We claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by Gardenfors [23] for defending the need of a conceptual, intermediate, representation level between the symbolic and the sub-symbolic one. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and reasoning in Cognitive Architectures

1. Conceptual Spaces for Cognitive Architectures:
A Lingua Franca for Different Levels of
Representation
Antonio Lieto* - Antonio Chella - Marcello Frixione
*University of Turin, Dept. of Computer Science, Italy
*ICAR-CNR, Palermo, Italy
18th July 2016, New York City, BICA Conference 2016

2. Representations in CAs
There are different representational assumptions available in
current Cognitive Architectures (CAs)
- Ex: Fully Connectionist Architectures: LEABRA (O’Reilly and
Munakata, 2000)
- Ex: Hybrid Architectures: ACT-R (Anderson et al. 2004), CLARION
(Sun, 2006)
- Ex. Fully Symbolic Architectures: SOAR (Laird 2012)
- Ex. Architectures integrating Diagrammatic Representations (e.g.
bi-SOAR (Kurup and Chandrasekaran, 2008).

3. Problem
no one of these representation can account for all aspects of cognition.
- Symbolic representations —-> LOGIC-ORIENTED, COMPOSITIONALITY,
an irrevocable trait of human cognition (Fodor and Pylyshyn, 88).
- Sub-symbolic representations (including deep nets) —-> LEARNING,
PERCEPTION, CATEGORIZATION.
- Diagrammatic representations —> VISUAL IMAGERY, SPATIAL
REASONING. Many types of representation proposed, which share some
characteristic with pictures or with diagrams and analog
representations.
We need, in computational systems, different levels of representation to
cover the full aspect of cognitive phenomena.

4. Proposal
We present, by extending the arguments proposed by
Gärdenfors (Gärdenfors, 1997), some advantages that CS can
offer w.r.t. symbolic, sub-symbolic and diagrammatic/analogical
representations.
Conceptual Spaces (Gärdenfors, 2000) as a Lingua Franca for
different levels of representations (all of them available in most
CAs).

5. Proposal
Diagrammatical/Analogical
Representations
typicality
opacity
unified framework

6. Conceptual Spaces (CS)
Conceptual Spaces (Gärdenfors, 2000), are geometrical
representational framework where the information is
organized by quality dimensions sorted into domains.
The chief idea is that knowledge representation can
benefit from the geometrical structure of conceptual
spaces: instances are represented as points in a
space, and their similarity can be calculated in the
terms of their distance according to some suitable
distance measure.

7. Conceptual Spaces - Concepts
Concepts corresponds to regions and regions with
different characteristics correspond to different type of
concepts.
Concepts are represented as sets of convex regions
spanning one or more domains. Each domain is made up
of a set of integral quality dimensions.

8. Domains and Quality Dimensions
Each quality dimension is endowed with a particular
geometrical structure.
Ex: dimensions of COLOR
Hue- the particular shade of colour
Geometric structure: circle
Value: polar coordinate
Chromaticity- the saturation of the colour; from grey to higher intensities
Geometric structure: segment of reals
Value: real number
Brightness: black to white
Geometric structure: reals in [0,1]
Value: real number

9. Ex. CS for “Color”
Intensity
Hue
Brightness
Green
Red
Yellow
Blue

10. Prototypes and Operations
The convexity of conceptual regions allows one to
describe points in the regions as having degrees of
centrality, which aligns this representational
framework with prototype theory (Rosch, ’75).

11. CS Advantages
The different proposals that have been advanced can be grouped in three main classes: a) fuzzy approaches, b) probabilistic and Bayesan approaches, c) approaches based on
non-monotonic formalisms.
W.r.t. Symbolic Representations (SR) => allow to deal with the
problem of compositionality with common-sense concepts.
W.r.t. Sub-symbolic Representations => alleviate the opacity
problem in neural networks (this problems explodes with
deep nets). An interpretation on neural nets in terms of
Conceptual Spaces can offer a more abstract and
transparent view of the underlying neural representations
and processes (compliance the Semantic Pointer Perspective
in SPAUN, Eliasmith 2012).
W.r.t. Diagrammatical/Analogical Representations =>
Conceptual Spaces can offer an unified framework for this
different families of analogical and diagrammatical
representations.

