=Paper=
{{Paper
|id=Vol-2378/longICFCA2
|storemode=property
|title=Conceptual Alignment with Formal Concept Analysis
|pdfUrl=https://ceur-ws.org/Vol-2378/longICFCA2.pdf
|volume=Vol-2378
|authors=Uta Priss
|dblpUrl=https://dblp.org/rec/conf/icfca/Priss19
}}
==Conceptual Alignment with Formal Concept Analysis==
https://ceur-ws.org/Vol-2378/longICFCA2.pdf
Conceptual Alignment with Formal Concept Analysis
Uta Priss
Zentrum für erfolgreiches Lehren und Lernen, Ostfalia University
www.upriss.org.uk
Abstract. Endres et al. (2009) use concept lattices as a model for neural decod-
ing. This paper continues with that idea and presents a thought experiment of how
certain structures (associative concepts) and certain processes (conceptual align-
ment and filtering) might be used to model mental concepts. Several examples
from the neuroscientific literature are discussed with respect to such a modelling
with FCA.
1 Introduction
Formal Concept Analysis1 (FCA) is a mathematical formalisation of concepts consist-
ing of extensions and intensions which lead to a conceptual hierarchy (Ganter & Wille
1999). Because the natural language word “concept” refers to mental representations
the question arises as to whether FCA can be used to model mental processes. There
are a few papers that discuss neural applications of FCA (e.g. Endres et al. 2009) and
also a few papers that discuss FCA and artificial neural networks, but so far not much re-
search has ventured in that direction. Belohlavek (2000) shows that concept lattices can
be learned by artificial neural networks of the type “bidirectional associative memory”.
In a similar manner, Kumar, Ishwarya & Loo (2015) use FCA for modelling cognitive
functions relating to memory. Kuznetsov (2004) discusses how FCA can be used to
model different types of artificial neural networks. There are a few more publications in
similar directions but as far as we know, the neuroscience community does not yet ap-
pear to see FCA as a crucial component for modelling mental processes. Furthermore,
the existing papers on neural applications focus on single lattices, not on networks of
lattices.
In this paper we are providing some suggestions for further possible connections
between neuroscience and FCA. This position paper is neither a mathematical nor a
computational paper. But we believe that it would be feasible to implement a neural
network simulation of what is described in this paper. Collecting neural data of the kind
needed for this paper is difficult because such data tends to be either detailed but with a
narrow focus or relating to the whole brain but not at a very detailed level. Furthermore
if it was possible to collect detailed neural data on a larger scale, the sheer amount of
Copyright c 2019 for this paper by its author. Copying permitted for private and academic
purposes.
1
The mathematical details of FCA are not discussed in this paper. They can be found in the
textbook by Ganter & Wille (1999).
2 U. Priss
data would be overwhelming and a challenge for FCA. It should be emphasised that
FCA is just used as a model in this paper. We are not suggesting that concept lattices
are actually computed in the brain.
This paper presents a thought experiment of how certain structures (associative con-
cepts) and certain processes (conceptual alignment and filtering) could constitute a pos-
sible mechanism that might contribute to how mental concepts are formed. We expect
many other mechanisms to occur in mental processes. The theory described in this pa-
per is somewhat inspired by Barwise & Seligman’s (1997) Information Flow Theory
(IF). Barwise & Seligman describe a network of classifications (equivalent to concept
lattices) that are connected through infomorphisms via a channel. In our opinion, a prob-
lem with IF is that once classifications and a channel exist, then infomorphisms can be
observed, but the theory does not prescribe an algorithm for how to build a channel. In
this paper associative concepts take on a similar role as an IF channel, but associative
concepts are generated by the contributing concept lattices. Thus it is not necessary to
build a channel in advance.
The next section recounts an example of an application of FCA to neural data and
explores consequences from that example. Section 3 introduces the notion of “associa-
tive concepts” which is discussed with respect to neuroscientific evidence. Section 4
defines “conceptual alignment” as a lattice infomorphism and describes its relevance
for associative concepts. The paper finishes with a conclusion.
