Tree-adjoining grammar (TAG) is a grammar formalism defined by Aravind Joshi. Tree-adjoining grammars are somewhat similar to context-free grammars, but the elementary unit of rewriting is the tree rather than the symbol. Whereas context-free grammars have rules for rewriting symbols as strings of other symbols, tree-adjoining grammars have rules for rewriting the nodes of trees as other trees (see tree (graph theory) and tree (data structure)).
History TAG originated in investigations by Joshi and his students into the family of adjunction grammars (AG), the "string grammar" of Zellig Harris. AGs handle exocentric properties of language in a natural and effective way, but do not have a good characterization of endocentric constructions; the converse is true of rewrite grammars, or phrase-structure grammar (PSG). In 1969, Joshi introduced a family of grammars that exploits this complementarity by mixing the two types of rules. A few very simple rewrite rules suffice to generate the vocabulary of strings for adjunction rules. This family is distinct from the Chomsky-Schützenberger hierarchy but intersects it in interesting and linguistically relevant ways. The center strings and adjunct strings can also be generated by a dependency grammar, avoiding the limitations of rewrite systems entirely.
Description
A TAG can be defined as a 5-tuple ⟨ Σ , N T , I , A , S ⟩ {\displaystyle \langle \Sigma ,NT,I,A,S\rangle } with:
Σ {\displaystyle \Sigma } as the finite set of terminal symbols;
N T {\displaystyle NT} as the finite set of non-terminal symbols, disjunct from Σ {\displaystyle \Sigma } ;
I {\displaystyle I} as a finite set of finite trees called initial trees; Initial trees have non-terminals as inner nodes. The frontier can consist of terminals and non-terminals. Non-terminals at the frontier are marked for substitution (typically by adding the symbol ' ↓ {\displaystyle \downarrow } ' after the non-terminal symbol). Nodes marked for substitution cannot be adjoined to.
A {\displaystyle A} as a finite set of finite trees called auxiliary trees; Auxiliary trees have a special leaf node known as the foot node (typically marked by ' ∗ {\displaystyle \ast } ') which needs to have the same non-terminal symbol as the root of the tree. Foot nodes cannot be substituted; all other non-terminal nodes at the frontier are marked for substitution. As in initial trees, inner nodes have non-terminal symbols.
S {\displaystyle S} as the special start symbol, belonging to the set of non-terminals. Additionally, TAGs with adjunction constraints on nodes have been introduced. An adjunction constraint on a node can: completely disallow adjunction (NA, for null adjunction); make it obligatory (OA); or only allow selected auxiliary trees to adjoin (SA). The two types of basic tree in TAG—initial trees (often denoted by ' α {\displaystyle \alpha } ') and auxiliary trees (' β {\displaystyle \beta } ')—are together called elementary trees. Initial trees represent basic valency relations, while auxiliary trees allow for recursion. A derivation starts with an initial tree, which is combined with further trees via either substitution or adjunction. Substitution replaces a frontier node with an initial tree whose root node has the same label as the leaf for which it is substituted. Adjunction inserts an auxiliary tree—at either a frontier or an internal node—whose root and foot labels both match the label of the node whereat it adjoins. Adjunction can thus have the effect of inserting an auxiliary tree into the center of another tree, which operation may be applied recursively.
Complexity and application For every context-free grammar, a tree-adjoining grammar can be generated which accepts the same string-language. Thus, TAGs can generate all context-free languages; they can generate, as well, some—but not all—context-sensitive languages. Two examples of context-sensitive/non-context-free languages that TAGs (with adjunction constraints) can generate are:
The copy language (i.e. the language of squares), in which an arbitrary string is repeated: { w w ∣ w ∈ Σ ∗ } {\displaystyle \left\{ww\mid w\in \Sigma ^{*}\right\}}
The count-4 language: { a n b n c n d n | 1 ≤ n } {\displaystyle \{a^{n}b^{n}c^{n}d^{n}|1\leq n\}} Tree-adjoining grammars are more powerful (in terms of weak generative capacity) than context-free grammars, but less powerful than linear context-free rewriting systems, indexed, or context-sensitive grammars. Two examples of context-sensitive languages that TAGs cannot generate are:
The language of triplicated strings (i.e. the language of cubes): { w w w ∣ w ∈ Σ ∗ } {\displaystyle \left\{www\mid w\in \Sigma ^{*}\right\}}
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