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| In these types of situation the letter <math>S,\!</math> that signifies the type of a sentence in the language of interest, is called the ''initial symbol'' or the ''sentence symbol'' of a candidate formal grammar for the language, while any number of letters like <math>T,\!</math> signifying other types of strings that are necessary to a reasonable account or a rational reconstruction of the sentences that belong to the language, are collectively referred to as ''intermediate symbols''. | | In these types of situation the letter <math>S,\!</math> that signifies the type of a sentence in the language of interest, is called the ''initial symbol'' or the ''sentence symbol'' of a candidate formal grammar for the language, while any number of letters like <math>T,\!</math> signifying other types of strings that are necessary to a reasonable account or a rational reconstruction of the sentences that belong to the language, are collectively referred to as ''intermediate symbols''. |
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− | Combining the singleton set <math>\{ S \}\!</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}</math> that assembles together all of the intermediate symbols results in the set <math>\{ S \} \cup \mathfrak{Q}</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ S \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ S \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ S \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> as the ''augmented strings'', in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings is becomes desirable to separate the augmented strings that contain the symbol <math>S\!</math> from all other sorts of augmented strings. In these situations, the strings in the disjoint union <math>\{ S \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar. | + | Combining the singleton set <math>\{ S \}\!</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}</math> that assembles together all of the intermediate symbols results in the set <math>\{ S \} \cup \mathfrak{Q}</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ S \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ S \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ S \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> as the ''augmented strings'', in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings it becomes desirable to separate the augmented strings that contain the symbol <math>S\!</math> from all other sorts of augmented strings. In these situations the strings in the disjoint union <math>\{ S \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar. |
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| In forming a grammar for a language statements of the form <math>W :> W',\!</math> | | In forming a grammar for a language statements of the form <math>W :> W',\!</math> |
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| <pre> | | <pre> |
− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | The combined effect of several typos in my typography
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− | along with what may be a lack of faith in imagination,
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− | obliges me to redo a couple of paragraphs from before.
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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− | A notation in the style of "S :> T" is now introduced,
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− | to be read among many others in this manifold of ways:
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− | | S covers T
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− | | S governs T
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− | | S rules T
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− | | S subsumes T
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− | | S types over T
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− | The form "S :> T" is here recruited for polymorphic
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− | employment in at least the following types of roles:
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− | 1. To signify that an individually named or quoted string T is
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− | being typed as a sentence S of the language of interest !L!.
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− | 2. To express the fact or to make the assertion that each member
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− | of a specified set of strings T c !A!* also belongs to the
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− | syntactic category S, the one that qualifies a string as
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− | being a sentence in the relevant formal language !L!.
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− | 3. To specify the intension or to signify the intention that every
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− | string that fits the conditions of the abstract type T must also
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− | fall under the grammatical heading of a sentence, as indicated by
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− | the type name "S", all within the target language !L!.
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− | In these types of situation the letter "S", that signifies the type of
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− | a sentence in the language of interest, is called the "initial symbol"
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− | or the "sentence symbol" of a candidate formal grammar for the language,
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− | while any number of letters like "T", signifying other types of strings
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− | that are necessary to a reasonable account or a rational reconstruction
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− | of the sentences that belong to the language, are collectively referred
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− | to as "intermediate symbols".
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− |
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− | Combining the singleton set {"S"} whose sole member is the initial symbol
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− | with the set !Q! that assembles together all of the intermediate symbols
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− | results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing
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− | the package, the alphabet !A! of the language is also known as the set
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− | of "terminal symbols". In this discussion, I will adopt the convention
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− | that !Q! is the set of intermediate symbols, but I will often use "q"
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− | as a typical variable that ranges over all of the non-terminal symbols,
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− | q in {"S"} |_| !Q!. Finally, it is convenient to refer to all of the
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− | symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
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− | prospective grammar for the language, and accordingly to describe
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− | the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
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− | in effect, expressing the forms that are superimposed on a language
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− | by one of its conceivable grammars. In certain settings is becomes
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− | desirable to separate the augmented strings that contain the symbol
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− | "S" from all other sorts of augmented strings. In these situations,
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− | the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
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− | as the "sentential forms" of the associated grammar.
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| o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
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