2 Concept lattices in the visual cortex?
In this section we use the notion “natural language concept” for formal concepts which
correspond to words of a natural language. For example, natural language concepts oc-
cur in the well known lattice of “bodies of water” (Wille 1984) which, for example,
defines a puddle as “a temporary, stagnant, natural body of water”. While lattices de-
scribing physical objects and their characteristics tend to contain at least some natural
language concepts, we would assume that many FCA applications are too abstract to
produce many (if any) natural language concepts. But we would argue that such con-
cepts do occur in the example presented by Endres et al. (2009).
Endres et al. apply FCA to neuroscientific data consisting of stimuli that were pre-
sented to a macaque monkey (used as formal objects) and discretised spike trains from
individual high-level neurons in its visual cortex (used as formal attributes). Tradition-
ally such data is analysed with tree-based clustering methods. But the examples from
Endres et al. show that using concept lattices is quite promising. Endres et al.’s lattices
contain concepts corresponding to natural language concepts such as “face/frontal view
of head”, “sideview of head”, “head from behind” and “round items” which indicates
that such concepts might emerge within that area of the visual cortex of monkeys. The
bottom concepts tend to relate to individuals, such as other monkeys and caregivers
known to the monkey, and the top concepts correspond to abstract, general categories.
In summary, we would argue that such lattices contain at least some natural language
concepts.
So, what are the conditions that neurons must meet so that the resulting lattices
contain natural language concepts? In Endres et al.’s examples the higher level concepts
Conceptual Alignment with Formal Concept Analysis 3
all tend to have large extensions which cannot easily be characterised by an obvious
single feature. This means that individual neurons respond to a wide range of stimuli.
The criteria that neurons use for their activations are most likely fuzzy, but appear not
to be totally random. The fact that some concepts can be labelled (such as “round” or
“face”) shows that there must be some neurons which react to all stimuli of a certain
kind. Thus, there must be a precise core group for the activation of many of the neurons.
Nevertheless each neuron also reacts to some other stimuli which are not in the core
group. The core groups of different neurons appear to overlap to a large degree leading
to closely related subconcept clusters in the lattices. But no two neurons have exactly the
same activation pattern. The lower concepts indicate that all neurons react to important,
most frequently encountered stimuli. Nevertheless, because the bottom concept has an
empty extension, for each stimulus there appears to be at least one neuron that does not
react to it.
In summary, the conditions for neurons are that each neuron appears to have a spe-
cial activation focus yet also some other stimuli it reacts to. Important stimuli lead to
the activation of most, but not all neurons. There are large overlaps between the sets of
stimuli of different neurons but none are exactly the same. Thus natural language con-
cepts appear to be formed not only in higher association or language processing areas,
but for example already in the visual cortex. As mentioned before, FCA is just used as a
(very suitable) model - we are not suggesting that lattices are actually computed in the
brain. FCA suggests a kind of naturally occurring subconcept-superconcept hierarchy
corresponding to the relationships between sets of stimuli and sets of neurons.
Kriegeskorte et al. (2008) show that categorical object representations in the inferior
temporal cortex of humans and macaque monkeys are similar. Thus, it can be assumed
that the functionality of brains of humans and other monkeys is similar in this respect
and similar concept lattices as the ones described by Endres at al. (2009) could in theory
be constructed for human data. Unfortunately such data is not normally available for the
following reasons: Retrieving data from individual neurons involves invasive methods
which cannot normally be performed on humans. Non-invasive neural imaging meth-
ods are ethically preferable because they can be performed on humans, but they tend to
collect data from brain regions instead from individual cells. Furthermore any type of
current neuroscientific data collection method tends to be tedious and expensive. There-
fore most studies only focus on very specific aspects of cognition. Last but not least, a
fair amount of processing is required in order to discretise any data. Therefore detailed,
large scale neuron/stimulus data sets are not available. Nevertheless apart from neuro-
science, robotics may be a possible application area for this kind of analysis. It may be
possible to collect or generate data from artificial neural networks of humanoid robots
or to reverse engineer large scale datasets that are available (knowledge graphs, ontolo-
gies, thesauri, web-based data collections, and so on) in order construct hypothetical
concept lattices that contain natural language concepts in a similar manner as in the
examples of Endres et al. (2009).
