PAPER

Geometric and Quantum Methods for
Information Retrieval
Yaoyong Li, Hamish Cunningham
Department of Computer Science, University of Sheffield, Sheffield, UK{yaoyong;hamish}@dcs.shef.ac.uk
Abstract
This paper reviews the recent developments in applying geometric and quantum mechanics methods for information retrieval and natural language processing. It discusses the interesting analogies between components of information retrieval and quantum mechanics. It
then describes some quantum mechanics phenomena found in the conventional data analysis
and in the psychological experiments for word association. It also presents the applications of
the concepts and methods in quantum mechanics such as quantum logic and tensor productto document retrieval and meaning of composite words, respectively. The purpose of thepaper is to give the state of the art on and to draw attention of the IR community to thegeometric and quantum methods and their potential applications in IR and NLP.

1Introduction

Search engine algorithms like PageRank and HITS exploit the link structure of the web,besides the lexical content in web pages (see e.g. [1]). Essentially they mine human decisions
about what pages are good to link to for a particular subject. For many information spaces,
however, this type of data is not available. Vector space retrieval has proven an effective
substitute where the requisite behavioural data to support ranking algorithms is not present
and is in widespread use in systems like Apache Lucene and (at least in early versions)
Inktomi-derived systems. Vector space model retrieval methods are potentially richer andmore sensitive than the key words matching.
In recent works vector space models have been extended to incorporate an apparently
productive analogy with parts of the mathematical apparatus of quantum theory and therepresentation and manipulation of word meaning. Keith Van Rijsbergen in [10] established
the analogies between information retrieval and quantum mechanics by extending the Euclidean space based vector model used in information retrieval (IR) to Hilbert space based
one. Hilbert space is the mathematical foundation of quantum mechanics. It also discussed
the applications of the standard theorems in quantum mechanics to address problems in IRsuch as relevance feedback and ostensive retrieval. Dominic Widdows in [12] explored thegeometry for natural language understanding, in particular the word meaning and context.
Some promising results have been obtained. For example, [7] used the tensor product tostudy the correlation of the user behaviour when accessing one document with the usefulness (or relevancy) of document for the user’s task, namely the implicit feedback for IR. [11]ACM SIGIR Forum

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applied the quantum logic, in particular vector negation to document retrieval and lexical
priming. For the latest developments of using quantum methods for information retrieval andnatural language processing, see the presentations and papers of the International Quantum
Interaction (QI) Symposium1 .
Actually vector space models of semantics are not new; see for example latent semantic
indexing [3]. LSI suffered from high computational cost and the lack of incremental update
capability which limited its applicability, but recent work on Random Indexing promises tosolve these problems [6].
In this paper we review the recent developments in applying the geometric and quantum methods for IR and natural language processing (NLP). The main purpose is to draw
attention of the IR community to the geometric and quantum methods and their potential
applications in IR and NLP. We first explain the information retrieval and quantum mechanics and present the analogies between the two. We then discuss some quantum mechanics
phenomena in information retrieval, and some recent applications of quantum mechanics
methods to IR and NLP. In the last section we give some discussions and future works.

2

Information retrieval

An Information retrieval (IR) system finds relevant information from a collection of
information objects. IR involves three elements:• A collection of information objects, which may be documents, web pages, images,
videos, audios, or RDF triples. The conventional IR system uses documents. Internet
search engines search web pages, mainly based on the text content and/or hyper-links
that web pages contain. Searching over multi-media objects such as images, video andaudio is a new domain. Selecting the relevant RDF triples is a new task for semantic
web applications [4]. Note that the collection may be closed, in the sense that theobjects in it do not change, or dynamic, in the sense that new objects may be added.
• An information need from a user. It may be a simple query containing one or moreterms, or a more complicated information need described by a paragraph of text, orsome typical examples (in this case the user wants to obtain more objects from thecollection which are similar to the given examples).
• The search procedure, which finds objects from the collection which are relevant to theinformation need.
Vector space models (VSMs) have been widely used in IR [1]. A VSM represents
the objects for retrieval and the information query in the same multi-dimensional Euclidean
space (or vector space). The coordinates of the vector space correspond to the content words(or terms) for text retrieval – one coordinate for each particular content word. The set ofcontent words for one application can be pre-defined [11], or can consist of all the words
contained in the documents except the function words and connectives (such as articles andprepositions, which are typically regarded as ‘stopwords’ in IR). Hence the vector space may
be of very high dimensionality if there are many content words. For one document or query,the component of its vector representation for one particular coordinate is the number ofoccurrences of the term corresponding to that coordinate (TF), which may be modified by1

