Regular Tree Grammar - Examples

Examples

Let, where

  • is our set of nonterminals,
  • is our ranked alphabet, arities indicated by dummy arguments (i.e. the symbol has arity 2),
  • is our starting nonterminal, and
  • the set consists of the following productions:

An example derivation is:

BList \Rightarrow_{G_1} cons(Bool,BList) \Rightarrow_{G_1} cons(false,BList) \Rightarrow_{G_1} cons(false,cons(Bool,BList)) \Rightarrow_{G_1} cons(false,cons(true,BList)) \Rightarrow_{G_1} cons(false,cons(true,nil)) .

The tree language generated by is the set of all finite lists of boolean values, that is, happens to equal . The grammar corresponds to the algebraic data type declarations

datatype Bool = false | true datatype BList = nil | cons of Bool * BList

in the Standard ML programming language: every member of corresponds to a Standard-ML value of type BList.

For another example, let, using the nonterminal set and the alphabet from above, but extending the production set by, consisting of the following productions:

The language is the set of all finite lists of boolean values that contain at least once. The set has no datatype counterpart in Standard ML, nor in any other functional language. It is a proper subset of .

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