  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 LL-regular
grammars have simple language-preserving translations from CFGs to PEGs.
