=Paper= {{Paper |id=Vol-1510/paper6 |storemode=property |title=Pattern-Recognition: a Foundational Approach |pdfUrl=https://ceur-ws.org/Vol-1510/paper6.pdf |volume=Vol-1510 |dblpUrl=https://dblp.org/rec/conf/aic/AugelloGOP15 }} ==Pattern-Recognition: a Foundational Approach== https://ceur-ws.org/Vol-1510/paper6.pdf

Pattern-Recognition: a Foundational Approach

Agnese Augello3 , Salvatore Gaglio1,3 , Gianluigi Oliveri2,3 , and Giovanni Pilato3
1
DICGIM - Università di Palermo
Viale delle Scienze, Edificio 6 - 90128, Palermo - ITALY
2
Dipartimento di Scienze Umanistiche - Università di Palermo
Viale delle Scienze, Edificio 12 - 90128, Palermo - ITALY
3
ICAR - Italian National Research Council
Viale delle Scienze - Edificio 11 - 90128 Palermo, Italy
{gaglio,oliveri}@unipa.it
{agnese.augello,giovanni.pilato}@cnr.it

Abstract. This paper aims at giving a contribution to the ongoing at-
tempt to turn the theory of pattern-recognition into a rigorous science.
In this article we address two problems which lie at the foundations of
pattern-recognition theory: (i) What is a pattern? and (ii) How do we
come to know patterns? In so doing much attention will be paid to trac-
ing a non-arbitrary connection between (i) and (ii), a connection which
will be ultimately based on considerations relating to Darwin’s theory of
evolution.

1 Introduction

As is well known, the main aim of pattern-recognition theory is to determine
whether, and to what extent, what we call ‘pattern-recognition’ can be accounted
for in terms of automatic processes. From this it follows that two of its central
problems are how to: (i) describe and explain the way humans, and other biologi-
cal systems, produce/discover and characterize patterns; and how to (ii) develop
automatic systems capable of performing pattern recognition behaviour.
Having stated these important facts, we need to point out that at the foun-
dations of pattern-recognition theory there are two more basic questions which
we can formulate in the following way: (a) what is a pattern? (b) how do we
come to know patterns? And it is clear that, if we intend to develop a science of
pattern recognition able to provide a rigorous way of achieving its main aim, and
of pursuing its central objects of study, it is very important to answer questions
(a) and (b).
After having addressed the problem of providing a definition of the concept
of pattern in §2, a case-study of a particular type of finite geometry is discussed
in §3 in the hope that by so doing we might obtain a rigorous characterization
of the concept of mathematical pattern.
Section 4 is then dedicated to the examination of some of the interesting
lessons that can be learned from the case-study in §3. In particular, one of these
has to do with the characterization of the concept of mathematical pattern in
terms of mathematical structure; and another concerns the possibility of gener-
alizing the view of mathematical patterns as structures to patterns belonging to
fields different from mathematics.
Finally, sections 5 and 6, armed with the notion of pattern developed so far,
bring the paper to a close by addressing question (b) above: how do we come to
know patterns?

2 Searching for a definition
A potentially fruitful approach to the problem ‘What is a pattern?’ is that of
Daniel Dennett. For Dennett, who in his discussion of the concept of pattern is
concerned with issues belonging to the philosophy of mind and action,
[W]e are to understand the pattern to be what Anscombe called the
“order which is there” in the rational coherence of a person’s set of
beliefs, desires, and intentions. [[6], §IV, p. 47.]
However, although taking into account final causes, beliefs and intentions
often can both reveal an order existing among a certain individual’s actions
and explain his behaviour in terms of giving an account not only of how, but
also of why he did what he did, it must be admitted that talking about ‘the
order which is there in the rational coherence of a person’s set of beliefs, desires,
and intentions’ is too vague to shed light on the notion of pattern. This is, in
particular, the case when the accounts of the order which is there . . . etc. are
several, radically differ from one another, and all seem to agree with the facts.
Moreover, since patterns do not occur only within the context of human
actions and beliefs, what happens when we are dealing with patterns displayed
by crystals of snowflakes? Of course, also in the case of crystals of snowflakes
(see Fig. 1)

