Left Recursion in Parsing Expression Grammars

TL;DR

This paper introduces a conservative extension to Parsing Expression Grammars (PEGs) that enables the correct handling of left recursion, a feature traditionally unsupported, thereby broadening PEGs' expressive power.

Contribution

It proposes a novel semantic extension for PEGs to support left recursion, including proofs of correctness and applicability to existing PEG implementations.

Findings

Extension supports direct and indirect left recursion in PEGs

Proves the extension is conservative and correct

Enables PEGs to express left-recursive CFG idioms

Abstract

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.