QI-2007 (http://ir.dcs.gla.ac.uk/qi2007/),
QI-2008(http://www-ags.dfki.uni-sb.de/ klusch/qi2009/index.html)

ACM SIGIR Forum23(http://ir.dcs.gla.ac.uk/qi2008/),

and

QI-2009Vol. 42 No. 2 December 2008

the inverse document frequency (IDF) of the term in some way, and the vector may benormalised.
The VSM measures the similarity between object and query by computing the cosine of
the angle or inner product between the object vector and query vector. Then it ranks theobjects according to the similarities to the query and presents the ranked list to the user.
Note that only part of user’s information need is expressed through the query. A user may
give the IR system feedback about which objects are relevant and irrelevant, and the system
may use the feedback to refine the query and make a new search. The VSM can handle therelevancy feedback and query refinement by using Rocchio’s method [1], for example. User
feedback is one form of the interaction between users and IR systems. In another form, theuser changes the information need after obtaining more information by e.g. reading some of
the retrieved documents.
The changes in both the collection and the information need reflect the dynamic natureof IR. An IR system should address those issues.
The VSM uses the Euclidean space, a concept from geometry. Based on the VSM, Keithvan Rijsbergen proposed the geometry of IR in [10], which uses the Hilbert space. In fact
only the inner product of two vectors is important in the VSM. The mathematical space with
inner products is Hilbert space. Therefore the VSM method can be generalised to a Hilbert
space.
Hilbert space is more abstract (or general) than Euclidean space. For example, Hilbert
space may have a finite or infinite dimension and can use real or complex numbers, while a
Euclidean space has only finite dimension and uses real numbers. Hence the Hilbert space
may provide more possibilities for improving IR methods. For example, [10] mentioned touse complex numbers instead of real numbers in the vector representation, and also proposed
to regard a retrieval object as an abstract object (or element) in Hilbert space and a set ofbases in the space as corresponding to one particular point of view to represent the retrieval
objects.
Moreover, as Hilbert space is the mathematical foundation for quantum mechanics (QM),basing IR on Hilbert space creates an analogy between IR and QM and may usefully bringsome concepts and methods from QM into IR.

3

Quantum Mechanics

Quantum mechanics (QM) is the study of mechanical systems whose scales are close toor below the atomic scale, such as molecules, atoms, electrons, protons and other subatomic
particles. QM reveals some important properties of quantum system that are very different
from the normal macroscopic mechanical system that the classical Newtonian mechanics
studies and we experience with in our daily life. We will discuss some of the properties lateron.
QM is based on complex Hilbert space. A quantum system is associated with a complex
separable Hilbert space. The possible states of the system are represented by unit vectors inthe Hilbert space. An observable (or measurable properties) of the quantum system, such asenergy, position, and momentum, corresponds to a self-adjoint linear operator in the Hilbert
space2 . The possible values of the observable are the eigenvalues of the corresponding self2

It is worth noting that QM asserts only the existence of the correspondence between a quantum system anda Hilbert space, but does not provide a general solution for finding out the corresponding Hilbert space for any