Fig. 1. A crystal of a snowflake

the patterns they display are related to the order in which the components of the
crystals of snowflakes are to one another. But, whereas in the case of the crystals
of snowflakes, if we use a microscope, we can actually see them, when we turn to
actions the verb ‘seeing’ appears to let us down. For an action, in contrast to the
crystal of a snowflake, is not just a brute physical fact and, therefore, the order
manifested by a sequence of actions ‘which is there in the rational coherence of a
person’s set of beliefs, desires, and intentions’ is not something we can perceive
by simply keeping our eyes wide open, and using instruments of observation.
This is an important point, because, if there is some truth in Dennett’s
definition of pattern, it means that what we might call ‘brute seeing’, that is,
the mere act of representing within visual perceptual space a given input — like
what happens with a photo-camera when we take a picture — cannot provide a
satisfactory account of what happens when we perceive a pattern.
Therefore, if we intend to give an account of perceiving a pattern which is in
accord with Dennett’s definition, we should appeal to a concept of seeing which
is much richer than brute seeing. A good candidate for such a concept of seeing
is the concept that in the Philosophical Investigations Wittgenstein famously
called ‘seeing something as’ or ‘aspect seeing’.4
Notice, for example, that in seeing something as a square the perception of
the square-pattern is not brute, because it presupposes, among other things,
that the observer has a grasp of the concept of square.
However, independently of questions relating to the nature of the ‘order which
is there . . . ’ in different contexts, and any consideration concerning what we must
mean by ‘seeing an aspect’ or ‘perceiving a pattern’, Dennett proposes a very
interesting general test for the existence of patterns. Basing himself on Chaitin’s
definition of randomness:

A series of numbers is random if the smallest algorithm capable of
specifying it to a computer has about the same number of bits of infor-
mation as the series itself. [[2], p. 48.]

Dennett asserts that:

A pattern exists in some data — is real — if there is a description of
the data that is more efficient than the bit map, whether or not anyone
can concoct it. [[6], §II, p. 34.]

Although that offered by Dennett is a very plausible criterion which, in some
cases, reveals the presence of patterns in a data-set, it is not specific to them.
To see this consider the definite description ‘The satellite of the Earth’. Such
a definite description certainly provides an enormous compression of data with
respect to the bit map of a computer visual representation of the Moon. But, it
is a description which uniquely identifies an object not a pattern/structure.
Lastly, the phenomenon of seeing something as a square appears to hint at a
structural feature of perception, where the pattern perceived is that of a square.
In fact if, by zooming in or out on the object we perceive as a square, we change
4
See on this [17], Part II, §XI, pp. 213e –214e .
(within a certain range) the magnitude of the picture of the object, we would
still see the object as a square.
The structural character of the pattern perceived is particularly evident in
the case of the crystal of a snowflake. Indeed, when we observe a crystal of
a snowflake through a microscope or when we look at a photograph or at an
artist’s accurate impression of that very crystal of a snowflake, etc. in spite of
being presented in each single case with a different object — the actual crystal,
the photograph of the crystal, and the artist’s accurate impression of the crystal
— we recognize the presence of the same pattern in all these objects. Of course,
the next question is ‘What is a structural feature of an object?’ or, in more
general terms, ‘What is a structure?’ The latter is, indeed, the problem which is
going to be at the heart of the next section.

3 Mathematical Patterns. A case study

If we are presented with objects a and b (see Figures 2 and 3), it is very difficult
to see what interesting mathematical feature they might have in common, if any,
let alone that they exemplify the same mathematical pattern.

