Left Recursion in Parsing Expression Grammars

Parsing Expression Grammars (PEGs) are a formalism that can describe all deterministic context-free languages through a set of rules that specify a top-down parser for some language. PEGs are easy to use, and there are efficient implementations of PEG libraries in several programming languages. A frequently missed feature of PEGs is left recursion, which is commonly used in Context-Free Grammars (CFGs) to encode left-associative operations. We present a simple conservative extension to the semantics of PEGs that gives useful meaning to direct and indirect left-recursive rules, and show that our extensions make it easy to express left-recursive idioms from CFGs in PEGs, with similar results. We prove the conservativeness of these extensions, and also prove that they work with any left-recursive PEG. PEGs can also be compiled to programs in a low-level parsing machine. We present an extension to the semantics of the operations of this parsing machine that let it interpret left-recursive PEGs, and prove that this extension is correct with regards to our semantics for left-recursive PEGs.Comment: Extended version of the paper "Left Recursion in Parsing Expression Grammars", that was published on 2012 Brazilian Symposium on Programming Language

Discriminative estimation of probabilistic context-free grammars for mathematical expression recognition and retrieval

[EN] We present a discriminative learning algorithm for the probabilistic estimation of two-dimensional probabilistic context-free grammars (2D-PCFG) for mathematical expressions recognition and retrieval. This algorithm is based on a generalization of the H-criterion as the objective function and the growth transformations as the optimization method. For the development of the discriminative estimation algorithm, the N-best interpretations provided by the 2D-PCFG have been considered. Experimental results are reported on two available datasets: Im2Latex and IBEM. The first experiment compares the proposed discriminative estimation method with the classic Viterbi-based estimation method. The second one studies the performance of the estimated models depending on the length of the mathematical expressions and the number of admissible errors in the metric used.This research has been developed with the support of Grant PID2020-116813RBI00a funded by MCIN/AEI/ 10.13039/501100011033 and FPI grant CIACIF/2021/313 funded by Generalitat Valenciana. Universitat Politecnica de Valencia Grant No. SP20210263Noya García, E.; Benedí Ruiz, JM.; Sánchez Peiró, JA.; Anitei, D. (2023). Discriminative estimation of probabilistic context-free grammars for mathematical expression recognition and retrieval. Pattern Analysis and Applications. 26:1571-1584. https://doi.org/10.1007/s10044-023-01158-81571158426Bahl LR, Jelinek F, Mercer RL (1983) A maximum likelihood approach to continuous speech recognition. IEEE Trans Pattern Anal Machine Intell 5(2):179–190Koehn P (2009) Statistical Machine Translation. 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