Ambiguity and sharing in Natural Language Parsing
Ambiguity and sharing in general
Given the generality of your question, I am trying to match that
generality.
The concept of ambiguity arises as soon a you consider a mapping or function f: A -> B which is not injective.
An injective function (also called one-to-one function) is one such
that when a≠a' then f(a) ≠ f(a'). Given a function f, you are often
interested in reversing it: given an element b of the codomain B of f,
you want to know what element a of the domain A is such that f(a)=b.
Note that there may be none if the function is not surjective
(i.e. onto).
When the function is not injective, there may be several values a in A
such that f(a)=b. In other words, if you use values in B to actually
represent values in A through the mapping f, you have an ambiguous
representation b that may not determine the value a uniquely.
From this you realize that the concept of ambiguity is so general that
it is unlikely that there is a unified body of knowledge about it,
even when limiting this to computer science and programming.
However, if you wish to consider reversing a function creating such
ambiguities, for example to compute the set f'(b)={a∈A | f(a)=b}, or
the best element(s) in that set according to some optimality criterion,
there are inded some techniques that may help you in situations where
the problem can be decomposed into subproblems that often re-occur
with the same arguments. Then, if you memorize the result(s) for the
various combinations of arguments encountered, you never compute twice
the same thing (the subproblem is said to be memo-ized). Note that
ambiguity may exist for subproblems too, so that there may be several
answers for some subproblem instances, or optimal answers among
several others.
This amount to sharing a single copy of a subproblem between all the
situations that require solving it with this set of parameters. The
whole technique is called dynamic programming, and the difficulty is
often to find the right decomposition into subproblems. Dynamic
programming is primarily a way to share the repeated subcomputation for a
solution, so as to reduce complexity. However, if each subcomputation
produces a fragment of a structure that is reused recursively in
larger structures to find an answer which is a structured object (a
graph for exmple), then the sharing of subcomputation step may result in
also sharing a corresponding substructure in all the places where it is
needed. When many answers are to be found (because of ambiguity for
example), these answers can share subparts.
Rather than finding all the answers, dynamic programming can be used
to find those satisfying some optimality criterion. This requires that
an optimal solution of a problem uses optimal solutions of
subproblems.
The case of linguistic processing
Things can be more specific in the case of linguistics and language
processing. For that purpose, you have to identify the domains you are
dealing with, and the kind of functions you use with these domains.
The purpose of language is to exchange information, concepts, ideas
that reside in our brains, with the very approximate assumption that
our brains use the same functions to represent these ideas
linguistically. I must also simplify things considerably (sorry about
it) because this is not exactly the place for a full theory of
language, which would be disputed anyway. And I cannot even consider
all types of syntactic theories.
So linguistic exchange of information, of an idea, from a person P to a person Q
goes as follow:
idea in P ---f--> syntactic tree ---g--> lexical sequence ---h--> sound sequence
|
s
|
V
idea in Q <--f'-- syntactic tree <--g'-- lexical sequence <--h'-- sound sequence
The first line is about sentence generation taking place in person P,
and the second line is about sentence analysis taking place in person
Q. The function s stands for speech transmission, and should be the
identity function. The functions f', g' and h' are supposed to be the
inverse of the functions f,g, and h that compute the successive
representations down to the spoken representation of the idea. But
each of these functions may be non injective (usually is) so that
ambiguities are introduced at each level, making it difficult for Q to
inverse then to retrieve the original meaning from the sound sequence
it receives (I am deliberately using the word sound to avoid getting
into details). The same diagram holds, with some variations in details,
for written communication.
We ignore f and f' since they are concerned with semantics, which may
be less formalized, and for which I do not have competence. Syntax
trees are often defined by grammatical formalisms (here I am skipping
over important refinements such as feature structures, but they can be
taken into account).
Both the function g and the function h are usually not injective, and
thus are sources of ambiguity. There are actually other sources of
ambiguity due to all kind of errors inherent to the speech chain, but
we will ignore it for simplicity, as it does not much change the
nature of problems. The presence of errors, due to sentence generation
or transmission, or to language specification mismatch between the
speaker and the listener, is an extra source of ambiguity since the
listener attempts to correct potential errors without knowing what
they may have been or whether they exist at all.
We assume that the listener does not make mistakes, and that he
attempts to best "decode" the sentence according to his own linguistic
standards and knowledge, including knowledge of error sources and
statistics.
Lexical ambiguity
Given a sound sequence, the listening system has to inverse the effect
of the lexical generation function g with a function g'. A first
probem is that several different words may give the same sound
sequence, which is a first source of ambiguity. The second problem is
that the listening system actually receives the sequence corresponding
to a string of words, and there may be no indication of where words
begin or end. So they may be different ways of cutting the sound
sequence into subsequences corresponding to recognizable words.
