A PROGRESS REPORT ON 
THE USE OF SLANT GRAMMAR CALCULUS FOR 
AUTOMATIC ANALYSIS 
Ferenc Kiefer 
The KVAL-approach to syntactic analysis has been based 
on categorial grammar as propounded by Bar-Hillel and Lambek. 
It was clear at the very outset that the original form of categorial 
grammar is not suitable for the purpose of automatized syntactic 
analysis in practice. The first attempts at developing a more 
adequate notation as well as at the closer examination of various 
properties of categorial grammar go back to 1965 and 1966 
/KVAL PM Z37, 248, 298, 302/. In order to distinguish our 
approach from original categorical grammar we have termed it 
slant grammar calculus. 
Our practical concern has been to write a grammar for 
Swedish nominal phrases which would enable us to automatize 
some linguistic aspects of documentation research. Such a gram- 
mar has been compiled by Bengt Svensson. The rules utilize 
lexical andd morphological information. In other words, no strict 
line is drawn between morphological and syntactic rules. A sub- 
stantial part of this preliminary grammar has already been 
checked generatively by means of a special string processing 
algorithm /STRIP/. 
As known one of the most tedious problems in connection 
with categorial grammar is the great number of categories 
assignable to most "of the units in the string to be analyzed. This 
is a par excellence :practical problem but it does not lack theor- 
etical implications. The most interesting of these can be roughly 
phrased as follows: If syntax is to be kept as simple as possible, 
then the lexicon will get complicated. If we aim at the optimal 
simplification of the lexicon, then the burden of our grammar 
will be taken over by syntax. (Recall that in categorial grammar 
we have onlytwo rules for the bidirectional case. Therefore, all 
possible syntactic functions for each lexical entry must be listed 
in the lexicon.) Now we may ask questions about the optimal 
distribution of tasks of a grammar between syntax and lexicon in 
order to achieve an overall simplification in grammatical descrip- 
tion. (Notice that the same question can be asked with respect to 
syntax and semantics, or still better, with respect to syntax, se- 
mantics and the lexicon.) This problem has, of course, not yet 
been solved and consequently, one is forced to work in general 
with several alternatives. 
On the other hand, one can neglect this theoretical issue 
P 
and concentrate on strategies that would lead to an essential re- 
duction of the possible ambiguities for a given string. Some pro- 
posals to this effect can be found in KVALPM 327 and 373. One 
of these strategies follows the usual path: the point is being made 
that the establishment of the correct syntactic structure /or struc- 
tures/ should be carried out in several /at least two/ steps. 
Thus, instead of having rules of the form 
a/b b -~ a 
/i/ 
a a\b -~ b 
we write rules like 
/Z/ x o y -~ z 
which may mean either 
Z/X X -~ Z 
or 
X X\ Z -* Z 
in/z/ we neglect the dominance relation /see below/ hold- 
ing between the two constituents and also their linear order. For 
the latter point notice that /Z/ corresponds to two phrase structure- 
type rules: 
z ~ a+b 
/3/ 
z ~ b + a 
where + stands for concatenation. 
As to the dominance relation it stands to reason to interpret 
z/x in a string z/x x as being the head or governor of the cons- 
truction. Similarly, for a string x x \ z we would say that x 
is governed by x I z. In view of the way the cancellation rules of 
the slant gr&mmar calculus are built up we may say that the 
"more complex" or "longer" category is the governor of a given 
expression. This information is clearly lost in /2/. 
However, as specified in Interim Report No |I, we are not 
forced to make this interpretation. The slant calculus as such 
lends itself to phrase structure grammar too, context-free or 
context- sensitive. 
Instead of /l/ our grammar would now contain a set of 
statements of the form /Z/: 
a I o b I = c 1 
/4/ a z o b z : c z 
,.* 
a o b c n n n 
Each statement in /4/ corresponds to a syntagm type in the given 
grammar. Of course, it is not necessary to have binary rules 
only. One might also have 
a I o a 2 o ... am = cl 
etc 
Now a string of categories in terms of our slant grammar 
calculus can cancel to a "simpler" category if, and only if, one of 
the rule schemata in /4/ holds. But ~t set of such schemata will 
yield many different categorial grammars, varying as to the inter- 
pretation of dependency, 
With the help of /4/ one can determine a sort of "deep" 
structure for a given sequence of morphemes. 
4 
After having determined this deep structure we can proceed 
by finding out more about the actual structure. 
Another approach consists of combining the aforementioned 
multi-level method with probabilistic considerations. For details 
see KVAL Interim Report No 12. 
Since we have interpreted categories in terms of dependencies 
it is readily seen that slant grammar calculus can be considered 
as a dependency-type grammar. The dependency statement for the 
rules /1/ can be rendered as /5/: 
a/b (~ \[b\]). 
a\b ( \[a\] x) 
We fully subscribe to the view that the determination of the 
head or governor of a construction is an important and indispensable 
task. /See, also, Jane Robinson/ Therefore, we think that depend- 
ency grammar in whatever notational conventions it is expressed, 
is superior to phrase structure grammar. Jane Robinson has ar- 
gued that many transformations need a reference to the head of 
construction which in usual phrase structure grammar can only be 
Provided by some ad hoc device. All considerations with respect to 
dependency grammar hold with equal force with respect to slant 
grammar calculus. The latter can be used as a base for transform- 
ations in the same way as dependency grammar can. Following 
Robinson we may make use of the following notational concentions 
that differentiate dependency structures from phrase structures: 
we may use asterisks to mark governing occurrences, parentheses 
to mark boundaries and a special pair of symbols /labels/, de- 
noting a variable depth of nesting. For example, 
(A* B ~() => 2 I 3 
1 2 3 
.In other words, the transformation rule applies to a family of 
trees with the structure 
A 
/ 
e/~,\ X i S`` . 
/" ", /s • /s "~ 
and as a result we get a family with the structure 
A 
I S • / ~. / • 
/Robinson, op. cit. pp. 26-27/. 
It is now of little importance what the symbols A, B and X 
denote, i.e. whether they denote categories characteristic of depend- 
ency grammar or of slant grammar calculus. We can thus conceive 
of a grammar that has as its base /"categorial component"/ a de- 
pendency-tTpe grammar which is context-free and a transformational 
component that operates on structures generated by the categorial 
----. . " .. 
component. In this case we can expect from slant grammar calculus 
a notational advantage at best. 
In fact, this seems to be the case. Thus, in many cases 
one can take advantage of the resemblance between categories of 
the slant grammar calculus and ordinary fractions. Under certain 
conditions we can determine the type of syntagm by assigning to 
each symbol in the sequence of categories a prime number and 
then carry out the cancellation in an arithmetical sense. Then, 
sequences of category symbols in the slant grammar calculus can 
easily be handled without any reference to their meanings. Several 
proposals have been worked out that take advantage of exactly 
this trait /KVAL PM 367/. One definement is proposed in Interim 
Report 5, where relatively prime 2 x Z matrixes with integer ele- 
ments are assigned to the atoms of any categorial grammar, so 
that a string is grammatical if, and only if, the product of the 
assigned matrixes is equal to some "unit". 
On the other hand, we may ask to what extent transforma- 
tional rules are really necessary for our purpose. Could they not 
all be replaced by context-sensitive rules? This problem has been 
examined in some detail in Interim Report if. Another kind of 
categorial symbols, with selectors as well as numerators and de- 
nominators and the cancellation rule /5/ were proposed in order 
to cover the context-sensitive case instead of /l/: 
x lly y-x y 
151 
y Yllx -.y x 
and in "mixed" cases: 
/6/ u z z ullxlyllv V y -~ U X V 
Some formal questions concerning the various grammars in 
the framework of slant grammar calculus have been tackled in 
H. Karlgren: Multi-index Syntactic Calculus. 
So far it is not clear whether rules of type /5/ or /6/ will 
solve all our problems. But it is in this direction that we want 
to work next. 

