On the relation between context-free grammars and parsing expression grammars

Abstract

Context-Free Grammars (CFGs) and Parsing Expression Grammars (PEGs) have several similarities and a few differences in both their syntax and semantics, but they are usually presented through formalisms that hinder a proper comparison. In this paper we present a new formalism for CFGs that highlights the similarities and differences between them. The new formalism borrows from PEGs the use of parsing expressions and the recognition-based semantics. We show how one way of removing non-determinism from this formalism yields a formalism with the semantics of PEGs. We also prove, based on these new formalisms, how LL(1) grammars define the same language whether interpreted as CFGs or as PEGs, and also show how strong-LL(k), right-linear, and LLregular grammars have simple language-preserving translations from CFGs to PEGs. Once these classes of CFGs can be automatically translated to equivalent PEGs, we can reuse classic top-down grammars in PEG-based tools.

More About Converting BNF to PEG

“…When there is nothing more to replace, we are left with the sets of "first letters" used in checking the LL(1) condition. For such sets, (4) means X ∩ Y = ∅, which confirms the result about LL(1) grammars from [5,6]. Note that (8) above is not LL(1), and not even LL(k) for any k.…”

“…The grammar G is left-recursive if e ∈ First(e) for some e ∈ E. If the grammar is left-recursive, its PEG parser may be involved in an infinite descent, meaning that a proof of PEG may not exist. On the other hand: This is proved in [5,6] by first verifying that the natural semantics given there is equivalent to the formal definition from [4], and then using a result from the same paper that grammar without left-recursion is "complete". In the Appendix, we give an independent proof that makes our presentation self-contained.…”

“…Following [5,6], we formally define the BNF interpretation by means of a relation BNF ⊆ (E×Σ * )×Σ * . We write [e] x BNF y to mean that the relation holds for e ∈ E and x, y ∈ Σ * .…”

“…This was successfully attempted by Medeiros in his Ph.D. thesis [6]. The thesis is in Portuguese; an extended English version is now available on Computing Research Repository [5]. The PEG and BNF are very similar in appearance, but their formal definitions are worlds apart.…”

“…Because the backtracking is limited, some simple grammars cannot act as their own PEG parsers. In [5,6], one shows how to construct PEG parsers for such grammars using the PEG's "and-predicate" to look as far ahead as needed. The constructions presented there apply to strong LL(k) and LL-regular grammars.…”