Towards a Geometrical Cognitive Framework
Ingo Frommholz1, Keith van Rijsbergen1, Fabio Crestani2, and Mounia
Lalmas1
1 Department of Computing Science, University of Glasgow, Scotland
2 Faculty of Informatics, University of Lugano, Switzerland
Abstract. Ingwersen’s cognitive framework is regarded as the begin-
ning of a turn which eventually should bring together classical system-
oriented and user-oriented IR communities. One of the consequences of
this framework is the polyrepresentation principle. The Logical Uncer-
tainty Principle (LUP) is regarded as a compatible model with the cogni-
tive framework. Recently it was shown how LUP can be expressed using
the mathematics of Hilbert spaces. This formalism, which is applied in
quantum mechanics, harmonises geometry, probability theory and logics.
Apart from being a way to express LUP, a further potential arises from
a quantum perspective of IR. We present an interactive framework as an
example of a quantum-inspired approach which also supports polyrepre-
sentation.
1 Cognitive Framework in IR
One of the main assumptions behind Ingwersen’s cognitive framework for IR
interaction, as introduced in [3], is that processing takes place on a symbolic or
sign level, whereas communication between humans may in addition take place
on a cognitive level. This inevitably leads to a cognitive “free fall” as during the
translation of a message into signs, any of its presuppositions, meaning and in-
tentionality is constantly lost. Conversely, a human’s interpretation of a message
may restore the meaning and intention at the cognitive level, but there is increase
in uncertainty which is an inherent feature of any communication process and
hence also IR.
To tackle this problem, the principle of polyrepresentation exploits different
cognitive and functional representations within the information and the cogni-
tive space. In an information space, different actors (e.g., author, indexer, user)
with different tasks and goals in mind (e.g., tagging, indexing, commenting,
reviewing) provide various document representations. In a cognitive space, dif-
ferent representations comprise the user’s information need, the problem state,
the current cognitive state and the work task. Information needs are often (but
not always) unstable and may be ill-defined and only vaguely formulated.
Polyrepresentation makes use of different representations in different spaces,
which are all a result of different interpretations by the actors involved. As a
consequence, the polyrepresentation hypothesises, that the more representations
point to a set of documents, the higher the probability that these documents are

relevant. Thus the basic idea of polyrepresentation is to facilitate a multitude
of cognitively and functionally different representations, provided by different
actors. This is done by determining the so-called cognitive overlap, which is the
intersection of various sets of documents which are relevant with respect to their
single representations.
2 Geometry, Logics and Probabilities in IR
Fig. 1: Probability theory,
logics and geometry
Ingwersen’s cognitive framework is regarded as
the beginning of a turn which eventually should
bring together the classical system-oriented and
the user-oriented IR communities in order to de-
velop an integrated view of information seeking
and retrieval [4]. Such a view needs a strong
methodological framework or a new “language” go-
ing beyond classical prevalent models. A step to-
wards this goal is the formulation of the Logical
Uncertainty Principle (LUP) [7]. The LUP starts
from the consideration that logic by itself cannot
fully model IR. In determining the relevance of a
document d to a query q the success or failure of
the logical implication d ! q is not enough. It is
necessary to take into account the uncertainty in-
herent in such an implication. To cope with uncertainty a logic for probabilistic
inference was introduced. If d ! q is uncertain, then we can measure its degree
of uncertainty by Pr(d ! q). In [7] van Rijsbergen proposed the use of a non-
classical conditional logic for IR. This would enable the evaluation of Pr(d ! q)
using the LUP, that was defined as follows:
“Given any two sentences x and y; a measure of the uncertainty of
y ! x related to a given data set is determined by the minimal extent to
which we have to add information to the data set, to establish the truth
of y ! x.”