4 U. Priss
3 Associative concepts
3.1 Results from neuroscience
The previous section describes how concept lattices may be relevant for modelling in-
dividual brain areas. This raises the question as to how concepts can be modelled that
combine information from different brain areas. Bauer & Just (2019) provide a review
of the neuroscientific literature on the topic of how conceptual knowledge is represented
in the brain. This section summarises that article but with one modification: we use the
term “associative concept” for concepts in the sense of Bauer & Just in order to distin-
guish them from “formal concepts” in the sense of FCA. According to Bauer & Just,
associative concepts are spatially distributed neural representations. This means that
multiple brain areas are involved in any associative concept. For example an associative
concept of “knife” activates visual areas (what a knife looks like), sensory areas (what
a knife feels like), motor areas (how a knife is held and used) and association areas
(further information about knifes). From a modelling view, the different areas can be
considered dimensions. The formal concepts mentioned in the previous section would
then be a component of an associative concept in one dimension. Bauer & Just state that
in addition to the dimensions themselves, the relations between the dimensions are also
represented in “specific high-order brain areas or convergence zones”. Thus, associative
concepts form relations which are represented or computed somewhere.
According to Bauer & Just, associative concepts can be elicited by pictures or
words. The picture or word of an associative concept is stored in sensory brain regions
separately from the concept itself. Apparently it does not matter whether an associative
concept is elicited by a picture or by a word, although pictures evoke potentially more
detailed descriptions whereas words evoke the most generic properties of an associative
concept. Neural representations of associative concepts appear to be largely indepen-
dent of the language which is used to describe them and, furthermore, fine-grained
activation patterns for associative concepts appear to be common across different peo-
ple. More concrete associative concepts correspond to more activation in sensorimo-
tor areas whereas more abstract concepts evoke other verbal concepts. Abstract scien-
tific concepts repurpose neural structures originally evolved for more general purposes.
Clearly, this is a very reduced and simplified summary of Bauer & Just’s already quite
condensed review article. But if structures within single brain areas are modelled as
concept lattices as suggested in the previous section, associative concepts then require
some sort of combination or further processing of concept lattices as discussed in the
next section.
3.2 Modelling with FCA
In Section 2 we have argued that individual brain areas might be modelled as concept
lattices. Section 3.1 poses the question as to how associative concepts can be formed by
combining the individual lattices and possibly other information. Different functional
brain areas have different biological configurations (size, shape, type of neurons, con-
nectivity). Thus, it cannot be assumed that they all function in the same manner and
should be modelled in exactly the same manner. Other models than FCA might be more
Conceptual Alignment with Formal Concept Analysis 5
suitable for planing, logical thinking, motor functions, emotions and so on. Therefore
the definition below does not state that dimensions must be lattices, but they can be.
Whereas the formal concepts discussed in Section 2 are most likely part of the uncon-
scious realm of thought, associative concepts as discussed in Section 3.1 are evoked
by pictures or words and therefore belong to the realm of conscious thought. Therefore
associative concepts are a type of mental construct which is formed in some high level
functional areas by relying on information provided by lower areas. Based on the infor-
mation from the previous sections, we suggest the following definition of associative
concepts:
Definition 1. For a many-valued formal context with a set C of objects and sets of
attributes D1 , ..., Dn : the elements of C are called associative concepts if the d ∈ Di
themselves are concepts (modelled using FCA or some other theory) and a common
null element d⊥ exists (i.e., ∀1≤i≤n : d⊥ ∈ Di ). For each ci ∈ C the non-null entries
in its row form a vector (di1 , di2 , ..., dik ) of length k ≤ n with components dij and
signature (Di1 , Di2 , ..., Dik ).
The definition indicates that some dimensions can be concept lattices or substruc-
tures of concept lattices. Using standard FCA techniques for many-valued contexts
(such as conceptual scaling or pattern structures) concept lattices can be computed in
any case. But we are arguing below that also other approaches might be of interest for
associative concepts. Dimensions are not meant to correspond to functional brain areas
but to subareas which contribute one specific aspect to an associative concept. Not ev-
ery associative concept uses all of the dimensions. For example, colour concepts most
likely do not activate areas related to movement or touch.
