| { | |
| "paper_id": "C67-1008", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T12:35:26.452333Z" | |
| }, | |
| "title": "", | |
| "authors": [], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "", | |
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| "paper_id": "C67-1008", | |
| "_pdf_hash": "", | |
| "abstract": [], | |
| "body_text": [ | |
| { | |
| "text": "By HANS KARLGREN KVAL, Fack, Stockholm 40, Sweden Summary Bar-Hillel and Lambek have outlined a syntactical description where the syntactical category symbols are written as fractions and where the analysis of a given sentence is performed according to rules very similar to common arithmetical reduction of fraction expressions with factors that cancel out. Are there algorithms for applying such a calculus to normally complexnatural language structures ?", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "SLANT GRAMMAR CALCULUS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Aim", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "We consider only one method, named that of a categorial grammar (Bar-Hillel and Lambek).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Aim", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "We make the following assumptions:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Aim", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "(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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Aim", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "(h) A sequence of elements is grammatical if there exists at least one wordfor-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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Aim", | |
| "sec_num": "1." | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Z. Shrinking Procedure", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Z. Shrinking Procedure", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The work reported in this paper has been sponsored by Humanistiska forsk-ningsr~det, Tekniska forskningsr~det and Riksbankens Jubileumsfond, Stockholm, Sweden.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Z. Shrinking Procedure", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Z. Shrinking Procedure", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "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: \\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. 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 theoriginal or the so far reduced -sequence into one symbol:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "a)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "x/y y~ x y y\\x-* x where x and Z are atomic or complex symbols.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "8) a grammatical sentence is a syntagm of a type which belongs to a short 1 -7", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "list of possible types of patterns, say type s.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "The categorial notation seems helpful in establishing a recognition calculus. Some programmable algorithms will be discussed in this paper.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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 \"equivalent\" to a recognition grammar).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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) .", | |
| "cite_spans": [ | |
| { | |
| "start": 251, | |
| "end": 259, | |
| "text": "(Marcus)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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 algo-rithm 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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Slant Grammar Calculus", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Reduction Procedure", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "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. 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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "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.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "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 ....", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "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 ....", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "where the underlined characters are numerator atoms and all others are denominator atoms.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "(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 numer-at0r, 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.) we have to decide which one is intended by a given denominator.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "How do we do this ?", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "The problem is in this form a purely computational one -and no easy such problem.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Substituting Complex Symbols", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "One algorithm can be summarized as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "We have a given string of words and want to ascertain whether its type is one of the set T = Is, t ....", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "]. For each word we have a given set of alternative category symbols.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "We first consider the simpler problem of analyzing a given string of category symbols and see if it can be reduced to s.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "We join the (not underlined symbol) s followed by space to the beginning of the given symbol string.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "When the reduction rules are applied the resultant string should vanish; we say it is reduced to unity.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "The string now contains exactly as many, say n, numerator and denominator atoms.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| }, | |
| { | |
| "text": "Every numerator should be paired with one other denominator; the whole probhefn is to decide which denominator. To analyze a string of words we assign to each word one symbol of the type ABCDE, where C is the set of numerators in all the word ts category symbols, B is the set of nominator atoms appearing immediately before the numerator in any one of the word's category symbols, etc. On the string of these new symbols the above procedure, mutatis mutandis, is applied. Thus, instead of the condition in (I) above that two atoms should be the \"same\" characters, it is required that one underlined atom A and a non-underlined atom B appearing separated only by space should fulfill the condition A n B ~ 0.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Algorithm with Stacking", | |
| "sec_num": "8." | |
| } | |
| ], | |
| "back_matter": [], | |
| "bib_entries": { | |
| "BIBREF0": { | |
| "ref_id": "b0", | |
| "title": "Bar-Hillel, A quasi-arithmetical notation for syntactic description", | |
| "authors": [ | |
| { | |
| "first": "I", | |
| "middle": [ | |
| "Y" | |
| ], | |
| "last": "", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1953, | |
| "venue": "Language", | |
| "volume": "9", | |
| "issue": "", | |
| "pages": "47--58", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "I. Y. Bar-Hillel, A quasi-arithmetical notation for syntactic description. Language Z9, 47-58 (1953).", | |
| "links": null | |
| }, | |
| "BIBREF1": { | |
| "ref_id": "b1", | |
| "title": "On the calculus of syntactic types", | |
| "authors": [ | |
| { | |
| "first": "J", | |
| "middle": [], | |
| "last": "Lambek", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1961, | |
| "venue": "Structure of Language and Its Mathematical Aspects", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "166--178", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "J. Lambek, On the calculus of syntactic types, in \"Structure of Language and Its Mathematical Aspects\". Proc. 12th S},mp. Appl. Mith., American Mathematical Society, Providence, R.I., 1961, pp. 166-178.", | |
| "links": null | |
| }, | |
| "BIBREF2": { | |
| "ref_id": "b2", | |
| "title": "Algebraic Linguistics; Analytical Models", | |
| "authors": [ | |
| { | |
| "first": "S", | |
| "middle": [], | |
| "last": "Marcus", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1967, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "S. Marcus, Algebraic Linguistics; Analytical Models, New York & London, 1967.", | |
| "links": null | |
| } | |
| }, | |
| "ref_entries": { | |
| "FIGREF0": { | |
| "text": "Example: b/(c/a) c/a b\\a/(b/c) d d\\(b/c) = b/y y a/x d d\\x.", | |
| "num": null, | |
| "uris": null, | |
| "type_str": "figure" | |
| } | |
| } | |
| } | |
| } |