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In computational linguistics, the term mildly context-sensitive grammar formalisms refers to several grammar formalisms that have been developed with the ambition to provide adequate descriptions of the syntactic structure of natural language. Every such formalism defines a class of mildly context-sensitive grammars (the grammars that can be specified in the formalism), and therefore also a class of mildly context-sensitive languages (the formal languages generated by the grammars).

Background
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. At the same time, the step to the next level of the Chomsky hierarchy, to context-sensitive grammars, appeared both unnecessary and undesirable. In an attempt to pinpoint the exact formal power required for the adequate description of natural language syntax, Aravind Joshi characterized ‘grammars (and associated languages) that are only slightly more powerful than context-free grammars (context-free languages)’. He called these grammars mildly context-sensitive grammars and the associated languages mildly context-sensitive languages.

Joshi’s characterization of mildly context-sensitive grammars was biased toward his work on tree-adjoining grammar (TAG). However, together with his students Vijay Shanker and David Weir, Joshi soon discovered that TAGs are equivalent, in terms of the generated languages, to the independently introduced head grammar (HG). This was followed by two similar equivalence results, for linear indexed grammar (LIG) and combinatory categorial grammar (CCG), which showed that the notion of mildly context-sensitivity is a very general one and not tied to a specific formalism.

The TAG-equivalent formalisms were generalized by the introduction of linear context-free rewriting systems (LCFRS). These grammars define an infinite hierarchy of languages in between the context-free and the context-sensitive languages, with the languages generated by the TAG-equivalent formalisms at the lower end of the hierarchy. Because of this, LCFRS is often regarded as the most general formalism for specifying mildly context-sensitive grammars. However, several authors have noted that some of the characteristic properties of the TAG-equivalent formalisms are not preserved by LCFRS, and that on the other hand there are languages that have the characteristic properties of mildly context-sensitivity but are not generated by LCFRS.

Characterization
There is no generally accepted formal definition of mildly context-sensitivity; Joshi only provides a ‘rough characterization’. According to this, a class of mildly context-sensitive grammars has the following properties: In addition to these, it is understood that every class of mildly context-sensitive grammars should be able to generate all context-free languages.
 * 1) limited cross-serial dependencies
 * 2) constant growth
 * 3) polynomial parsing

Cross-serial dependencies
The term cross-serial dependencies refers to certain characteristic word ordering patterns, in particular to the verb–argument patterns observed in subordinate clauses in Dutch and Swiss German. These are the very patterns that can be used to argue against the context-freeness of natural language; thus requiring mildly context-sensitive grammars to model cross-serial dependencies means that these grammars must be more powerful than context-free grammars.

Some authors identify the ability to model cross-serial dependencies with the ability to generate the copy language

$$\mathit{COPY} = \{\, ww \mid w \in \{a, b\}^* \,\}$$

This language is not context-free, which can be shown using the pumping lemma.

Constant growth
A formal language is of constant growth if every string in the language is longer than the next shorter strings by at most a (language-specific) constant. Languages that violate this property are often considered to be beyond human capacity, although some authors have argued that certain phenomena in natural language do show a growth that cannot be bounded by a constant.

Most mildly context-sensitive grammar formalisms (in particular, LCFRS) actually satisfy a stronger property than constant growth called semilinearity. A language is semilinear if its image under the Parikh-mapping (the mapping that ‘forgets’ the relative position of the symbols in a string, effectively treating it as a bag of words) is a regular language. All semilinear languages are of constant growth, but not every language with constant growth is semilinear.

Polynomial parsing
A grammar formalism is said to have polynomial parsing if its membership problem can be solved in deterministic polynomial time. This is the problem to decide, given a grammar G written in the formalism and a string w, whether w is generated by G – that is, whether w is ‘grammatical’ according to G. The time complexity of this problem is measured in terms of the combined size of G and w. Note that for practical applications one is interested not only in the yes/no question whether a sentence is grammatical, but also in the syntactic structure that the grammar assigns to the sentence.

Formalisms
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.

TAG-equivalent formalisms

 * Tree-adjoining grammar (TAG)
 * Head grammar (HG)
 * Linear indexed grammar (LIG)
 * Combinatory categorial grammar (CCG)

LCFRS-equivalent formalisms

 * Linear context-free rewriting systems (LCFRS)
 * Multicomponent tree-adjoining grammars (MCTAG)
 * Multiple context-free grammars (MCFG)
 * Minimalist grammars (MG)
 * Simple range concatenation grammars (SRCG)