This principle was the one of first attempts to make an explicit connection be-
tween non-classical logics and IR modelling. However, when proposing the above
principle, van Rijsbergen was not specific about which logic and which uncer-
tainty theory to use. As a consequence, various logics and uncertainty theories
have been proposed and investigated. The choice of the appropriate logic and
uncertainty mechanisms has been a main research theme in logical IR mod-
elling leading to a number of different approaches over the years (for a detailed
description of some of these models see [1]).
It turns out that the LUP is fully compatible with the cognitive framework.
In fact, Ingwersen described uncertainty as “which and how much context needs
to be added to retrieve those semantic values which provide the information
searched for” [3, p. 36].

A broader view is provided by looking at the mathematics of Hilbert spaces
and linear operators, as used in quantum theory (QT). This potentially com-
bines different directions underlying the most important models in IR, namely
geometry, logics and probability theory [8], as indicated in Figure 1. On the
geometry side, there are vectors, subspaces and projectors. While these can be
used to model many of the well-known retrieval approaches like the vector space
model, Gleason’s Theorem builds a bridge between geometry and a generalisa-
tion of probability theory. By applying this theorem, so-called density operators
are able to induce a probability distribution on subspaces [8, ch. 6] which gives, if
certain conditions (like vectors being normalised) are met, geometric approaches
to probabilistic interpretations. Moreover, a conditional logic can be established
based on the notion of subspaces [8, ch. 5], which together with Gleason’s The-
orem gives rise to a geometric interpretation of the LUP, seamlessly combining
geometry, probability theory and logics. For example, if d and q can be repre-
sented as subspaces, one way of estimating Pr(d ! q) is using the trace function
to compute an inner product between q and d [8, p. 96].
3 Quantum-Inspired Interactive Polyrepresentation
Framework
As an example of a quantum-inspired approach which also facilitates polyrepre-
sentation, we present an interactive geometrical framework called IQIR (Inter-
active Quantum Information Retrieval) [6]. Its aim is to address certain aspects
of the cognitive framework, applying the mathematical formalism known from
quantum mechanics, so that it supports interactive retrieval. In terms of the
Venn diagram in Fig. 1, the IQIR framework lies in the intersection of probabil-
ity theory and geometry.
The core assumption of the IQIR framework is that there exists a Hilbert
space H of so-called pure information needs (pure INs). This resembles the idea
from quantum mechanics, where physical systems are represented as vectors in
a Hilbert space. Each such system can be in a certain state, which is indicated
by a state vector ', a unit vector in the corresponding Hilbert space. In the IR
case, ' represents the system’s view of the user’s information need. It can be
shown that a state vector induces a probability distribution on the subspaces
of H [8]; each such subspace R represents an event, for instance the event that
we measure a certain physical property (like the location of a particle) or, more
IR-related, the event that a document d is relevant. Given a state ', we can now
compute Pr (Rjd; '), the probability that d is relevant given the current system
state, as the square of the length of the orthogonal projection of ' onto the
subspace R.
Usually a system cannot know precisely the information need, thus there will
always be uncertainty about the user’s intention (as a result of the cognitive free
fall and for other reasons). In quantum mechanics, we face a similar problem
as there is often uncertainty about the state a physical system is in, so it is
assumed that the system is in a certain state with a given probability. This

ϕ5
ϕ
3
ϕ
4
ϕ
2
ϕ
1
R
(a) Before observation
ϕ
5
ϕ
3
ϕ
2
ϕ
1
R
(b) After observation
Fig. 2: Example mixed state. The dashed arrow shows the projection of '1 onto R.
means that we deal with a set of state vectors and associated probabilities (the
so-called ensemble S of states); the system is said to be in a mixed state if
there is more than one possible state vector. An example can be seen in Fig. 2a,
where we find five possible state vectors, each of them has a probability assigned
that the system is in the respective state (omitted here). Transferred to an IR
scenario, the system would assume the user to have one of the 5 different pure
information needs each represented by a corresponding state vector. To compute
the probability of relevance Pr (Rjd; S), here a generalisation of the law of total
probability is applied using the squared length of the orthogonal projection of
each state vector onto R, which in this example represents document relevance
as a 2-dimensional subspace.
