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

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).

Problemno 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.

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).

Proposal

Diagrammatical/Analogical Representations

typicality

opacity

unified framework

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.

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.

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

Ex. CS for “Color”

Intensity

Hue

Brightness

Green

Red

Yellow

Blue

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).

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.

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.

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.

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).

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.

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.

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.

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.

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