nestedderivation

Nested derivation is a form of derivation in formal grammars and proof systems where the expansion of a nonterminal itself contains further derivations, creating nested levels of rule applications. It reflects recursive or hierarchical structure in the language being described.

In formal grammars, a derivation is a sequence of productions from the start symbol to a target string. Nested derivations occur when part of the string being derived contains a nonterminal that is expanded by its own sub-derivation, often represented by a derivation tree or parse tree with several levels. They arise naturally in context-free grammars with recursive productions and in proof systems that allow subproofs within larger proofs.

Representation of nested derivations is commonly done with parse trees or derivation trees, which illustrate nested derivations as subtrees beneath a higher-level branch. These trees help with modular reasoning, explaining the structure of strings, and analyzing ambiguity. They also align with recursive definitions found in many grammar formalisms and logical calculi.

Example: consider the context-free grammar with S -> a S b | ε. A nested derivation of the string aabb proceeds as: S => a S b => a (a S b) b => a a S b b => aabb. This demonstrates a nested expansion where the inner S is expanded before the outer portion is completed, yielding a pattern of nested dependencies that is characteristic of recursive grammars.

See also: formal grammar, parse tree, derivation, leftmost derivation, rightmost derivation.