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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 16:42, 26 April 2012→6.7. Basic Notions of Formal Language Theory: markup
In formal language theory one typically fixes the syntactic resource <math>\underline{\underline{X}}</math> as the primary reality, that is, as the ruling parameter of discussion, and then considers each formal language <math>\underline{X}</math> that can be generated on <math>\underline{\underline{X}}</math> as a particular subset of the maximal language that is possible on <math>\underline{\underline{X}}.</math> This direction of approach can be contrasted with what is more usual in algebraic studies, where the generated object <math>\underline{X}</math> is taken as the primary reality, and a basis <math>\underline{\underline{X}}</math> is defined secondarily as a minimal or independent spanning set, but generally serves as only one of many possible bases.
In formal language theory one typically fixes the syntactic resource <math>\underline{\underline{X}}</math> as the primary reality, that is, as the ruling parameter of discussion, and then considers each formal language <math>\underline{X}</math> that can be generated on <math>\underline{\underline{X}}</math> as a particular subset of the maximal language that is possible on <math>\underline{\underline{X}}.</math> This direction of approach can be contrasted with what is more usual in algebraic studies, where the generated object <math>\underline{X}</math> is taken as the primary reality, and a basis <math>\underline{\underline{X}}</math> is defined secondarily as a minimal or independent spanning set, but generally serves as only one of many possible bases.
The linguistic relation <math>\underline{\underline{X}} ~\text{is a resource for}~ \underline{X}</math> is thus exploited in the opposite direction from the algebraic relation <math>\underline{\underline{X}} ~\text{is a basis for}~ \underline{X}.</math> There does not appear to be any reason in principle why either study cannot be cast the other way around, but it has to be noted that the current practices, and the preferences that support them, dictate otherwise.
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By way of a general notation, I use doubly underlined capital letters to denote finite sets taken as the syntactic resources of formal languages, and I use doubly underlined lower case letters to denote their symbols. Schematically, this appears as follows:
By way of a general notation, I use doubly underlined capital letters to denote finite sets taken as the syntactic resources of formal languages, and I use doubly underlined lower case letters to denote their symbols. Schematically, this appears as follows: