledged the points made, conceding that the described transformations might not be universally applicable or optimal for all parsing situations. This reply clarified that the primary focus of the original post was on grammar manipulation and generation, rather than parsing specifically. It also admitted that for parsing, techniques like those mentioned (packrat parsing, operator precedence) are often more suitable. Finally, the reply suggested that the transformations might still be valuable in certain contexts beyond parsing, such as grammar analysis or transformation for other purposes.