image: visual/tactile/weight/size: colour: visual/tactile/sound: hand/arm movement:
blade thin, long handheld smooth surface cold stab cut
haft
silver metal
category: schema: phonetic:
weapon tool cutlery night
* eating knight
* cooking (k)ni knife (i)f(e)
* butchering nice
fork spoon * fighting strife
* injuring life
Fig. 1. An associative concept of “knife”
6 U. Priss
Fig. 1 shows a possible associative concept for the word “knife”. It consists of a
visual dimension that contains a prototypical picture of a knife. This dimension also
represents part-whole relationships. In accordance with Section 2, this dimension could
be modelled as a concept lattice where the pictures are formal objects corresponding to
visual stimuli. The reason why some of the other dimensions are labelled with combina-
tions (“visual/tactile/sound”) is explained further below. The different dimensions can
also be viewed as “facets” or “aspects”. They refer to different physical characteristics
of knifes: their colour, material, shape, size and typical usages. The characteristics are
described from an embodied perspective. For example “handheld” is meant to represent
a physical characteristic of something that can be held in a hand because of its size,
weight and shape. While the dimensions at the top half of Fig. 1 are perceptual and for-
mal concepts as described in Section 2, the dimensions at the bottom half are at a higher
functional level, such as categories, schemata (typical usage situations) or referring to
a representational form. If someone is asked to provide a definition of “knife” they will
most likely invoke a category (“A knife is a weapon that ...”, “ ... a tool for ...”, “a type
of cutlery”). Understanding a word requires phonetic processing of the consonants and
vowels. The schema and phonetic dimensions are not modelled as lattices in this exam-
ple. Schemata could be modelled as conceptual graphs; phonetic processing could be
modelled as a neural network that processes sequential inputs in a probabilistic man-
ner. Fig. 1 is just meant as a possible example for illustrating what associative concepts
might be like.
If all dimensions were concept lattices, then a direct product of the lattices could be
calculated and an associative concept would just be a concept in the product lattice. But
considering that there are billions of neurons in the human brain and single lattices al-
ready tend to have many more concepts than attributes (which correspond to neurons in
Section 2), the number of concepts in such a direct product would be huge. Furthermore,
although humans generate subconcept-superconcept relationships in their minds (of the
sort: “a poodle is a dog”), there is no evidence that a long chain of such relationships is
ever generated. Most people who are not trained biologists would generate a sequence
such as “poodle” → “dog” → “mammal” → “animal” → “thing”. At the level of “thing”
the relationships tend to become circular, for example, by defining a thing as an object
and then an object as a thing. Such circular top-level definitions can be found in any dic-
tionary. Thus, it is questionable whether associative concepts form a lattice at all. In our
opinion it is more likely that there are some local subconcept-superconcept-like rela-
tionships amongst associative concepts but also many other semantic relationships. The
hierarchical relationships are generated ad hoc when an associative concept is thought
about and never extend very far. Using Miller’s (1956) famous number, maybe at most
7 associative concepts and their relationships are ever considered at once. Such rela-
tionships will be locally meaningful and free of contradictions, but globally potentially
circular and contradictory. Thus, we would argue that there is no reason to calculate a
global structure that encompasses all possible associative concepts.
Instead of computing a global structure, it is of interest to observe the changes oc-
curring within and amongst the dimensions if one navigates from one associative con-
cept to another. A train of thought might correspond to navigating between associative
concepts. For example, navigating from “knife” to “cutlery” only changes components
Conceptual Alignment with Formal Concept Analysis 7
in some dimensions (image, movement, category, phonetic) while the other compo-
nents all stay the same. Apart from the phonetic dimension, the changes in the other
dimensions are all small. It is to be expected that there is a mutual interaction between
associative concepts and their components both while an associative concept is thought
about but also while the brain is learning new concepts. Somehow selecting one com-
ponent can influence how other components are selected and modifying a component
in one dimension might modify components in other dimensions. In our opinion, the
interesting questions about associative concepts are how they interact, how the different
dimensions shape and influence each other and how they contribute to and are modified
by other associative concepts. Section 4 discusses this further.