As discussed in [3, 4], information needs are usually unstable. To reflect this
situation, in the IQIR framework, we can borrow another notion from quantum
mechanics, the measurement postulate, which says that each observation of an
event usually implies a state change. In interactive IR, such events can, for
instance, be the submission of query by the user or a relevance judgement. Given
that we can describe such an event as a subspace in our Hilbert space, the
realisation of the event would cause the ensemble of state vectors to be projected
onto the corresponding subspace and to be renormalised. Suppose in the example
in Fig. 2a the event that a user judges a document as relevant is represented
by the subspace R. Fig. 2b describes the situation after this event was observed
by the system – the state vectors are now all within the 2-dimensional plane
determined by R, and one vector, '
4
even disappeared as it was orthogonal
to R. Two effects result from this kind of dynamics. First, the system gains
more certainty about the user’s IN as now all state vectors are bound to a lower-
dimensional plane, and some state vectors even disappeared. Second, slight shifts

in information needs are supported as well, for instance if the vectors in Fig. 2b
are later projected onto another 2-dimensional subspace which is similar to R:
The IN space discussed so far is very abstract. There are several ways to
create a concrete instantiation of the IN space. For example, in [5] it is shown
how a topical IN space can be constructed in a standard term space well-known
in IR, where queries are represented as state vectors and documents as sub-
spaces. But we are not bound to topical IN spaces. In fact, to satisfy different
aspects of information needs, one may need to refer to non-topical information
(like user-given ratings). In [2] it is shown how different topical and non-topical
representations can be expressed as IN spaces in the IQIR framework with the
aim to support polyrepresentation of documents. These component spaces can
be combined into a composite space by means of a tensor product to create
the cognitive overlap. This composite space can be regarded as an IN space in
its own right as discussed above, with all its properties regarding user interac-
tion. It further allows for the system state to become non-separable; the system
is said to be in an entangled state then. Entanglement and composite spaces
are further important mechanisms borrowed from quantum mechanics. In our
case, entanglement can be used to express possible interdependencies between
different representations in a polyrepresentation scenario.
4 Acknowledgements
We thank Benjamin Piwowarski and Birger Larsen for their valuable comments.
References
1. Fabio Crestani, Mounia Lalmas, and Keith van Rijsbergen. Information Retrieval:
Uncertainty and Logics. Kluwer Academic Publisher, Norwell, MA, USA, 1998.
2. Ingo Frommholz, Birger Larsen, Benjamin Piwowarski, Mounia Lalmas, Peter In-
gwersen, and C J van Rijsbergen. Supporting Polyrepresentation in a Quantum-
inspired Geometrical Retrieval Framework. In Proc. IIiX 2010, New Brunswick,
2010.
3. Peter Ingwersen. Cognitive perspectives of information retrieval interaction: Ele-
ments of a cognitive IR theory. The Journal of Documentation, 52:3–50, 1996.
4. Peter Ingwersen and Kalvero Järvelin. The turn: integration of information seeking
and retrieval in context. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2005.
5. Benjamin Piwowarski, Ingo Frommholz, Mounia Lalmas, and Keith van Rijsber-
gen. Exploring a Multidimensional Representation of Documents and Queries. In
Proceedings of the 9th RIAO Conference (RIAO 2010), Paris, France, 2010.
6. Benjamin Piwowarski and Mounia Lalmas. A Quantum-based Model for Interactive
Information Retrieval. In Proceeedings of the 2nd International Conference on the
Theory of Information Retrieval (ICTIR 2009), pages 224–231, Cambridge, UK,
2009.
7. C J van Rijsbergen. A Non-Classical Logic for Information Retrieval. The Computer
Journal, 29:481–485, 1986.
8. C J van Rijsbergen. The Geometry of Information Retrieval. Cambridge University
Press, New York, NY, USA, 2004.