<?xml version="1.0" standalone="yes"?>
<Paper uid="C67-1008">
  <Title>SLANT GRAMMAR CALCULUS</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1. Aim
</SectionTitle>
    <Paragraph position="0"> We seek a formal recognition procedure that will enable us to decide for any given sequence of elements from a given language whether or not the sequence is grammatical.</Paragraph>
    <Paragraph position="1"> We consider only one method, named that of a categorial grammar (Bar-Hillel and Lambek).</Paragraph>
    <Paragraph position="2"> We make the following assumptions: (a) The knowledge we want to utilize for recognition can without residue be summarized in a list, giving for each word the grammatical categories the word belongs to; a set of combination rules for the category symbols. (h) A sequence of elements is grammatical if there exists at least one word-for-word translation of it into grammatical category symbols which yields a symbol sequence that is permitted according to the set of combination rules. We say that a symbol sequence which agrees with the combination rules is a grammatical symbol sequence.</Paragraph>
    <Paragraph position="3"> Z. Shrinking Procedure We assume that it is possible to verify the grammaticality of a symbol sequence by reduction of it to simpler and shorter sequences step by step. In each step one or more symbols in the sequences are replaced by one new symbol. The string replaced by one other symbol will - to begin with without linguistic interpretation - be called a syntagrn; the replacing symbol will be called the name of the syntagm.</Paragraph>
    <Paragraph position="4"> The work reported in this paper has been sponsored by Humanistiska forskningsr~det, Tekniska forskningsr~det and Riksbankens Jubileumsfond, Stockholm, Sweden.</Paragraph>
    <Paragraph position="5"> By successive application of rewriting rules, the original sequence is shrunk to a no longer reducible residuej whichmay be just one symbol. If this residue is contained in a given list of permissible sentence patterns, the sentence is grammatical.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3. Slant Grammar Calculus
</SectionTitle>
    <Paragraph position="0"> The kind of grammar under study we shall simply call slant grammar from its salient trait, the notation. It is characterized by the following properties: a) IThe category symbols are all of the form a (atomic symbols) % or x/y~ (complex symbols) \xj where x and y in their turn have the same form ~atomic or complex) as the categor--y symbols. We shall call a_ and x numerators and X a denominator in such cases.</Paragraph>
    <Paragraph position="1"> b) Combinatorics is condensed to the following a) a sym-bol sequence is a grammatical syntagm Of type t if and only if it can be reducedtot by successive application of one of-the following t~o cancellation rules for contracting two neighbouring symbols of the original or the so far reduced - sequence into one symbol:</Paragraph>
    <Paragraph position="3"> where x and Z are atomic or complex symbols.</Paragraph>
    <Paragraph position="4"> 8) a grammatical sentence is a syntagm of a type which belongs to a short 1 -7 list of possible types of patterns, say type s.</Paragraph>
    <Paragraph position="5"> The categorial notation seems helpful in establishing a recognition calculus. Some programmable algorithms will be discussed in this paper.</Paragraph>
    <Paragraph position="6"> It is easily seen that slant grammars of the type discussed are equivalent to context-free phrase structure grammars (as far, i.e., as any generative grammar can be &amp;quot;equivalent&amp;quot; to a recognition grammar).</Paragraph>
    <Paragraph position="7"> The cancellation rules presuppose that if the symbols a/b and b are reduced to a0 there must not stand anything between the syntagms a/b and b - i.e., the~e must be no hole in the syntagm a - although the symbols a/b--and b may in the original sequence stand widely apart.</Paragraph>
    <Paragraph position="8"> A slant grammar for one given language may be written in many different ways. Thus one may design the grammar so that the category symbols hav__~e at the most one ~denominator and even so that they have only left denominator or only right denominator (Marcus).</Paragraph>
    <Paragraph position="9"> A natural way to design the grammar would be to let governed syntagms have simple symbols (~ and the governingo--~complex symbols x/y, or inversely, so that the relation operator/operand would imply dependency relation.</Paragraph>
    <Paragraph position="10"> However, the number of alternative symbols for each word will tend to increase if such a priori rules should apply to the whole set of symbols. Given the algorithm for recognition, one may ask how the categorial grammar should be designed so as to give the minimum number of operations, e.g., so as to yield on an average, the minimum number of possible word-for-word translations into grammatical symbols.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4. Reduction Procedure
</SectionTitle>
    <Paragraph position="0"> To begin with, we shall inve~stigate some procedures for analysis of a given sequence of category symbols. Then, cf. 8 below, we turn to the practically more important problem when not a sequence o~r-symbols but a sequence of words is given, each word having several potential categories.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5. Substituting Complex Symbols
</SectionTitle>
    <Paragraph position="0"> We make a preliminary simplification of the problem by replacing every complex denominator in the sequence by a new, arbitrary atomic symbol. Simultaneously, we make corresponding substitutions of numerators: if we replace b/c by x as a denominator, we also replace b/c by, x at some other place, where b/c appears as a numerator.</Paragraph>
    <Paragraph position="1"> Example: b/(c/a) c/a b\a/(b/c) d d\(b/c) = b/y y a/x d d\x.</Paragraph>
    <Paragraph position="2"> Now, if b/c should happen to appear in numerator position more often in the given sequence than it does qua denominator, this replacement can be performed in more than one way. We then do perform it in more than one way~ thus generating a number of alternative symbol strings to be processed. Through this artifice, we have sequences where all denominators are certain to be atoms, a fact which radically simplifies the analysis. Instead we have made the symbol qelection procedure more difficult.</Paragraph>
    <Paragraph position="3"> Since now all denominators are atoms and since {b\a)/c is equivalent to b/(a/c) and to b~a/c, the brackets are now redundant and can be omitted.</Paragraph>
    <Paragraph position="4"> The symbols, then consist of a kernel atom, possibly neighboured at one side or both by a slant__ and another atom, in its turn possibly neighboured by slant plus atom, and so on, a11 slants to the left of the kernel being tilted to the left and those to the right tilted in the opposite direction:  r-a, a/c, b\alc, g\f\e\d\alblc I ....</Paragraph>
    <Paragraph position="5"> If one knows which element is the kernel.one does not even need the slants and we can proceed to simplify one step further: a, ac, bac, gfeda_bc ....</Paragraph>
    <Paragraph position="6"> where the underlined characters are numerator atoms and all others are denominator atoms.</Paragraph>
    <Paragraph position="7"> (If the language has no rules for the relative order of syntagms, grammaticality is rapidly teste_~l~Just check that to each denominator atom corresponds one numerator atom of the same name, leaving without a match just one numerat0r, which then denotes the type of the syntagm. In this case we are permitted to treat the atoms as numerators and denominators in the arithmetical sense. If we assign prime numbers to each atom and reduce in the standard arithmetical way, we end up with the numerical value of the type of the syn~agm. This simple test may be worth considering as a first check, even though the strUcture of the language be far more complex.)</Paragraph>
  </Section>
class="xml-element"></Paper>