WordNet: Wikidata:
arm, weapon edge tool edge and bladed weapon
to knife blade
to stab knife haft, helve knife
point use: cutting
murder
bread knife collection
carving knife used by: butcher
slicer .... cook
has part: blade
handle
Fig. 2. Associations of “knife” in lexical or semantic databases
Because, as mentioned before, it is difficult to obtain neural data, it is of interest to
use databases that are available, such as lexicons, dictionaries, ontologies or knowledge
bases. Unfortunately, their structures are somewhat different from Fig. 1. Fig. 2 shows
entries for “knife” in a lexical database (WordNet2 ) and a knowledge base (Wikidata3 ).
The focus of WordNet and Wikidata is more on verbal relationships and ignores some
of the more basic perceptual dimensions. It might be possible to extract perceptual di-
mensions but this would require some additional processing, for example, supported
by formal ontologies. Lexical databases might be more suitable for more abstract as-
sociative concepts (such as “democracy”) which are according to Bauer & Just (2019)
represented in a more verbal form in the brain as well.
4 Conceptual Alignment
4.1 Synesthesia as a form of conceptual alignment
The previous section suggests that the components of an associative concept may influ-
ence each other. Ludwig, Adachi & Matsuzawa (2011) observe that not just humans but
2
https://wordnet.princeton.edu/
3
https://www.wikidata.org
8 U. Priss
also chimpanzees associate high pitched sounds with high luminant colours. They argue
that such a relationship of synesthesia is not learned but “constitutes a basic feature of
the primate sensory system”. In our opinion this is an example of alignment amongst
visual, tactile, auditory or motoric conceptual dimensions which we define as follows:
Definition 2. Conceptual alignment is an order-preserving partial isomorphism which
maps formal concepts of one concept lattice onto formal concepts of another concept
lattice. Two concept lattices are in total conceptual alignment if a total order-preserving
isomorphism exists between them.
A partial isomorphism is only a mapping between substructures of the lattices. If
two concept lattices that are not aligned at all are combined by concatenating their
formal contexts, then the combination lattice is in the worst case the direct product
of the initial lattices. Contrary, if two totally aligned concept lattices are combined
into one lattice (by concatenating their reduced and isomorphically ordered contexts),
then the combination is also isomorphic to the initial lattices. Thus, a larger degree of
conceptual alignment reduces the complexity of combining lattices. It follows that if
two dimensions which belong to an associative concept are strongly aligned, then it is
possible to combine them into one dimension (without changing the original lattices
much) in order to reduce the number of dimensions of the associative concept.
When a baby first explores the world, it simultaneously uses multiple senses while it
is building its conceptual hierarchies. For example, a round shape can be simultaneously
seen and felt. That is the reason why in Fig. 1 we labelled some of the dimensions as
“visual/tactile/weight/size” and so on. Fig. 3 shows how that dimension is potentially
itself constructed from further lower-level dimensions. For small physical objects that
can be held in one hand, a mental, conceptual representation of shape should be the
same no matter whether the object is seen or felt. Such an object may have additional
visual features (colour) and tactile features (temperature) but the joint visual-tactile
features should have a similar conceptual representation. Therefore we would argue
that there are either lattices for parts of the visual and tactile areas that are strongly
aligned (as in the left hand side of Fig. 3) or only one joint lattice is built for visual and
tactile inputs. A similar situation occurs for weight and size (right hand side of Fig. 3).
For most everyday objects there is a correlation between weight and size. Thus, the two
lattices should be aligned. There are exceptions: a lead ball is heavy, a polystyrene ball
is lightweight. But picking up such an object produces a surprise. Therefore exceptions
must be dealt with in a different manner, but in non-exceptional cases, a correspondence
between weight and size can be expected. There may also be some connections between
the dimensions of shape and of size/weight. But while the concepts for weight/size
are graded and interval-based (heavy/large, medium, light/small), concepts for shape
are feature-based and not graded. Thus, lattices for weight/size and shape cannot be
conceptually aligned apart from a few connections.