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adjoint linear operator3 . Note that a linear operator may have discrete (and even limited)eigenvalues, meaning that the corresponding observable has only discrete values. During ameasurement for one observable, the system collapses from a given initial state to a particulareigenstate corresponding to one eigenvector of the linear operator for the observable. The
probability of collapsing to the eigenstate is the square of the absolute value of the innerproduct between the initial state and the eigenstate. The value of the observable for themeasurement is the eigenvalue associated with the eigenvector (eigenstate).
The Hilbert space of a composite system is the Hilbert space tensor product of the state
spaces associated with the component systems. The state of a composite system may or
may not be the tensor product of the states of the component systems. If it is not, it is anentangled state, as explained below.
QM incorporates the following four types of phenomena that classical physics cannot
account for.
• the quantization (discretization) of certain physical quantities, in the case that theassociated self-adjoint linear operator has discrete eigenvalues.
• wave-particle duality, a concept that all forms of matter and energy exhibit wave-like
properties (such as filling in the space and having a certain wavelength) and particle-likeones (such as having a position in space).
• the uncertainty principle, which states that locating a particle in a small region of
space makes the velocity of the particle uncertain; and conversely, that measuring thevelocity of a particle precisely makes the position uncertain. This principle can be
explained by the wave-particle duality. In fact the uncertainty principle holds for any
two observables if the associated self-adjoint operators A and B are non-commutative,
namely AB 6= BA.
• quantum entanglement, a phenomenon in which the quantum states of two or moreobjects are linked together — even though the individual objects are spatially separated.
This interconnection leads to correlations between observable physical properties ofremote systems.
Some of the QM phenomena have been found in IR and other information processing
tasks. Before we describe those phenomena, we will discuss the analogy between QM andIR.

4Analogy between QM and IR

There are some interesting correspondences between the components of QM and IR, assummarised in Table 1. QM uses a complex Hilbert space to represent a particular quantum
system, while IR needs some information space to represent the objects in one collection for
retrieval. The state vector is the complete and maximal summary of the characteristics of thequantum system at a moment in time, which changes at different time. Objects in collection
contain all the information available for retrieval. We may make an analogy of one object inquantum system. Given a quantum system and some observable, you have to work out the particular Hilbert spacecorresponding to the quantum system and the particular self-adjoint linear operator for the observable.
3
It has been proved mathematically that all the eigenvalues of a self-adjoint linear operator in complex Hilbert
space are real numbers, which guarantees that the values of an observable are all real numbers.

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Table 1: Analogy between QM and IR.
QM
a quantum systemcomplex Hilbert spacestate vector
observablemeasurement
eigenvalues
probability of getting one eigenvalueIRa collection of object for retrieval
information space
objects in collection
query
searchrelevant or not for one object
relevance degree of object to query

Table 2: Comparison of VSM with QM.
QM
complex Hilbert spacestate vector
self-adjoint linear operator for observablemeasurement – interaction between measurement
device and quantum systemeigenvalues of the operator
probability of obtain one eigenvalue

VSM for IR
real Euclidean space
vectorquery vector
search – may involve interaction
between user and system.
relevant or not for one object
relevance degree of object to querycollection as a state vector of the quantum system at one particular time (or the quantum
system at one particular state). Observables are the physical quantities that one can measureon quantum system. Different quantum systems may have different types of observables.
Query represents a question for which a user may obtain answer from one collection. Different
collections have different types of sensible questions to ask. Measurement is a procedure to
determine the value of an observable for one quantum system in one particular state, while
search is a procedure to determine the relevance of an object in collection to a query.
Since the VSM is a common method for IR, Table 2 compares the VSM with QM, basedon the analogy between IR and QM discussed above. We can see that the QM use moregeneral model than the VSM, which may inspire new methods beyond the VSM. For example,
complex Hilbert space is more general and expressive than the Euclidean space. We may
use complex numbers to represent quantities in IR. Hilbert space will enable us to represent
the retrieval objects as abstract objects and to use any base and base transformation. In
particular one may construct some base according to the query and use the base to represent
each object from query’s point of view. One may also represent the query using some subspaceor operator in Hilbert space.

5

QM Phenomena in IR

Since the QM is based on Hilbert space, and the Euclidean space used in the VSM of IR
and natural language processing (NLP) is a specific type of Hilbert space, it is not surprisingthat there are QM phenomena in IR and NLP.