ABCDE FG
BCDE FGA
DEFGABC

Fig. 2. Object a

A
# A
P2 P7A P6
HHA
HHA
"! HA
P3 P4 P5
Fig. 3. Object b

Indeed, whereas object a is a 3 × 7 matrix whose elements are the first seven
letters of the Italian alphabet, object b is a geometrical entity consisting of 7
lines and 7 points. The lines of object b are: the sides of the triangle drawn in
Figure 3, the bisecting segments, and the inscribed circle. On the other hand,
the 7 points are the points of intersection of three lines.
However, the situation radically changes if we introduce the following formal
system T with the appropriate interpretations.
Let a formal system T be given such that the language of T contains a
primitive binary relation ‘x belongs to a set X’ (x ∈ X), and its inverse ‘X
contains an element x’ (X 3 x).
Furthermore, let us assume that D is a set of countably many undefined
elements a1 , a2 , . . .; call ‘m-set’ a subset X of D; and consider the following as
the axioms of T:
Axiom 1 If x and y are distinct elements of D there is at least one m-set
containing x and y;
Axiom 2 If x and y are distinct elements of D there is not more than one m-set
containing x and y;
Axiom 3 Any two m-sets have at least one element of D in common;
Axiom 4 There exists at least one m-set.
Axiom 5 Every m-set contains at least three elements of D;
Axiom 6 All the elements of D do not belong to the same m-set;
Axiom 7 No m-set contains more than three elements of D.5
Now, the language of T contains two different sorts of variables: x, y, . . . and
X, Y, . . . Let us assume that the variables x, y, . . . range over D1 = {A, . . . , G};
and that the variables X, Y, . . . range over D∗1 , where D∗1 is a set whose elements
are the subsets of D1 the elements of which appear in the columns of the matrix
in Figure 1, that is:

D∗1 = {{A, B, D}, {B, C, E}, {C, D, F}, {D, E, G}, {E, F, A}, {F, G, B}, {G, A, C}}.

(The elements of D∗1 are the m1 -sets.)
It turns out that D1 ∪ D∗1 is the domain of the model of T represented in
Figure 2. To see this, using the interpretation suggested above, it is sufficient
to verify that Axioms 1 − 7 are true of the matrix in Figure 2. We call such a
model ‘M1 (T).’
On the other hand, if we change interpretation making: (a) the variables
x, y, . . . range over D2 = {P1 , . . . , P7 }, where P1 , . . . , P7 are the 7 distinct points
indicated in Figure 3; and (b) the variables X, Y, . . . range over D∗2 whose ele-
ments are the m2 -sets, that is, the sets of three Pi points, for 1 6 i 6 7, lying
on the sides, the bisectrices, and the circle inscribed in the triangle represented
in Figure 3: D2∗ = {{P6 , P2 , P4 }, {P2 , P7 , P5 }, {P5 , P4 , P3 },
{P4 , P7 , P1 }, {P3 , P7 , P6 }, {P3 , P2 , P1 }, {P1 , P6 , P5 }}; we have that D2 ∪D∗2 is also
the domain of a model of T, a model represented in Figure 3. We call such a
model ‘M2 (T).’
To show that M2 (T) is a model of T, it is sufficient, using the interpretation
just provided, to check that Axioms 1 − 7 are true of the object represented in
Figure 3.
If we, now, compare M1 (T) with M2 (T), we realize that, among other things:
(1) (D1 ∪ D∗1 ) ∩ (D2 ∪ D∗2 ) = ∅; (2) the elements of D1 ∪ D∗1 are not homogeneous
5
These axioms have been taken, with some minor alterations, from [16], §2.10, p. 30.
with the elements of D2 ∪ D∗2 ; and that (3) M1 (T) and M2 (T), are isomorphic
to each other.
With regard to point (3) above, we notice that if f is the function f : D1 →
D2 such that:
f (A) = P6 ,
f (B) = P2 ,
f (C) = P5 ,
f (D) = P4 ,
f (E) = P7 ,
f (F ) = P3 ,
f (G) = P1 ;
and g is the function g : D1∗ → D2∗ such that:

g(X) = g({xi , xj , xk })
= {f (xi ), f (xj ), f (xk )}

for 1 6 i 6 j 6 k 6 7, then f induces a bi-univocal correspondence between D1
and D2 , whereas g induces a bi-univocal correspondence between the set D1∗ (of
m1 -sets) and the set D2∗ (of m2 -sets).
Now, it is clear that the function ψ, where ψ : D1 D1∗ → D2 D2∗ such
S S
that:

f (x) if λ = x
ψ(λ) =
g(X) if λ = X

shows that M1 (T) and M2 (T) are isomorphic
S to one another. In fact, ψ induces
a bi-univocal correspondence between D1 D1∗ and D2 D2∗ preserving the two
S
(primitive) relations ∈ and 3, that is:

x ∈ X iff ψ(x) ∈ ψ(X)
X 3 x iff ψ(X) 3 ψ(x).

The case relative to the existence of two isomorphic models M1 (T) and
M2 (T) of T brings out very clearly that the pattern described by the axioms
and theorems of T is independent of the nature of the objects present in D1 ∪ D∗1
(the first seven letters of the alphabet plus . . . ), and in D2 ∪D∗2 (the seven distinct
points highlighted in Figure 3 plus . . . ). The pattern described by the axioms
and theorems of T is an abstract mathematical structure realized by/present in
both M1 (T) and M2 (T).
At this point a legitimate problem that might arise is ‘How is the structure
common to M1 (T) and M2 (T) given to us?’ and another is ‘What sort of thing
is this structure?’ Let us address the second question first.
A structure/pattern is an ordered pair the first element of which is the domain
of the structure — in our case D1 ∪ D∗1 or D2 ∪ D∗2 — and whose second element
is a set of relations defined on this domain — in our case the relations are ∈ and
3 — relations the basic properties of which are implicitly defined by the axioms.
With regard to the question concerning the reality of the structure instanti-
ated by M1 (T) and M2 (T), consider that if objects a and b exist and, therefore,
are real then also the structure they realize exists and, therefore, is real.
The answer to the first question is more complicated, because there is no
unique way in which a pattern, even a mathematical one, becomes salient to an
observer. However, it is certainly the case that necessary conditions for seeing a
certain object as the realization of the pattern/mathematical structure we have
been talking about in this paper are: (1) the observer’s acquaintance with object
a or with object b, (2) the observer’s knowledge of T, and (3) the observer’s
knowledge of the appropriate interpretation of T.
Another non-psychological way of addressing the question ‘How is the struc-
ture common to M1 (T) and M2 (T) given to us?’ consists in transforming object
b into an object c isomorphic to object b such that object c is clearly isomor-
phic to object a (see on this Figures 4-6). For, since isomorphism is a transitive
relation this would show that object b is isomorphic to object a.

A
# A
P2 P7A P6
H A
HH A

"! HHA
P3 P4 P5
Fig. 4. Object b

P6 P2 P5 P4 P3 P3 P1
P2 P7 P4 P7 P7 P2 P6
P4 P5 P3 P1 P6 P1 P5

Fig. 5. Object c

ABCDE FG
BCDE FGA
DEFGABC

Fig. 6. Object a

Notice that the procedure illustrated above is non-psychological, because,
although we always assume that the observer finds himself in ‘normal conditions’,
the procedure acts on the objects observed and not on the observer. Indeed, in
constructing object c, we have simply ‘opened’ b in such a way as to obtain a
3 × 7 matrix which has as columns the sets of points contained in each line. (The
order in which the points actually occur in the respective lines is not relevant
for our purposes.)
Several are the things that interest us in this example. We shall briefly com-
ment on some of them in the next section.6

4 Some comments on the case study

Among the necessary conditions for ‘seeing a certain object as . . . ’ that we have
mentioned in the previous section the first is the observer’s acquaintance with
object a and/or with object b. Now, the possibility for an observer of being
acquainted with a and/or b depends, among other things, on:

[T]he particular pattern-recognition machinery hard-wired in our vi-
sual systems — edge detectors, luminance detectors, and the like . . . [T]he
very same data (the very same streams of bits) presented in some other
format might well yield no hint of pattern to us ([6], p. 33).

Other important conditions upon which the possibility of an observer being
acquainted with a and b depends are the size and position of objects a and b
relative to the observer. To see this, immagine that objects a and b are micro-
scopic and the observer is an average human being without any support provided
by technology; or that a and b are too far from the observer to be surveyable
by him, etc.
Secondly, in the absence of the formal system T and of the relevant inter-
pretations of T, the observer cannot see the pattern/structure instantiated by a
and b. This is because, in the absence of the formal system T and of the relevant
interpretations of T, he is in no position for making the observations concerning
the salient features of the pattern/structure in question, observations such as
those which have to do with the part/whole distinction, etc. This shows that T,
together with the relevant interpretations, does not simply power a deductive
engine, but is also a system of representation.
From the considerations above, we can conclude that necessary conditions for
pattern recognition in mathematics are the existence of: (1) an observer O; (2)
a domain of objects D; and of (3) a system of representation Σ, i.e. (O, D, Σ).7
Thirdly, the mathematical structure which becomes salient when we observe
objects a and b through T depends not only on T, but also on a and b —
this is where the realism concerning mathematical structures comes in. In fact,
6
A discussion of whether mathematics as a whole is conceivable as a science of pat-
terns/structures can be found in: [14], [10], [11], [15], [Resnik, 2001], [12], [13], [1]
7
Actually, the system of representation Σ is an ordered pair Σ = (T, I), where T
is a set containing (as a subset) a recursive set of axioms A and all the logical
consequences of A, and I is an interpretation of T on to D.
given that we can prove in T that there exist exactly seven elements in D and
seven m-sets, if, for instance, the number of letters of the Italian alphabet we
considered as elements of our matrix were different from seven, the matrix could
not be a model of T (the same applies mutatis mutandis to the number of points
of intersection of three lines in b).
Fourthly, we have a criterion of identity for the structure/pattern described
by T, criterion of identity represented by model isomorphism, i.e., a and b
instantiate the same structure, because they are isomorphic models of T. This is
a very important condition, because it guarantees that the concept of structure
is well defined.
Fifthly, we should notice that the definition of structure we offered in §4
— a structure S is an ordered pair whose first element is a domain of objects
D, and second element is a set < of relations defined on D — together with
the criterion of identity for structures (isomorphism) provide both a rigorous
characterization of what falls under the concept of pattern in mathematics, and
the possibility of operating a natural generalization of this concept to fields
different from mathematics.
With regard to the second point above, notice that both the examples of
patterns examined in §3 can be accounted for in terms of structures. In the
philosophy of mind and action case, the structure S1 = (D1 , <1 ) is such that
D1 contains beliefs, whereas <1 contains relations defined on D1 such as |=pd
— the plausible deontic consequence relation, where B1 , . . . , Bn |=pd B means:
someone who believes B1 , . . . , Bn plausibly ought to believe B. (The turnstile
|=pd is typical of a non-monotonic logic.)
The case of a structuralist account of patterns displayed by crystals of snowfl-
akes (see Fig. 1) is even simpler than that discussed above. The pattern/structure
S2 = (D2 , <2 ) of a crystal of a snowflake consists of a domain D2 , the elements
of which are the molecules of water contained in the snowflake, and of a set <2
whose elements are the physical laws determining how the molecules of water in
D2 are related to one another in the crystal.
But, of course, if the definition of structure we offered in §4 is applicable to
both the examples of patterns examined in §3, so does also the identity condition
for structure: structure isomorphism.
From here on, as a consequence of what we have been arguing so far, we are
going to consider the two words ‘pattern’ and ‘structure’ as synonyms.