The synesthetic connections observed by Ludwig, Adachi & Matsuzawa (2011) for
chimpanzees are not between vision and touch as in Fig. 3 but between vision and sound
which might seem surprising. But we would argue that they are actually also based in
Conceptual Alignment with Formal Concept Analysis 9
visual (shape): tactile (shape): tactile (weight): tactile (size):
Fig. 3. Conceptual alignment
physics4 . From a physical viewpoint dark colours and soft surfaces collect heat whereas
bright colours and smooth surfaces reflect light and heat. This even extends to sounds:
soft surfaces produce a muffled sound when tapped whereas solid/smooth surfaces pro-
duce a sharp sound. Therefore synesthetic connections can develop between these per-
ceptual dimensions reflecting what babies (human or chimpanzee) observe when they
explore physical objects. It should be mentioned that this account of the relationship be-
tween perception and physics is related to the well-known idea of “embodied cognition”
(Varela, Thompson, & Rosch, 1992). But the neural basis of embodied cognition is still
being investigated (Caramazza et al. 2014). Our suggestion is that low level synesthesia
is related to conceptual alignment. It can be speculated that conceptual alignment is
such a strong driving factor of early developmental learning processes that it even over-
generalises and creates more connections than what is actually observed by a baby. This
could be an explanation of unusual synesthesia in some people.
We thus argue that a process of alignment occurs between some of the concept lat-
tices that constitute dimensions of an associative concept as shown in Fig. 3. We further
hypothesise that the associative concepts themselves may be what drives the process of
conceptual alignment. Therefore, if a baby repeatedly conjures sets of associative con-
cepts which appear to have some regular combination of components then the lattices
may have a tendency to arrange themselves so that their corresponding concepts align.
For example, if two domains can be modelled with intervals, then these might attempt
to align themselves. Furthermore if there is a subconcept-superconcept relationship for
two associative concepts in two of the lattices whereas in a third lattice only the sub-
concept exists but not the superconcept, then maybe that concept is added to that lattice
so that the subconcept-superconcept relationship holds in all three lattices. We propose
the following principle:
Principle 1 Reduction through Alignment: if two dimensions of an associative concept
are partially aligned, they may have a tendency to increase their alignment so that they
can be combined into one dimension thus reducing the number of dimensions of the
associative concept.
This process does not need to lead to total alignment. Lattices of perceptual di-
mensions are most likely never totally aligned because different types of perception
4
Some forms of synesthesia may not be based in physics. For example, some people associate
colours with graphemes (i.e. letters or numbers). Since graphemes are learned at a later stage
than basic shapes, it is likely that higher level brain areas are involved in that type of synesthe-
sia. Thus, it is a different type of synesthesia than the one discussed in this paper.
10 U. Priss
always have subtle differences. In the example in Fig. 1, there is a relationship between
“silver” and “metal” (because metals tend to have a silver or copper colour) but such
relationships cannot be found between all kinds of colours and materials. There is also a
relationship between the weight/size of an object and the types of hand/arm movements
that can be performed with an object.
4.2 Lack of conceptual alignment
Further evidence for alignment between low level perceptual concepts arises from con-
ditions where such alignment fails to occur. Autism is a potential example for such
a condition. Autism involves impairment in social interaction but also unusual per-
ceptual abilities and deficits most likely due to cerebellar abnormalities (Fatemi et al.
2012) which appear to obstruct fast neural communication between certain brain ar-
eas, for example, motor and vision. In our terminology, conceptual alignment of low
level perceptual brain areas may be lacking. Asperger’s syndrome and high function-
ing autism (HFA) are milder forms of the condition. In this paper we are using the
“speech-in-noise” problem as an example which refers to difficulties in understanding
speech in noisy situations, for example, in a restaurant setting. Adults with HFA tend to
have more difficulties with speech-in-noise understanding than people without autism
(i.e. neurotypical or NT people) because their brains have difficulties filtering the rele-
vant sound from the background noise (Smith & Bennetto 2007). Lip reading improves
speech-in-noise understanding for NT people but not for adults with HFA (Smith &
Bennetto 2007).
Fig. 4 shows a modelling of the conceptual alignment required for speech-in-noise
understanding. The upper half of the figure shows potential lattices involved in speech-
in-noise perception in an NT person. By using “theory of mind” which involves under-
standing and mapping someone else’s actions, a connection is established between the
other person’s lip movements and the person’s own lip movements. Concepts relating
to seeing consonants being formed are aligned with concepts of oral muscle movement.