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[9] used a recommender system as example to demonstrate some QM phenomena in data
analysis. As we know, such a system recommends the similar new objects (e.g. films, books)
to the user based on his previous experience.
• In the VSM, the similarity of a new object to the previously seen objects is computed
based on the correlation matrix of the vectors of the previously seen examples. The
correlation matrix is a self-adjoint linear operator in the vector space, corresponding to
an observable in QM.
• [9] obtained a Bell’s inequality of similarities and found some examples violating theinequality. Violation of Bell’s inequality means that one cannot predict the future using
the user’s profile derived from the past in the framework adopted. It can be viewed as
an indicator of a quantum statistical correlation.
• [9] also pointed out that the concept lattice based on the taxonomy of concepts is not
distributive, namely,x ∧ (y ∨ z) 6= (x ∧ y) ∨ (x ∧ z)
For example, in a taxonomy of animals, take x =“bird”, y =“human” and z =“lizard”.
Then both x ∧ y and x ∧ z are empty, so that (x ∧ y) ∨ (x ∧ z) remains empty. On theother hand, y ∨ z =“vertebrates”, because vertebrates are the smallest class including
both humans and lizards. Hence x ∧ (y ∨ z) = “bird”, which is not empty. The nondistributive property does not hold in the classic set based logic but occurs in quantum
logic (which will be discussed in the following section).
[8] discussed several psychological experiments for human’s word association and described some interesting findings from the experiments. In one experiment, giving a cue
word to a human, they measured the likelihood that she recall one related word called target
word. Note that one word may have several associative words. For example, given the word“planet”, a human may recall the related words such as “earth”, “moon” and “universe”.
Figure 1 shows hypothetically a cue word, a target word and two associative words with thepre-existing links. There are three links from cue word to the target word and two associates, one associate-to-associate link, and one associate-to-target link from the Associate 2
to target. The values on the links indicates relative link strengths. Besides the links between
the cue word and target word, the associate-to-associate links are also useful for the recallof target word from cue word. One explanation for why associate-to-associate links benefit
recall uses the spreading activation equations
R(T ) = Sct +n
XSci Sit +

i=1

n X
n
X

Sci Sij Sjt = 0.8 + 0.1 ∗ 0.7 + 0.2 ∗ 0.6 ∗ 0.7 = 0.954

i=1 j=1

which is based on the classic idea that activation spreads through a fixed associative network,
weakening with conceptual distance by multiplying link strengths. Another explanation isbased on the assumption that each link in the associative network contribute additively to therecall strength. It uses the activation at a distance equation to compute the recall likelihood
R(T ) = Sct +

n
X
i=1

Sci +

n
X
i=1

Sit +

n X
n
X

Sij = 0.8 + 0.2 + 0.1 + 0.7 + 0.6 = 2.4

i=1 j=1

It was shown in [8] (in which the target word was the same as the cue word and Sct =0.0) that the psychological experimental results were much more consistent with the second

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Figure 1: A hypothetical cue and target with two associates, and cue-to-target link, cue-toassociate links, associate-to-associate link, and associate-to-target links with link strengths (modified from [8]).
explanation than the first one. The authors explained the additive effects of associate-toassociate and associate-to-target links on the recall level by the entanglement of the target
and its associates. [8] speculated that there were entanglements between the target and its
associates and because of the entanglements, the target word and the associated words acted
as correlated instead of separated entities.

6 Quantum Logic and its application in document
retrieval
Quantum logic is defined on the sub-spaces of Hilbert space (see [15], for example). It is also
called vector logic. Given a Hilbert space H, the logical connectives of the quantum logic
defined in the sub-spaces of H are• The negation of a sub-space A is the orthogonal sub-space of A.
• The conjunction of the two sub-spaces A and B is the sub-space consisting of the vectors
belonging to both A and B.
• The disjunction of the two sub-spaces A and B is the sub-space consisting of the vectors
each of which is a linear combination of two vectors respectively from A and B.
In comparison with classic set logic (namely considering one sub-space as a sub-set of thevector set H), the conjunction is the same but the negation and disjunction are different in
the two logics. In set logic the negation of one sub-set contains all the elements not being inthe sub-set, while in quantum logic the negation of one sub-space contains only the vectorsthat are orthogonal to the sub-space. The conjunction of the two sub-sets in set logic isthe union of the elements in each of the two sub-sets, while the conjunction of the two subspaces in quantum logic contains not only all the vectors of the two sub-spaces but also thosevectors between the two sub-spaces. In other word, the quantum logic utilizes the structureof Hilbert space, whereas the set logic just uses the set relationships among the sub-sets ofHilbert space.
Quantum logic can be used in the VSM based methods. VSM has been widely used in
IR and NLP applications. [11] applied the orthogonal negation in vector space for modelling
word-meaning and document retrieval and compared the results with those using set logic.
In the following we discuss the document retrial experiments presented in [11].