5 Patterns’ morphogenesis and cognitive architectures

If we consider the pattern/structure instantiated in object a (Fig. 2), we realize
that this is a complex entity composed out of simpler entities. The simplest, or
atomic entities, are the first 7 letters of the Italian alphabet A, B, . . . , G, and
then we have the molecular entities represented by the subsets of three elements
of the set {A, B, . . . , G} which appear as the columns of the 3 × 7 matrix in Fig.
2.
Notice that the atomic entities mentioned above can be thought as pat-
terns/structures of points, as is shown by observing the obvious isomorphism
existing among any two of the following different objects:

A , A, A , A , A.
Moreover, molecular expressions such as {A, B, D}, {B, C, E}, . . . , {G, A, C}
(the columns of the matrix) can also be seen as patterns of patterns. Indeed,
the structural rôle of these three-element sets (of patterns) is revealed by the
fact that they are obviously isomorphic to the following three-element sets of
patterns: {a, b, d}, {b, c, e}, . . . , {g, a, c}.
All these considerations lead us, in a very natural way, to speak of a mor-
phogenetic process which, starting from atomic patterns A, B, . . . , G (patterns of
type 0), produces molecular patterns {A, B, D}, {B, C, E}, . . . , {G, A, C} (these
are patterns of type 1, because their elements are patterns of type 0), molecular
patterns which then give origin to the pattern realized in object a (Fig. 2). (The
latter is a pattern of type 2, because its elements are patterns of type 1).
Now, from the brief account of the patterns’ morphogenetic process described
above, it should be clear that such a process is capable of generating patterns of
arbitrarily large complexity. Therefore, to answer the problem ‘How do we come
to know patterns?’ on the part of a finite cognitive system which is dependent
on a limited amount of resources, resources for which he is in competition with
other finite cognitive agents, we are going to suggest that such an agent must
be endowed with a biologically inspired cognitive architecture (described in §6)
which consists of different systems for the representation and the manipulation
of information.
To see this, let A, B, . . . , G be the shortest neural network algorithms for the
recognition of A, B, . . . , G, within the set of the alphabet letters {A, B, . . . , Z}.8
The shortest neural network algorithm for the recognition of {A, B, D} will have
a length much longer than the sum of the lengths of A, B and D, because,
among other things, lacking a concept of set, our neural network will have to
treat {A, B, D} as a plurality of individual patterns and, if we exclude pluralities
containing repetitions of letters such as {A, A, B}, etc., our algorithm will have
to deal with a domain D represented by the power set of {A, B, . . . , G} which
contains 27 elements.
Furthermore, the next step, that is, the recognition of a, becomes already
computationally onerous. For, if ABD, BCE, . . . , GAC are the shortest neural
network algorithms for the recognition of, respectively, the following patterns:
{A, B, D}, {B, C, E}, . . . , {G, A, C}, the length k of the shortest neural network
algorithm for the recognition of a will be quite formidable, because, having to
recognize a out of 7! possible 3×7 matrices the columns of which are the possible
permutations of {A, B, D}, {B, C, E}, . . . , {G, A, C}, k will be much greater than
the sum of the lengths of ABD, BCE, . . . , GAC.
8
We mention here neural network algorithms, because such algorithms are so far the
most basic biologically inspired general procedures for pattern-recognition.
But, of course, in order to individuate the relevant structure realized in a,
we should now concatenate to our neural network algorithm for the recognition
of a another neural network algorithm of S length k ∗ for
S the individuation of the
∗ ∗
isomorphism inducing function ψ : D 1 D 1 → D 2 D2 (see §3). And, since
both D1 D1∗ and D2 D2∗ contain 14 elements each, our algorithm will have
S S
to recognize ψ out of a set of 1414 functions. A tall order indeed!
All these considerations make us suspect that if a finite cognitive agent depen-
dent on a limited amount of resources, resources for which he is in competition
with other finite cognitive agents, has in its cognitive architecture systems for
the representation of information which use only neural networks, it could not go
very far in its pattern recognition activity. And this would not be a consequence
of the fact that there are certain patterns for which in principle there is no neural
network based algorithm capable of recognizing them, but of the consideration
that these algorithms, if they exist, would have to be unfeasibly long, given the
computational limitations of our agent.

