By 1985, several researchers in descriptive and mathematical linguistics had provided evidence against the hypothesis that the syntactic structure of natural language can be adequately described by context-free grammars.
[1][2] At the same time, the step to the next level of the Chomsky hierarchy, to context-sensitive grammars, appeared both unnecessary and undesirable.
However, together with his students Vijay Shanker and David Weir, Joshi soon discovered that TAGs are equivalent, in terms of the generated string languages, to the independently introduced head grammar (HG).
The TAG-equivalent formalisms were generalized by the introduction of linear context-free rewriting systems (LCFRS).
Independently of and almost simultaneously to LCFRS, Hiroyuki Seki et al. proposed the essentially identical formalism of multiple context-free grammar (MCFG).
[11] Recent years have seen increased interest in the restricted class of well-nested linear context-free rewriting systems/multiple context-free grammars,[10][12] which define a class of grammars that properly includes the TAG-equivalent formalisms but is properly included in the unrestricted LCFRS/MCFG hierarchy.
Despite a considerable amount of work on the subject, there is no generally accepted formal definition of mild context-sensitivity.
[14] Most mildly context-sensitive grammar formalisms (in particular, LCFRS/MCFG) actually satisfy a stronger property than constant growth called semilinearity.
Over the years, a large number of grammar formalisms have been introduced that satisfy some or all of the characteristic properties put forth by Joshi.