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For document retrieval the experiments were done on the British National Corpus, onenews article corpus, and one medical document corpus. To obtain vector representation for
document and query, 1000 content words were selected. Each document was represented as a
1000-dimensional vector — each coordinate corresponded to one content word and the tf-idf
weighting scheme was used for the value of each coordinate. A query had the form a NOT b,
where a and b are two terms. Each term was also represented as a 1000-dimensional vector
by counting the occurrences of the selected content words in a 15 words context window
centred around the term in document and using the tf-idf weighting scheme. Each vector
was normalised. Suppose that the terms a and b have the vectors va and vb respectively,
according to the orthogonal negation, the query a NOT b has the vector representation
vq = va − < va , vb > vb
Then the document retrieval was performed by computing the similarity between the query
vector and document vectors and ranking documents according to the similarity.
The author also implemented the document retrieval using the negation of set logic. It
first retrieved the documents by using the term vector va and then removing those documents
containing the unwanted term b.
Experimental results showed that, in comparison to the set logic negation, the vector
negation method retrieved the documents containing a much lower number of negative synonyms and negative neighbours but, unfortunately, also a lower number of positive terms.
For more detailed discussions about the experimental results, see [11].

7 Tensor product and its application in combination of meaning of individual words
QM represents composite quantum system using the tensor product of the Hilbert spaces ofindividual quantum systems. The tensor product of two objects consists of the combinationsof any two components of the two objects, respectively. For example, given two matrices A
and B
"
#
"
#
a11 a12b11 b12A=B=
a21 a22b21 b22the tensor product of A and B is


"

A⊗B=

a11 B a12 B
a21 B a22 B#

a11 b11
 a b

=  11 21
 a21 b11
a21 b21a11 b12a11 b22a21 b12a21 b22

a12 b11a12 b21a22 b11a22 b21

a12 b12a12 b22 
a22 b12 
a22 b22

Similarly, the tensor product of two vectors is a matrix (or a long vector).
In the VSM methods for IR and NLP application, we often need the composite of two
or more elements. [2] investigated the composite meaning of a sentence using the semantic
representations of individual words. The method used was the tensor product of the semantic
vectors of individual words. For example, given a sentence “John likes Mary”, and the vector
representation of the semantics of the three words, vJohn , vlikes , and vM ary , the sentence wasrepresented as tensor productvJohn ⊗sv vlikes ⊗vo vM ary

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Table 3: Verb noun similarities for several word pairs using three different vector composition
methods.
First Pair
Second Pair Vector Sum Direct Sum Tensor Productearn money pay wages
0.66
0.41
0.16
earn money wages pay
0.66
0.51
0.26
eat tomato
0.73
0.65
0.29
eat apple
eat apple
throw apple
0.57
0.52
0.04
where ⊗sv and ⊗vo represent tensor products for subject and verb and for verb and object,
respectively. Using the tensor product operations on semantic vectors, one sentence is represented as a matrix. Then we may compute the similarity of two sentences by using the innerproduct of the two matrices.
[13] used the vector composition to model semantic composition. In one experiment theauthor computed the similarity between verb-noun pairs using vector composition methods.
It is expected that eating a tomato is relatively similar to eating an apple, whereas throwing
an apple is quite different. This experiment first represents each word as a semantic vector
using the 1000 content-bearing words, as described in Section 6. Then it computed a vector
for a pair of words using three different methods. Given the two vector a and b with the same
dimension n, the vector addition operation on them resulted in a vector of the dimension n,
a + b. The direct sum of the two vectors was the concatenation of the two vectors, a vector ofthe dimension 2n. The tensor product of the two vectors was a vector of dimension n ∗ n. As
one pair of words was represented by a vector, the similarity of two pairs can be computed
as the inner product of two corresponding vectors.
Table 3 shows some results of the comparison of the similarities of noun verb pairs. We
can see that the tensor product numbers gave the best discrimination.

