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| 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. |
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− | <pre>
| + | 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 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. | |
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− | 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". | |
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− | 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: | |
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− | | 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> |
| + | |} |
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| + | <pre> |
| 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: |