Dyck language

Summary

In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of brackets. The set of Dyck words forms a Dyck language. The simplest, D1, uses just two matching brackets, e.g. ( and ).

Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions.

Formal definition edit

Let   be the alphabet consisting of the symbols [ and ]. Let   denote its Kleene closure. The Dyck language is defined as:

 

Context-free grammar edit

It may be helpful to define the Dyck language via a context-free grammar in some situations. The Dyck language is generated by the context-free grammar with a single non-terminal S, and the production:

Sε | "[" S "]" S

That is, S is either the empty string (ε) or is "[", an element of the Dyck language, the matching "]", and an element of the Dyck language.

An alternative context-free grammar for the Dyck language is given by the production:

S → ("[" S "]")*

That is, S is zero or more occurrences of the combination of "[", an element of the Dyck language, and a matching "]", where multiple elements of the Dyck language on the right side of the production are free to differ from each other.

Alternative definition edit

In yet other contexts it may instead be helpful to define the Dyck language by splitting   into equivalence classes, as follows. For any element   of length  , we define partial functions   and   by

  is   with " " inserted into the  th position
  is   with " " deleted from the  th position

with the understanding that   is undefined for   and   is undefined if  . We define an equivalence relation   on   as follows: for elements   we have   if and only if there exists a sequence of zero or more applications of the   and   functions starting with   and ending with  . That the sequence of zero operations is allowed accounts for the reflexivity of  . Symmetry follows from the observation that any finite sequence of applications of   to a string can be undone with a finite sequence of applications of  . Transitivity is clear from the definition.

The equivalence relation partitions the language   into equivalence classes. If we take   to denote the empty string, then the language corresponding to the equivalence class   is called the Dyck language.

Properties edit

  • The Dyck language is closed under the operation of concatenation.
  • By treating   as an algebraic monoid under concatenation we see that the monoid structure transfers onto the quotient  , resulting in the syntactic monoid of the Dyck language. The class   will be denoted  .
  • The syntactic monoid of the Dyck language is not commutative: if   and   then  .
  • With the notation above,   but neither   nor   are invertible in  .
  • The syntactic monoid of the Dyck language is isomorphic to the bicyclic semigroup by virtue of the properties of   and   described above.
  • By the Chomsky–Schützenberger representation theorem, any context-free language is a homomorphic image of the intersection of some regular language with a Dyck language on one or more kinds of bracket pairs.[1]
  • The Dyck language with two distinct types of brackets can be recognized in the complexity class  .[2]
  • The number of distinct Dyck words with exactly n pairs of parentheses and k innermost pairs (viz. the substring  ) is the Narayana number  .
  • The number of distinct Dyck words with exactly n pairs of parentheses is the n-th Catalan number  . Notice that the Dyck language of words with n parentheses pairs is equal to the union, over all possible k, of the Dyck languages of words of n parentheses pairs with k innermost pairs, as defined in the previous point. Since k can range from 0 to n, we obtain the following equality, which indeed holds:
 

Examples edit

 
Lattice of the 14 Dyck words of length 8 - [ and ] interpreted as up and down

We can define an equivalence relation   on the Dyck language  . For   we have   if and only if  , i.e.   and   have the same length. This relation partitions the Dyck language:  . We have   where  . Note that   is empty for odd  .

Having introduced the Dyck words of length  , we can introduce a relationship on them. For every   we define a relation   on  ; for   we have   if and only if   can be reached from   by a series of proper swaps. A proper swap in a word   swaps an occurrence of '][' with '[]'. For each   the relation   makes   into a partially ordered set. The relation   is reflexive because an empty sequence of proper swaps takes   to  . Transitivity follows because we can extend a sequence of proper swaps that takes   to   by concatenating it with a sequence of proper swaps that takes   to   forming a sequence that takes   into  . To see that   is also antisymmetric we introduce an auxiliary function   defined as a sum over all prefixes   of  :

 

The following table illustrates that   is strictly monotonic with respect to proper swaps.

Strict monotonicity of  
partial sums of          
    ] [  
    [ ]  
partial sums of          
Difference of partial sums 0 2 0 0

Hence   so   when there is a proper swap that takes   into  . Now if we assume that both   and  , then there are non-empty sequences of proper swaps such   is taken into   and vice versa. But then   which is nonsensical. Therefore, whenever both   and   are in  , we have  , hence   is antisymmetric.

The partial ordered set   is shown in the illustration accompanying the introduction if we interpret a [ as going up and ] as going down.

Generalizations edit

There exist variants of the Dyck language with multiple delimiters, e.g., D2 on the alphabet "(", ")", "[", and "]". The words of such a language are the ones which are well-parenthesized for all delimiters, i.e., one can read the word from left to right, push every opening delimiter on the stack, and whenever we reach a closing delimiter then we must be able to pop the matching opening delimiter from the top of the stack. (The counting algorithm above does not generalise).

See also edit

Notes edit

  1. ^ Kambites, Communications in Algebra Volume 37 Issue 1 (2009) 193-208
  2. ^ Barrington and Corbett, Information Processing Letters 32 (1989) 251-256

References edit

  • Dyck language at PlanetMath.
  • A proof of the Chomsky Schützenberger theorem
  • An AMS blog entry on Dyck words