8

Conclusion and future work

The geometry in Hilbert space provides a general framework for IR. Using the framework,
IR can be studied from a broader view. There are many similarities between IR and QM.
They both are based on Hilbert space and involve the interaction between system and user (orobserver). From physics’ point of view, QM studies the phenomena at sub-atomic level, which
are very different from those physical phenomena at a human perception level. However,
QM phenomena, such as entanglement, the uncertainty principle and state collapse, exist
in information-rich areas such as information processing, biology, geosciences and economics[14]. The concepts and methods in QM may inspire new concepts and methods in thoseareas. For example, as discussed earlier, quantum logic and the tensor product operation
have been used in IR and NLP and obtained promising results. [5] proposed a theoreticalframework for representing documents, inspired by physical measurement on quantum state,
and their potential uses in clustering documents.
On the other hand, the quantum system is simpler than IR and natural language understanding because it involves a small number of types of elements and a few operations.
In order to model the complicated phenomena in IR and natural language, one may haveto employ more complicated mathematical tools. Differential geometry studies the objects
with a more complicated structure than Hilbert space. In particular, it does not presuppose

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the existence of a global set of coordinates or attributes, is not constrained to being flat or
linear, and pays much attention to the local context. Hence it may be more suitable formodelling the meaning of text [12]. For example, instead of specifying some attributes as
global coordinates to represent all queries and retrieval objects, the information space for IR(or semantic space for NLP) may be segmented into different areas in a hierarchical fashion
to represent different contexts or semantic types in natural language, and each context area
is characterised by the attributes and structure relevant to the context.
Acknowledgements: We would like to thank our colleague Adam Funk for help improving the English of the manuscript. This work was supported by the EU-founded project
LarKC, http://www.larkc.eu/.

References[1] S. Chakrabarti. Mining the Web – Discovering Knowledge from Hypertext Data. Morgan
Kaufmann Publishers, 2003.
[2] S. Clark, B. Coecke, and M. Sadrzadeh. A compositional distributional model of meaning. In Proceedings of the Second Quantum Interaction Symposium (QI-2008). College
Publications, 2008.
[3] S. Deerwester, S. Dumais, G. Furnas, T. Landauer, and R. Harshman. Indexing bylatent semantic analysis. Journal of the American Society for Information Science,
41:391–407, 1990.
[4] D. Fensel, F. van Harmelen, B. Andersson, P. Brennan, H. Cunningham, E. D. Valle,
F. Fischer, Z. Huang, A. Kiryakov, T. K. Lee, L. School, V. Tresp, S. Wesner, M. Witbrock, and N. Zhong. Towards larkc: a platform for web-scale reasoning. In Proceedingsof the IEEE International Conference on Semantic Computing (ICSC 2008), Santa
Clara, CA, USA, 2008. IEEE Computer Society Press.
[5] A. F. Huertas-Rosero, L. A. Azzopardi, and K. van Rijsbergen. Characterising througherasing. In Proceedings of the Second Quantum Interaction Symposium (QI-2008). College Publications, 2008.
[6] J. Karlgren and M. Sahlgren. From words to understanding. In Y. Uesaka, P. Kanerva,and H. Asoh, editors, Foundations of Real-World Intelligence, pages 294–308. Stanford:
CSLI Publications, 2001.
[7] M. Melucci. Towards modeling implicit feedback with quantum entanglement. In Proceedings of the Second Quantum Interaction Symposium (QI-2008). College Publications, 2008.
[8] D. Nelson and C. McEvoy. Entangled Associative Structures and Context. In Proceedingsof the AAAI Spring Symposium on Quantum Interaction. AAAI Press, 2007.
[9] D. Pavlovic. On quantum statistics in data analysis. In Proceedings of the Second
Quantum Interaction Symposium (QI-2008). College Publications, 2008.
[10] K. van Rijsbergen. The Geometry of Information Retrieval. Cambridge, 2004.
[11] D. Widdows. Orthogonal negation in vector spaces for modelling word-meanings anddocument retrieval. In The 41st Annual Meeting of the Association for Computational
Linguistics, pages 136–143, 2003.

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[12] D. Widdows. Geometry and Meaning. CSLI Publications, 2004.
[13] D. Widdows. Semantic vector products: Some initial investigations. In Proceedings of
the Second Quantum Interaction Symposium (QI-2008). College Publications, 2008.
[14] D. Widdows and P. Bruza. Quantum information dynamics and open world science. In
Proceedings of the First Quantum Interaction Symposium (QI-2007). AAAI Press, 2007.
[15] D. Widdows and S. Peters. Word vectors and quantum logic: Experiments with negationand disjunction. In Eighth Mathematics of Language Conference, pages 141–154, 2003.

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