For HFA people lip reading does not help because they tend to have problems with
“theory of mind” cognition (Baron-Cohen et al. 1997). Thus, the connection labelled
“A” in Fig. 4 is difficult for people with HFA. Furthermore, because they generally
have a problem with the alignment of perceptual inputs from different channels, the
connection labelled “B” is also difficult for them. Therefore, in the case of autism, the
low level perceptual lattices are not aligned. In fact it is known that people with autism
perceive and produce too much pitch in spoken language (Nadig & Shaw 2012) which
in NT people is only relevant for emotional connotations and not for phoneme per-
ception. DePape et al. (2012) show that people with HFA have “less specialisation for
native speech sound categories than controls”. Or in other words they have too many
categories for speech sounds. But people with HFA tend to produce less diverse mouth
movements than NT adults (Parish-Morris et al. 2018). Translated into our terminology,
in autism, the lattices of pronunciation related motor neurons have too few concepts for
consonants whereas the phoneme perception lattices have too many concepts. There-
fore they are not aligned. Both lattices contain irrelevant concepts pertaining to pitch
because of the insufficient connection A. Because speech understanding without noise
is similar to NT people, HFA people must be using higher level processing for language
Conceptual Alignment with Formal Concept Analysis 11
understanding that overcomes low level difficulties but without ever leading to an align-
ment of the lower level lattices. The lower level deficiency only becomes apparent in
speech-in-noise situations.
native language:
theory of
mind
A B
lip reading pronunciation phoneme recognition
(visual cortex) (motor neurons) (perception)
foreign language: autism:
A B A B
missing missing too few too many
concepts concepts concepts or irrelevant
concepts
Fig. 4. Speech-in-noise processing for people with and without autism
NT people also find it more difficult to understand speech in a noisy situation if the
speech is not in their native language but in a second language that they learned at a later
age. Because this depends on the type of noise it cannot be solely due to better semantic
networks in the native language (Broersma & Scharenborg 2010). For NT people con-
nection A is not affected and there is also no reason to assume that they have difficulties
in mapping muscle movements for producing consonants onto the sounds produced by
consonants (connection B). Their problem is most likely that they are missing concepts
both for their motor neurons as well for their phoneme perception. Apparently such con-
cepts can only be learned at an early age. For example, Japanese people have a problem
hearing the difference between “l” and “r” in English and tend to pronounce “l” instead
of “r”. Thus, they are missing the concept of “r” both as a phoneme as well as a pro-
nunciation. In some cases, higher level processing can support speech understanding
and production. For example, with adequate training, it is possible for movie actors to
learn to pronounce a foreign language with a high degree of accuracy. But this involves
higher levels of concentration. Another example for high level processing is the fact that
most people can hear the tonal differences of Chinese words if their attention is drawn
to them. But associating (lower level) meaning differences with tonal differences may
be impossible to learn at a later age. In summary, we are arguing that the notion of “con-
ceptual alignment” can serve as a model for explaining low level perceptual processing.
In particular, a lack of alignment may cause deficits which cannot be fully compensated
for by higher level processes.
12 U. Priss
4.3 Filtering
We suspect that associative concepts and their dimensions mutually influence each
other: if concepts from different dimensions co-occur then they give rise to an associa-
tive concept that combines them more solidly. One the other hand once the associative
concept has been formed, it encourages its dimensions to conceptually align with each
other if at all possible. From the viewpoint of the associative concept it is advantageous
to have fewer dimensions because that reduces its cognitive load. We believe that apart
from alignment there is a second possibility of reducing dimensions as described in the
following definition and principle:
Definition 3. The strength of a dimension is the degree to which it is a result of combi-
nations of other conceptually aligned dimensions.
Principle 2 Filtering: associative concepts prefer stronger dimensions. Very weak di-
mensions may be omitted from an associative concept thus reducing the number of
dimensions.
Thus, both the Principle of Reduction through Alignment as well as the Principle of
Filtering lead to a reduction of dimensions and therefore reduce the computational load
related to an associative concept. Low-level conceptual alignment as in Fig. 3 is ad-
vantageous because it strengthens the individual dimensions and reduces dimensions.
Dimensions that do not fit at all are considered noise and are filtered out. There are
many known examples of mental filtering. For example, there tend to be many conti-
nuity errors in films5 , such as objects that are suddenly in a different place or changes
in clothing of the actors in the middle of a scene, which most people do not notice.
With respect to linguistic processing, people tend to overlook missing letters or words
in written language and do not hear certain consonant variations in spoken language.
