In the theory of formal languages of computer science, mathematics, and linguistics, the Dyck language (Dyck being pronounced "deek") is the language consisting of those balanced string (computer science)#Formal theory of parentheses [ and ]. It is important in the parsing of expressions that must have a correctly nested sequence of parentheses, such as arithmetic or algebraic expressions. ==Formal definition== Let ''Σ = { [, ] }'' be the alphabet consisting of the symbols [ and ] and let ''Σ*'' denote its Kleene closure. For any element ''u ∈ Σ*'' with length |''u''| we define partial functions ''insert: Σ* × ('' N ''∪ {0}) → Σ*'' and ''delete: Σ* ×'' N → Σ*'' by :''insert(u, j) = '' ''u'' with "[]" inserted into the ''j''th position :''delete(u, j) = '' ''u'' with "[]" deleted from the ''j''th position with the understanding that ''insert(u, j)'' is undefined for ''j'' > |''u''| and ''delete(u, j)'' is undefined if ''j'' > |''u''| ''− 2''. We define an equivalence relation ''R'' on ''Σ*'' as follows: for elements ''a, b ∈ Σ*'' we have ''(a, b) ∈ R'' if and only if there exists a finite sequence of applications of the ''insert'' and ''delete'' functions starting with ''a'' and ending with ''b'', where the empty sequence is allowed. That the empty sequence is allowed accounts for the reflexive relation of ''R''. symmetric relation follows from the observation that any finite sequence of applications of ''insert'' to a string can be undone with a finite sequence of applications of ''delete''. transitive relation is clear from the definition. The equivalence relation partitions the language ''Σ*'' into equivalence classes. If we take ''ε'' to denote the empty string, then the language corresponding to the equivalence class ''Cl(ε)'' is called the Dyck language. ==Properties== *The Dyck language is closed under the operation of concatenation *By treating ''Σ*'' as an algebraic monoid under concatenation we see that the monoid structure transfers onto the quotient ''Σ*/R'', resulting in the syntactic monoid of the Dyck language. The class ''Cl(ε)'' will be denoted 1. *The syntactic monoid of the Dyck language is not commutative: if ''u = Cl(''['')'' and ''v = Cl('']'')'' then ''uv = Cl(''[]'') = 1 ≠ Cl('']['') = vu''. *With the notation above, ''uv = 1'' but neither ''u'' nor ''v'' are invertible in ''Σ*/R''. *The syntactic monoid of the Dyck language is isomorphic to the bicyclic semigroup by virtue of the properties of ''Cl(''['')'' and ''Cl('']'')'' describe above. ==Reference== S.A. Rankin and I.J.W. Robinson, ''Mathematics for Computer Scientists, Volume 1: The Fundamentals''. Tenth Edition. Kinson Publishing, Ltd., 2004. Formal languages
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Dyck language
In the theory of formal languages of computer science, mathematics, and linguistics, the Dyck language (Dyck being pronounced "deek") is the language consisting of those balanced string (computer science)#Formal theory of parentheses [ and ]. It is important in the parsing of expressions that must have a correctly nested sequence of parentheses, such as arithmetic or algebraic expressions. ==Formal definition== Let ''Σ = { [, ] }'' be the alphabet consisting of the symbols [ and ] and let ''Σ*'' denote its Kleene closure. For any element ''u ∈ Σ*'' with length |''u''| we define partial functions ''insert: Σ* × ('' N ''∪ {0}) → Σ*'' and ''delete: Σ* ×'' N → Σ*'' by :''insert(u, j) = '' ''u'' with "[]" inserted into the ''j''th position :''delete(u, j) = '' ''u'' with "[]" deleted from the ''j''th position with the understanding that ''insert(u, j)'' is undefined for ''j'' > |''u''| and ''delete(u, j)'' is undefined if ''j'' > |''u''| ''− 2''. We define an equivalence relation ''R'' on ''Σ*'' as follows: for elements ''a, b ∈ Σ*'' we have ''(a, b) ∈ R'' if and only if there exists a finite sequence of applications of the ''insert'' and ''delete'' functions starting with ''a'' and ending with ''b'', where the empty sequence is allowed. That the empty sequence is allowed accounts for the reflexive relation of ''R''. symmetric relation follows from the observation that any finite sequence of applications of ''insert'' to a string can be undone with a finite sequence of applications of ''delete''. transitive relation is clear from the definition. The equivalence relation partitions the language ''Σ*'' into equivalence classes. If we take ''ε'' to denote the empty string, then the language corresponding to the equivalence class ''Cl(ε)'' is called the Dyck language. ==Properties== *The Dyck language is closed under the operation of concatenation *By treating ''Σ*'' as an algebraic monoid under concatenation we see that the monoid structure transfers onto the quotient ''Σ*/R'', resulting in the syntactic monoid of the Dyck language. The class ''Cl(ε)'' will be denoted 1. *The syntactic monoid of the Dyck language is not commutative: if ''u = Cl(''['')'' and ''v = Cl('']'')'' then ''uv = Cl(''[]'') = 1 ≠ Cl('']['') = vu''. *With the notation above, ''uv = 1'' but neither ''u'' nor ''v'' are invertible in ''Σ*/R''. *The syntactic monoid of the Dyck language is isomorphic to the bicyclic semigroup by virtue of the properties of ''Cl(''['')'' and ''Cl('']'')'' describe above. ==Reference== S.A. Rankin and I.J.W. Robinson, ''Mathematics for Computer Scientists, Volume 1: The Fundamentals''. Tenth Edition. Kinson Publishing, Ltd., 2004. Formal languages
Dyck language
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DA | DB | DC | DE | DF | DG | DH | DI | DJ | DK | DL | DM | DN | DO | DP | DR | DS | DT | DU | DW | DX | DY | DZ |Words begining with Dyck_language:
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