Changes

A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \emptyset.</math>  A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\emptyset) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\emptyset).</math>  This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math>  A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.

A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \emptyset.</math>  A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\emptyset) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\emptyset).</math>  This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math>  A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.

Only one thing remains to cast this description of the cactus language into a form that is commonly found acceptable.  As presently formulated, the principle PC&nbsp;4 appears to be attempting to define an infinite number of new concepts all in a single step, at least, it appears to invoke the indefinitely long sequences of operators, <math>\operatorname{Conc}^k</math> and <math>\operatorname{Surc}^k,</math> for all <math>k > 0.\!</math>  As a general rule, one prefers to have an effectively finite description of

Only one thing remains to cast this description of the cactus language

conceptual objects, and this means restricting the description to a finite number of schematic principles, each of which involves a finite number of schematic effects, that is, a finite number of schemata that explicitly relate conditions to results.

into a form that is commonly found acceptable.  As presently formulated,

the principle PC 4 appears to be attempting to define an infinite number

of new concepts all in a single step, at least, it appears to invoke the

indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0.

As a general rule, one prefers to have an effectively finite description of

conceptual objects, and this means restricting the description to a finite

number of schematic principles, each of which involves a finite number of

schematic effects, that is, a finite number of schemata that explicitly

relate conditions to results.

A start in this direction, taking steps toward an effective description

A start in this direction, taking steps toward an effective description of the cactus language, a finitary conception of its membership conditions, and a bounded characterization of a typical sentence in the language, can be made by recasting the present description of these expressions into the pattern of what is called, more or less roughly, a ''formal grammar''.

of the cactus language, a finitary conception of its membership conditions,

and a bounded characterization of a typical sentence in the language, can be

made by recasting the present description of these expressions into the pattern

of what is called, more or less roughly, a "formal grammar".

A notation in the style of "S :> T" is now introduced,

A notation in the style of <math>S :> T\!</math> is now introduced, to be read among many others in this manifold of ways:

to be read among many others in this manifold of ways:

| S covers T

{| align="center" cellpadding="4" width="90%"

|

| S governs T

| <math>S\ \operatorname{covers}\ T</math>

|

|-

| S rules T

| <math>S\ \operatorname{governs}\ T</math>

|

|-

| S subsumes T

| <math>S\ \operatorname{rules}\ T</math>

|

|-

| S types over T

| <math>S\ \operatorname{subsumes}\ T</math>

|-

| <math>S\ \operatorname{types~over}\ T</math>

The form "S :> T" is here recruited for polymorphic

The form "S :> T" is here recruited for polymorphic

employment in at least the following types of roles:

employment in at least the following types of roles: