# 1. Notation

The grammar, then, will be a set of transformation statements each of which transforms a given representation of a sentence into a more specific one.^{10} if α, β, γ, with or without subscripts and primes, stand for any sequences (or zero, henceforth ø) of the elements appearing in statements (e.g., sequences of phonemes, morphemes, phrases, etc., including brackets, dots, etc.), then the basic transformation statements of the grammar will be of the form:

(1) α → β, where …

where α and β contain no notational elements but are simply sequences of the elements set up to represent parts of sentences (phonemes, morphemes, etc.). This means that a is transformed by this statement into a, when conditions … obtain.

If α = α_{l} β_{l} γ and β= α_{l} β_{l}, γ, we rewrite (1) as:

β_{l} → β_{l}, in environment α_{l} ____ γ, where …

The notational devices which will actually be used should be introduced definitionally (by so-called ‘contextual definitions’) be describing a procedure to convert each expression using these notations into a sequence of simple expressions of the form (1) or (2) (which is reductible to (1)) where no notational elements appear. Two kinds of brackets--{}, []--and two kinds of parentheses--(), <>--will be employed as follows:

N1. A statement ‘…

…’ is an abbreviation for

(i) ‘…α_{1}…’, (ii) ‘…α_{2}…’, …, (n) ‘…α_{n}…’,

in that order. if two sets of brackets with a different number of row appear, either can be expanded first. if two or more sets of brackets of this form with the same number of rows appear, then they are expanded simultaneously, the k^{th} row of the first concurring with the k^{th} row of the second. For example

stands for

(i) ‘α_{1}α_{2}α_{3} → β_{1}β_{2}, where … γ_{1} …’

(ii) ‘α_{1}α_{3}α_{4} → β_{1}β_{3}, where … γ_{2} …’

in that order. To indicate how many rows a given set of brackets have, ‘---’ is written where no element α occurs.

Thus ‘

’ has three rows.

N2. N1 holds in exactly the same form for [].

N3. A statement containing one or more elements in main parentheses () is an abbreviation for two statements, one in which all of the parenthesized elements appear, and one in which none of the parenthesized elements appear, in that order. For example

‘α_{1}(α_{2})α_{3} → β_{1}β_{2}(γ_{1}(γ_{2})), where … (---) …’

stands for

‘α_{1}α_{2}α_{3} → β_{1}β_{2}γ_{1}(γ_{2}), where …---…’

‘α_{1}α_{3} → β_{1}β_{2}, where ……’

with (i) preceding (ii), and (i) in turn standing for two statements by the same process of development.

N4. A statement containing one or more elements in parentheses <> is an abbreviation for the conjunction, in any order, of all statements with zero or more of the parenthesized elements omitted. For example

‘α<β>γ → α_{1}β_{1}’

stands for

(i)‘αβγ → α_{1}β_{1}’ | (iii)‘αγ → α_{1}β_{1}’ |

(ii)‘αβγ → α_{1}’ | (iv)‘αγ → α_{1}’ |

taken in any order. Order does not happen to be important in the statements of the grammar in which <> is used. But it could be, and an order could be imposed. Note that the appearance of a single <> is just like that of a single() (except that order is imposed in the second case).

It remains to give the interpretation for the cases where several of these four notations co-occur in one statement. To do this we have to give an order of priority, stating which of co-occurring notations is to be expanded first, so as to have a unique interpretation for each statement. The order of development follows these two principles:

N5. No brackets or parentheses are expanded if enclosed within brackets or parentheses. I.e., at each step in the development of a sentence only main brackets or parentheses may be developed.

N6. If there is more than one set of main brackets or parentheses, they are developed in the order (i) {}, (ii) [], (iii) (), (iv) <>; i.e., in exactly the order in which they were introduced by Nl–4.

This now gives us an explicit step by step procedure for converting each statement of the grammar into an ordered sequence of statements of form (1) or (2). Notice that the case of co-occurring brackets of the same kind with the same number of rows is analogous in interpretation to matrix multiplication, while co-occurring brackets of different kinds give essentially the Cartesian product.

One other point concerning co-occurrence of various notations needs clarification, namely, occurrence of brackets and parentheses within other brackets or parentheses.

N7. Each set of brackets or parentheses is treated as a single element when inside of a containing set. E.g., the main bracket in ‘

’ has two rows.

N8. In accordance with customary practice, a set of brackets with a single row is used to give the membership of a class. Thus {α_{l}, α_{2}, …, α_{n}} is the class containing as members α_{l}, α_{2}, …, α_{n}. A statement of the form ‘α={α_{l}, α_{2}, …, α_{n}}’ is inter...