6 The cognitive architecture. An evolutionary account.

Given what we said in the previous section about the connection existing be-
tween patterns’ morphogenesis and the cognitive architecture of a finite cognitive
agent who is dependent on a limited amount of resources, resources for which
he is in competition with other finite cognitive agents, in what follows in this
section we are going to illustrate a cognitive architecture (see figure 7) consisting
of three levels of information-representation: a subconceptual level, in which data
coming from the environment (sensory input) are processed by means of a neural
network based system; a conceptual level, where data are represented and concep-
tualized independently of language; and, finally, a symbolic level which makes it
possible to manage the information through symbolic/linguistic representations
and computations.
Notice that all three levels for the representation and processing of information
mentioned above are present in humans, and that the first two levels may be
found in most higher animals, etc.
We have already come across the sub-conceptual level of representation (the
Sub-conceptual Tier) in §5 when we discussed the possibility of recognizing type
0 patterns (atomic patterns) by means of algorithms based on neural networks.
What we need to do now is providing a brief description of the conceptual and
symbolic levels of representation of the cognitive architecture sketched in Figure
7.
The conceptual level of the cognitive architecture of our agent consists of
the so-called ‘Gärdenfors conceptual spaces’. According to Gärdenfors, concep-
tual spaces are metric spaces which represent information exploiting geometrical
structures rather than symbols or connections between neurons. This geomet-
rical representation is based on the existence/construction of a space endowed
with a number of what Gärdenfors calls ‘quality dimensions’ whose main func-
Fig. 7. A sketch of the cognitive architecture

tion is to represent different qualities of objects such as brightness, temperature,
height, width, depth.
Moreover, for Gärdenfors, judgments of similarity play a crucial role in cog-
nitive processes and, according to him, the smaller is the distance between the
representations of two given objects (in a conceptual space) the more similar to
each other the objects represented are.
For Gärdenfors, objects can be represented as points in a conceptual space,
points which we are going to call ‘knoxels’,9 and concepts as regions (in a concep-
tual space). These regions may have various shapes, although to some concepts
— those which refer to natural kinds or natural properties — correspond regions
which are characterized by convexity.10 According to Gärdenfors, this latter type
of region is strictly related to the notion of prototype, i.e., to those entities that
may be regarded as the archetypal representatives of a given category of objects
(the centroids of the convex regions).
Finally, the symbolic level (the Symbolic Tier) of the cognitive architecture
consists, instead, of language-based systems of information representation and
computation.

9
The term ‘knoxel’ originates from [7] by the analogy with “pixel”. A knoxel k is a
point in Conceptual Space and it represents the epistemologically primitive element
at the considered level of analysis.
10
A set S is convex if and only if whenever a, b ∈ S and c is between a and b then
c ∈ S.
To see the three levels of the cognitive architecture at work, and assess their
relative merits, consider the following problem: to recognize the pattern exem-
plified by object A.
If we assume that the algorithms that follow can all be expressed in a given
language L, then the advantage of using algorithms based on tools character-
istic of the sub-conceptual level (neural networks) to solve the problem above
is that . . . an algorithm is better than nothing! On the other hand, the obvious
disadvantage is that neural network based algorithms can be relatively long.
Imagine now a 2-d Gärdenfors conceptual space, CSA, related to the letters
of the alphabet. This is a CSA tessellated by means of prototypes of such letters
using the well-known Voronoi’s procedure. The pattern-recognition algorithm
relating to A is quite simple: determine to which of the finitely many points
belonging to CSA which represent the prototypes of the letters of the alphabet
the point representing A in CSA is nearest.
Although the use of conceptual spaces is able to produce pattern recognition
algorithms much more compressed than neural network based algorithms for
the recognition of the same patterns, it has a serious defect: conceptual spaces
are ‘in the head’ in the sense that they ultimately have perceptual space as a
‘vehicle’. And, therefore, a finite cognitive agent dependent on a limited amount
of resources, resources for which he is in competition with other finite cognitive
agents, will have difficulties in exploiting the full potential of conceptual spaces.
However, the following ‘symbolic algorithm’: (1) list the letters of the alpha-
bet; (2) check whether A is an instance of the first letter; if yes (3) stop; if no
(4) check whether A is an instance of the second letter; . . . (n) stop; is certainly
shorter (and safer) than the CSA-algorithm mentioned above.
Other advantages of stepping up to the symbolic level are that:

