Eckhard Bick - VISL

number of rules Rn, but the number of context conditions in the grammar, Cn. While
obviously slowing down the parser, adding more absolute contexts does not change the
linear complexity characteristic as such, a good algorithm that avoids checking contexts
twice for different rules, may even make R grow slower than Rn. Unbounded context
conditions, however, force the parser to look, if necessary, at all words and their
readings in its half of the sentence. Processing time will therefore grow binomially
((n*a) 2 ) with sentence length for that proportion G% of contexts that is unbounded.
(3b) time ~ (n * a * C) * (n * a * G)
where
C = context number constant, depending on, but less than proportional to the
number of contexts Cn in the grammar
G = globality constant, depending on, but less than proportional to the proportion
of unbounded contexts, G%, in the grammar
Finally, processing time is also proportional to the proportion RM of REMOVE rules,
since REMOVE rules have to look at all readings, while SELECT rules, when hitting
the right reading (on average by trying half of them), discard all others automatically.
Therefore 120 , the variable a has to be replaced by a*(RM+1)/2 in the first parenthesis of
equation (3b). Likewise, a in the second parenthesis is influenced by the proportion SC
of safe context conditions (NOT and C) in unbounded contexts.
(3c) time ~ n * a * (RM+1)/2 * C * (n * a * (SC+1)/2 * G)
where:
RM = proportion of REMOVE rules
SC = proportion of safe unbounded context conditions
Binomial complexity growth is tolerable, and compares favourably with the exponential
complexity growth 121 seen when a parser has to look at all analysis paths for a sentence
parse (a n *C).
Having discussed a in the chapter on ambiguity, and G as well asRM earlier in
this chapter, I will now try to shed some light on rule complexity (C) and context
certainty (SC).
120
With SE for the SELECT rule proportion, the formula would be a*RM + a*SE/2, with SE =1-RM we get a*RM + a*(1-
RM)/2, which can be transformed into a*(RM+1)/2.
121
In a probabilistic HMM PoS tagger this problem can be solved by not "remembering" all paths, but only the highest
probability path when progressing from left to right through the sentence. Complexity will then grow in a linear way (~ n * a
* N), with N being the constant reflecting the size of the n-gram window. A probabilistic syntactic parser, evaluating whole
sentence paths, will, of course, have to deal with the above mentioned exponentiality problem.
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