References 

Kiefer, Ferenc: Bestimmung der syntaktischen Konnexlttlt 
yon Morphemesequenzen, KVAL PM 237 /t965/ 

Kiefer, Ferenc: Ein Algorithrnus for Konnexit~tsbestimmung, 
KVAL PM 248 /i965/ 

Kiefer, Ferenc: Lexical Redundancy Rules in Categorial 
Grammar, KVAL PM 302 /1966/ 

Kiefer, Ferenc: The Question of Ambiguity in Categorial 
Grammar, KVAL PM Z98 /1966/ 

Brodda, Benny: Om sekvenser av matriselement, KVAL 
Interim Report No 5 /i967/ 

Kiefer, Ferenc: The Possibility and/or Necessity of CS- 
rules in Categorial Grammar, KVAL interim Report 
No 6 /1968/ 

Karlgren, Hans: Slant Grammar Calculus, KVAL PM 367 
/1967/ 

Karlgren, Hans: Unique Labelling, KVAL PM 373 /1968/ 

Karlgren, Hans: Categorial Grammar Analysis of Context- 
Sensitive Languages, KVAL Interim Report No il /1968/ 

Karlgren, Hans: LSsande av kategorialuttryck, KVAL 
Interim Report No IZ /1968/ 

Karlgren, Hans: Multi-index Syntactic Calculus, to appear 
in Computational Linguistics, Vol VIII 

Robinson, J. Jane: Dependency Structures and Transforma- 
tional Rules, IBM, Thomas J. Watson Research Center, 
Scientific Report No 3, Yorktown Heights /1968/ 
