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the letter “W” is to itself or, even more surprising, that
the letter “C” is more similar to the letter “O” than
“W” is to itself. This hypothesis is made on the basis
of an experimental study in which the reaction time
has been used as a measure of similarity of two per-
ceptual inputs: in this context, longer reaction times
indicate a lower degree of similarity while shorter re-
action times indicate a degree of higher similarity.
In the SPMS the higher reaction time in response
to a perceptual input can be justified by the fact that
detailed and more complex perceptive inputs take
longer to be processed by the visual cortex. The letter
“W” is undoubtedly more graphically complex than
the letter “O” and it is plausible that, given the rich-
ness in the details, the decomposition of the starting
semantic pointer into its constituent elements is an
operation that takes a long time to perform. On the
other hand much simpler perceptual inputs can reach
a higher level of compression and can therefore be
processed faster.
In other words, there is a significant amount of
time it takes a semantic pointer to go through a trans-
formation in the brain. Therefore, depending on the
size and nature of the input, compressing or decom-
pressing some semantic pointers can take longer than
others.
3
Symmetric Property. The symmetric property in a
GMS has also often been criticized. This property
implies that, for any A, B ∈ E , the measure of simi-
larity of A towards B should be the same if computed
for B towards A. Obviously, in a geometric model this
property always holds since the distance from x to y
is the same as that from y to x. However some stud-
ies highlight that, from a perceptual point of view,
this property is not always valid. A famous exam-
ple in this direction is that presented in (Laakso and
Cottrell, 2005), according to which it is assumed that
North Korea is perceived much more like China than
China can be perceived similar to North Korea. An
other example is that presented in (Polk et al., 2002)
who found that when the frequency of colors is ex-
perimentally manipulated, rare colors are judged to be
more similar to common colors than common colors
are to rare colors.
Of course, such criticisms are based on the idea
that the judgments of similarity are all formulated in
the same mental space of comparison. By adopting
the SPMS and the idea that the spaces of comparison
may change with the context or with the size of the
semantic pointers, these problems can be overcome.
3
This also suggests that the reaction time (Podgorny and
Garner, 1979) is not the best measure of similarity for com-
puting the similarity between two concepts, at least not in
all cases.
Discussing the same example presented
in (Laakso and Cottrell, 2005), it can easily be
assumed that China is better known than North
Korea. In other words, people generally know much
more details about China (size, history, language,
currency, culture, etc.) than about North Korea.
China is therefore a more relevant concept than North
Korea and this implies that the semantic pointer
representing China can be much richer than the
semantic pointer representing the North Korea.
Referring to Example 5, we can assume that the
concept of China has representations starting from the
highest levels of the compression networks while the
concept of North Korea has only a more abstract rep-
resentation which resides in lower levels. Since in the
SPMS the similarity value depends on which of the
two concepts is introduced first, if the more relevant
concept is introduced first we start our computation
from an higher dimensional mental space and we need
to move to the lower levels of the network. Unlike if
the less relevant concept is introduced first, our com-
putation may start (and end) on the lower levels.
Triangle Inequality. Regarding the triangle inequal-
ity in (Tversky and Gati, 1982) they found some vi-
olations when it is combined with an assumption of
segmental additivity. Specifically it turns out that in
the standard GMS, given three concepts A, B and C,
we would aspect that δ(A,B) + δ(B, C) ≥ δ(A, C) (see
Example 6).
However, it is easy to find violations of these as-
sumptions in the common perception of similarity be-
tween concepts. Consider for instance the three con-
cepts life-ring, donut and cookie. We can assume
that life-ring and donut are judged similar due to their
common shape, while donut and cookie are judged
similar since they are both pastries. However, it is
difficult to assume that the distance between life-ring
and cookie is small enough to justify the triangular
inequality (see Example 7). Once again the problem
from which this objection arises is that it is assumed
that the comparison is made within the same geomet-
ric space. If, on the other hand, one accepts that dif-
ferent comparisons can take place in different mental
spaces, there is an easy justification for this inconsis-
tency.
Number of Closest Neighbors. in (Tversky and
Hutchinson, 1986) the authors suggested that a GMS
also imposes an upper limit on the number of points
that can share the same closest neighbor. A much
more restrictive limit is implied in the hypothesis that
the data points represent a sample of a continuous dis-
tribution in a multidimensional Euclidean space. For
instance, under the constraint that there must be a
minimum angle of ten degrees between two pointers,
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