Changes

So long as one considers only words or only sentences, that is, only one level of finite sequences of symbols, it does not matter essentially what the sequences are called.  Unless otherwise specified, a formal language is taken by default to be a ''one-level formal language'', containing only a single level of sequences.  If one wants to consider both words and sentences, that is, finite sequences of symbols and then finite sequences of these lower level sequences, all in the same context of discussion, then one has to move up to an essentially more powerful concept, that of a ''two-level formal language''.

So long as one considers only words or only sentences, that is, only one level of finite sequences of symbols, it does not matter essentially what the sequences are called.  Unless otherwise specified, a formal language is taken by default to be a ''one-level formal language'', containing only a single level of sequences.  If one wants to consider both words and sentences, that is, finite sequences of symbols and then finite sequences of these lower level sequences, all in the same context of discussion, then one has to move up to an essentially more powerful concept, that of a ''two-level formal language''.

Until further notice, the next part of this discussion strictly applies only to one-level formal languages.  When this project reaches the stage of dealing with higher level formal languages, a few of the following definitions and default assumptions will need to be adjusted slightly.

Until further notice, the next part of this discussion applies only to one-level formal languages.  When this project reaches the stage of dealing with higher-level formal languages, a few of the following definitions and default assumptions will need to be adjusted slightly.

 

It is convenient to have a generic term for referring to alphabets and lexicons, indifferently, without concern for their level of construction.  Therefore, I describe any finite set <math>\underline{\underline{X}}</math> as a '''syntactic resource''' for the syntactic domain <math>\underline{X},</math> provided that its elements are regarded as syntactic primitives that can be used to construct the signs and expressions in <math>\underline{X}.</math>  If the primitive signs in a syntactic resource are regarded as denoting primitive objects or primitive operations, then I refer to a collection of these objects or operations as an ''objective'' or ''operational'' resource, as the case may be.

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It is always tempting to seek analogies between formal language theory and algebraic studies, and it is often very useful to do so.  But if one tries to forge an analogy between the relation "X is a resource for X", in the formal language sense, and the relation "X is a basis for X", in the algebraic sense, then it becomes necessary to observe important differences between the two perspectives, as they are currently applied.

It is always tempting to seek analogies between formal language theory and algebraic studies, and it is often very useful to do so.  But if one tries to forge an analogy between the relation "X is a resource for X", in the formal language sense, and the relation "X is a basis for X", in the algebraic sense, then it becomes necessary to observe important differences between the two perspectives, as they are currently applied.