1. language enables many minds to be connected in what we might call a ‘world
wide web’ overcoming in this way the computational limitations of every
single mind;
2. language is not ‘in the head’, in the sense that language allows:
2.1 the storing of portable information in the form of articles, books, in-
scriptions, etc. information which, among other things, no longer needs
to occupy storing space in individuals’ minds;
2.2 objectivity in the treatment of information, because in language infor-
mation is conveyed by assertions for which there exist public criteria of
correctness which we all learn when we learn the language;
3. language extends our representational and computational capabilities. To
10
see this consider the natural number 1010 . There is no chance that we are
able to represent within our visual perceptual space such a multiplicity and
10
distinguish it, for example, from a multiplicity of 1010 ± 7 elements. And
yet, within number theory, not only there are many things we can prove
about such multiplicities, but we can also use their cardinal numbers in our
ordinary arithmetical computations. These considerations apply even more
so to transfinite cardinal numbers such as ℵ0 , ℵ1 , . . . and their arithmetic.
Many more are the things that could be said in favour of the great importance
of language for pattern-recognition. However, those which have already been
mentioned in this section are sufficient to show the crucial rôle that the symbolic
level has in the cognitive architecture of a finite cognitive agent who is dependent
on a limited amount of resources, resources for which he is in competition with
other cognitive agents.
But, before ending this section and the paper, we need to spend a few words
to justify the cognitive architecture here presented. To this end, let us consider,
as we have repeatedly said, that our cognitive agent is finite, dependent on a
limited amount of resources, and engaged in a constant struggle for life with
nature and other cognitive agents, and that:

Owing to this struggle for life, any variation, however slight and from
whatever cause proceeding, if it be in any degree profitable to an indi-
vidual of any species, in its infinitely complex relations to other organic
beings and to external nature, will tend to the preservation of that indi-
vidual, and will generally be inherited by its offspring. ([5], Chapter III,
p. 40.)

From this we have that, as a consequence of natural selection,11 our cognitive
agent not only develops a hard-wired pattern-recognition machinery in his visual
system — edge detectors, luminance detectors, and the like (see on this the
quotation from [6] on p. 7 of this article) — but also a multi-level cognitive
architecture for the representation and manipulation of information.
At this point it is clear that questions like ‘Why does the cognitive architec-
ture have three different levels?’, ‘How do conceptual spaces come about in the
cognitive architecture?’, etc. can only be give an ‘evolutionary answer’, that is,
the cognitve architecture we have illustrated above is the consequence of vari-
ations which come about in the system of representation and manipulation of
information of human beings. These are variations which have been preserved
as a consequence of their being greatly profitable for the crucially important
pattern-recognition activity of humans.

7 Conclusions

In this paper we intended to give a contribution to the foundations of pattern-
recognition theory; and, to do so, we decided to address two central questions:
(a) ‘What is a pattern?’ and (b) ‘How do we come to know patterns?’
Dealing with question (a), we produced a definition of mathematical pattern
which we then generalized to fields different from mathematics (philosophy of
mind and action, physics). But, when it came to answering question (b), we
thought of presenting a cognitive architecture for a finite cognitive agent who
is dependent on a limited amount of resources. This is a cognitive architecture
11
‘This preservation of favourable variations and the rejection of injurious variations,
I call Natural Selection.’ ([5], Chapter IV, p. 51).
which is, in principle, able to cope with some of the basic demands posed by the
process of pattern-recognition; and has developed as a consequence of Darwinian
natural selection.

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