12. Compositionality and Typicality (in SR)
(1) polka_dot_zebra(Pina) = .97
(2) zebra(Pina) = .2
∀x (polka_dot_zebra(x) ↔ zebra(x) ∧ polka_dot_thing(x))
the problem is that if we adopt the simplest and more widespread
form of fuzzy logic, the value of a conjunction is calculated as the
minimum of the values of its conjuncts.
This makes it impossible that at the same time the value of
zebra(Pina) is .2 and that of polka_dot_zebra(Pina) is .97.

13. Compositionality and Typicality (in CS)
According to the conceptual spaces approach, Pina should presumably turn
out to be very close to the center of polka dot zebra (i.e. to the
intersection between zebra and polka dot thing).
In other words, she should turn out to be a very typical polka dot zebra,
despite being very eccentric on both the concepts zebra and polka dot
thing; that is to say, she is an atypical zebra and an atypical polka dot thing.
This representation better captures our intuitions about typicality.

14. CS and Sub-symbolic Representations
The opacity of this class of representations is difficult to accept in CAs aiming at
providing transparent models of human cognition and that, as such, should be
able not only to predict the behavior of a cognitive artificial agent but also to
explain it.
CS offer a more transparent interpretation of underlying neural networks.
Ex. the operation of each layer may be described as a functional geometric space
where the dimensions are related to the transfer functions of the units of the
layer itself. In this interpretation, the connection weights between layers may be
described in terms of transformation matrices from one space to another.
Different works showing: i) how these transformation operations can be done
(also with convolutional neural networks, Eliasmith et al., 2015) and ii) how it
is possible to interpret Radial Basis Function networks in terms of CS (Balkenius,
1999).

15. More about Analog/Diagrammatic
Representations
A plethora of different kinds of diagrammatic representations (e.g. Mental Models
Johnson-Laird 2006).
Ex. The relation “to be on the right of” is usually transitive:
if A is on the right of B and B is on the right of C then A is on the right of C.
But in a round table situation it can happen that C is on the right of B, B is on the
right of A but A is on the left of C.
Complex to model in
symbolic terms.
Interpretable in terms
of CS.

16. CS for Unifying Analog and Diagrammatical
Representations
Conceptual spaces are useful also in representing non-specifically spatial
domains phenomena.
A typical problem of both symbolic and neural representations regards
the ability to track the identity of individual entities over time.
Conceptual Spaces suggest a way to face the problem: in a dynamic
perspective, objects can be rather seen as trajectories in a suitable
Conceptual Space indexed by time.
As the properties of an object are modified, the point, representing it in
the Conceptual Space, moves according to a certain trajectory (Chella,
Coradeschi, Frixione, Saffiotti, 2004).
Also in this case, crucial aspects of diagrammatic representations find a
more general and unifying interpretation in Conceptual Spaces.

17. CS for Unifying Analog and Diagrammatical
Representations/2
A plethora of different kinds of diagrammatic representations
has been proposed without the development of a unifying
theoretical framework.
Conceptual Spaces, thanks to their geometrical nature, allow
the representation of this sort of information and offer, at
the same time a general, well understood and theoretically
grounded framework that could enable to encompass most of
the existing diagrammatic representations.

18. Upshots
We have proposed Conceptual Spaces as a sort of lingua franca
allowing to unify and integrate on a common ground the symbolic, sub-
symbolic and diagrammatic approaches and to overcome some well
known problems specific to such representations.
By extending some arguments proposed by Gärdenfors we have shown
how Conceptual Spaces allow dealing with conceptual typicality
effects, which is a classic problematic aspect for symbolic and logic-
oriented symbolic approaches.
Moreover, Conceptual Spaces enable a more transparent interpretation
of underlying neural network representations and may constitute a
sort of blueprint for the design of such networks.
Finally, we have argued that Conceptual Spaces can offer a unifying
framework for interpreting many kinds of diagrammatic and analogical
representation.

19. Conceptual Spaces for Cognitive Architectures:
A Lingua Franca for Different Levels of
Representation
Antonio Lieto* - Antonio Chella - Marcello Frixione
*University of Turin, Dept. of Computer Science, Italy
*ICAR-CNR, Palermo, Italy
18th July 2016, New York City, BICA Conference 2016