Changes

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.

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.

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:

X   =   {x1, ..., xn}.

| <math>\underline{\underline{X}} ~=~ \{ \underline{\underline{x}}_1, \ldots, \underline{\underline{x}}_k \}.</math>

In a formal language context, I use singly underlined capital letters to indicate the various formal languages being considered, that is, the countable sets of sequences over a given syntactic resource that are being singled out for attention, and I use singly underlined lower case letters to indicate various individual sequences in these languages.  Schematically, this appears as follows:

In a formal language context, I use singly underlined capital letters to indicate the various formal languages being considered, that is, the countable sets of sequences over a given syntactic resource that are being singled out for attention, and I use singly underlined lower case letters to indicate various individual sequences in these languages.  Schematically, this appears as follows:

X   =   {x1, ..., xm, ...}.

| <math>\underline{X} ~=~ \{ \underline{x}_1, \ldots, \underline{x}_\ell, \ldots \}.</math>

|}

Usually, one compares different formal languages over a fixed resource, but since resources are finite it is no trouble to unite a finite number of them into a common resource.  Without loss of generality, then, one typically has a fixed set X in mind throughout a given discussion and has to consider a variety of different formal languages that can be generated from the symbols of X.  These sorts of considerations are aided by defining a number of formal operations on the resources X and the languages X.

Usually, one compares different formal languages over a fixed resource, but since resources are finite it is no trouble to unite a finite number of them into a common resource.  Without loss of generality, then, one typically has a fixed set X in mind throughout a given discussion and has to consider a variety of different formal languages that can be generated from the symbols of X.  These sorts of considerations are aided by defining a number of formal operations on the resources X and the languages X.