[Inquiry] Re: Theme One Program -- Exposition

[Jon Awbrey]

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TOP.  Expository Note 7

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3.1.  Lexical Cacti (cont.)

Example 2.  Two-Level Formal Language:  am

Consider a very small example of a 2-level formal language,
whose only word is the character sequence <"a", "m">, and
whose only phrase is the singleton sequence <"am">, where
we observe the convention that "am" = <"a", "m">.

Formally speaking, we have the following data:

   L_1 = {<"a", "m">} c !A!*.

   L_2 = {<"am">} = {<<"a", "m">>} c !A!**.

Theme One stores its data about the lexical level of a 2-level
formal language in the form of a "lexical cactus".  As always,
one may choose to view this information as an abstract graph,
as a concrete pointer graph, or as a text file consisting of
the characters that encode a traversal string for the graph.

Example 2.  Lexical Level:  am

Figure 4 shows the abstract lexical cactus
for the lexical level L_1 = {<"a", "m">}.

      o
      |
   mo-o o
    |/ /
   ao-o
    |/
    @

   ( a ( m , ( ) ) , ( ) )

   Figure 4.  Abstract Lexical Cactus:  am

Figure 5 shows the concrete lexical cactus
for the lexical level L_1 = {<"a", "m">}.

                                               o-----o
                                       o-------|--o  |
                                       |  o----o  |  |
                                       o->| )0 |--o  |
                                          o----o     |
                                          ^   o------o
                                          |  /    o-----o          o-----o
                  o-----------------------|-/-----|--o  |  o-------|--o  |
                  |  o----o  o----o  o----o< o----o  |  |  |  o----o  |  |
                  o->| m1 |->| ,1 |->| (0 |->| )0 |--o  |  o->| )0 |--o  |
                     o----o  o----o  o----o  o----o     |     o----o     |
                     ^   o------------------------------o     ^   o------o
                     |  /                                     |  /    o-----o
     o---------------|-/--------------------------------------|-/-----|--o  |
     |  o----o  o----o<              o----o              o----o< o----o  |  |
     o->| a1 |->| (1 |-------------->| ,0 |------------->| (0 |->| )0 |--o  |
        o----o  o----o               o----o              o----o  o----o     |
        ^  o----------------------------------------------------------------o
        | /
   o----o<
   | (1 |
   o----o
   ^
   |
   @

   (1 a1 (1 m1 ,1 (0 )0 )0 ,0 (0 )0 )0

   Figure 5.  Concrete Lexical Cactus:  am

Jon Awbrey

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