This problem may be worsened when noise creates more confusion
between words.
An example is the following holorime verses taken from the web, that
are pronounced more or less similarly:
Ms Stephen, without a first-rate stakeholder sum or deal,
Must, even with outer fur straight, stay colder - some ordeal.
The analysis of the sound sequence can be performed by a finite state
non-deterministic automaton, interpreted in a dynamic programming
mode, which produces a directed acyclic graph where the nodes
correspong the word separations and the edges to recongnized words.
Any longest path through the graph corresponds to a possible way
of analyzing the sound sequence as a sequence of words.
The above example gives the (fairly simple) word lattice (oriented
left to right):
the-*-fun
/ \
Ms -*-- Stephen \ without --*-- a first -*- ...
/ \ / \ /
* * *
\ / \ / \
must --*-- even with -*- outer fur -*- ...
So that the sound sequence could also correspond to the following word
sequences (among several others):
Ms Stephen, with outer first-rate ...
Must, even with outer first-rate ...
This make the lexical analysis ambiguous.
Probabilities may be used to choose a best sequence. But it is also
possible to keep the ambiguity of all possible reading and use it
as is in the next stage of sentence analysis.
Note that the word lattice may be seen as a finite state automaton
that generates or recognizes all the possible lexical readings of the
word sequence
Syntactic ambiguity
Syntactic structure is often based on a context-free grammar
skeleton. The problem of ambiguity of context-free languages is well
known and analyzed. A number of general CF parsers have been devised
to parse ambiguous sentences, and produce a structure (which varies
somewhat) from which all parses can be extracted. Such structure have
come to be known as parse forests, or shared parse forest.
What is known is that the structure can be at worst cubic in the length of
the analyzed sentence, on the condition that the language grammar is
binarized, i.e. with no more than 2 non-terminals in each rule
right-hand-side (or more simply, no more than 2 symbols in each rule
right-hand-side).
Actually, all these general CF parsing algorithms are more or less
sophisticated variations around a simple concept: the intersection of
the language L(A) of a finite state automaton A and the language L(G)
of a CF grammar G. Construction of such an intersection goes back to
the early papers on context-free languages (Bar-Hillel, Perles and
Shamir 1961), and was intended as proof of a closure property. It took
some thirty years to realize that it was also a very general parsing algorithm in a
1995 paper.
This classical cross-product construction yields a a CF grammar for the
intersection of the two languages L(A) and L(G). If you consider a
sentence w to be parsed, represented as a sequence of lexical elements,
it can also be viewed as finite state automaton W that generate only
the sentence w. For example:
this is a finite state automaton
=> (1)------(2)----(3---(4)--------(5)-------(6)-----------((7))
is a finite state automaton W accepting only the sentence
w="this is a finite state automaton". So L(W)={w}.
If the grammar of the language is G, then the intersection
construction gives a grammar G_w for the language
L(G_w)=L(W)∩L(G).
It the sentence w is not in L(G), then L(G_w) is empty, and the
sentence is not recognized. Else L(G_w)={w}. Furthermore, it is then
easily proved that the grammar G_w generates the sentence w with
exactly the same parse-trees (hence the same ambiguity) as the grammar
G, up to a simple renaming of the non-terminals.
The grammar G_w is the (shared) parse forest for w, and the set of
parse trees of w is precisely the set of derivations with this
grammar. So this gives a very simple view organizing the concepts, and
explaining the structure of shared parse forests and general CF parsers.
But there is more to it, because it shows how to generalize to
different grammars and to different structures to be parsed.
Constructive closure of intersection with regular sets by
cross-product constructions is common to a lot of grammatical
formalisms that extend the power of CF grammars somewhat into the
context-sensitive realm. This includes tree-adjoining grammars, and
linear context-free rewriting systems. Hence this is a guideline on
how to build for these more powerful formalisms general parsers that
can handle ambiguity and produce shared parse-forests, wich are simply
specialized grammars of the same type.
The other generalization is that, when there is lexical ambiguity so
that lexical analysis produces many candidate sentences represented
with sharing by a word lattice, this word lattice can be read as a
finite state automaton recognizing all these sentences. Then, the same
intersection construction will eliminate all sentences that are not in
the language (not grammatical), and produce a CF grammar that is a
shared parse forest for all possible parses of all admissible
(grammatical) sentences from the word lattice.
As requested in the question, all possible ambiguous readings are
preserved as long as compatible with available linguistic or utterance
information.
The handling of noise and ill-formed sentences is usually modelled also
with finite state devices, and can thus be addressed by the same
techniques.
There are actually many other issues to be considered. For example,
there are many ways of building the shared forest, with more or less
sharing. The techniques used to precompile pushdown automata to be
used for general context-free parsing my have an effect on the quality
of the sharing. Being too smart is not always very smart.
See also other answers I made on SE on this topic:
