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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.
Let be the alphabet consisting of the symbols [ and ]. Let denote its Kleene closure. The Dyck language is defined as:
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:
- S → ε | "[" 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.
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.
- 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.
- The Dyck language with two distinct types of brackets can be recognized in the complexity class .
- 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:
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.
|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.
There exist variants of the Dyck language with multiple delimiters, e.g., 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.
- Kambites, Communications in Algebra Volume 37 Issue 1 (2009) 193-208
- Barrington and Corbett, Information Processing Letters 32 (1989) 251-256