We would argue that this is a consequence of alignment (which overwrites erroneous
values) and filtering which ensures that an unusual perceptual input is not added as an
extra dimension to an associative concept but instead omitted. Not everything can be
filtered out, though. Inputs that interfere with normal processing (for example spoken
language with foreign accents or unusual rhythms such as stuttering) or contradict ex-
pectations (for example an incorrect note in a piece of music) are not filtered out but
instead consciously noticed. The same holds for observations that attract emotions (such
as curse words). Thus, as mentioned before, exceptions must be processed differently
from non-exceptional perceptive inputs.
With respect to the speech-in-noise perception mentioned in the previous section, it
is known that people with HFA have problems filtering. In our terminology this means
that people with HFA perceive many simultaneous, conflicting associative concepts of
few dimensions instead of one single coherent associative concept with many dimen-
sions. In summary, this section describes how conceptual alignment may be related
to the formation of associative concepts and how associative concepts may attempt to
reduce the number of their dimensions via alignment and filtering.
5
https://en.wikipedia.org/wiki/Continuity (fiction)
Conceptual Alignment with Formal Concept Analysis 13
5 Conclusion
This paper starts with an example of an application of FCA to neural data by Endres et
al. (2009) which describes potential concept lattices for single perceptional brain areas.
The paper then argues that individual lower level lattices are combined as dimensions
in higher level associative concepts. Relationships of mutual influence exist between
associative concepts and their component dimensions, for example, in the form of con-
ceptual alignment.
As a thought experiment, the paper poses many questions for further research. We
believe that in particular questions about different types of learning are of interest for
concept lattices. In early developmental phases the concept lattices must be built. This
could be modelled with bidirectional associative memory as suggested by Belohlavek
(2000). But learning does not stop at that point. Later in life conceptual structures are
still modified, but slowly and incrementally. In our opinion the mutual influences that
exist between associative concepts and their dimensions provide a model for such kind
of learning. We hope that this paper might encourage further research in these direc-
tions.
References
1. Baron-Cohen, S., Jolliffe, T., Mortimore, C., & Robertson, M. (1997). Another advanced test
of theory of mind: Evidence from very high functioning adults with autism or Asperger syn-
drome. Journal of Child psychology and Psychiatry, 38, 7, p. 813-822.
2. Barwise, Jon; Seligman, Jerry (1997). Information Flow. The Logic of Distributed Systems.
Cambridge University Press.
3. Bauer, Andrew J. & Just, Marcel A. (2019). Neural representations of concept knowledge. In:
Zubicaray, & Schiller (Eds.), The Oxford Handbook of Neurolinguistics.
4. Belohlavek, R. (2000). Representation of concept lattices by bidirectional associative memo-
ries. Neural Computation, 12, 10, p. 2279-2290
5. Broersma, M., & Scharenborg, O. (2010). Native and non-native listeners’ perception of En-
glish consonants in different types of noise. Speech Communication, 52, 11-12, p. 980-995.
6. Caramazza, A., Anzellotti, S., Strnad, L., & Lingnau, A. (2014). Embodied cognition and
mirror neurons: a critical assessment. Annual review of neuroscience, 37, p. 1-15.
7. DePape, A., Hall, G., Tillmann, B., & Trainor, L. (2012). Auditory Processing in High-
Functioning Adolescents with Autism Spectrum Disorder. PLoS ONE 7, 9, e44084.
https://doi.org/10.1371/journal.pone.0044084
8. Endres, Dominik M.; Foldiak, Peter; Priss, Uta (2009). An Application of Formal Concept
Analysis to Semantic Neural Decoding. Annals of Mathematics and Artificial Intelligence, 57,
3, Springer-Verlag, p. 233-248.
9. Fatemi, S. H., Aldinger, K. A., Ashwood, P., Bauman, M. L., Blaha, C. D., Blatt, G. J., ...
& Estes, A. M. (2012). Consensus paper: pathological role of the cerebellum in autism. The
Cerebellum, 11, 3, p. 777-807.
10. Ganter, Bernhard; Wille, Rudolf (1999). Formal Concept Analysis. Mathematical Founda-
tions. Berlin-Heidelberg-New York: Springer.
11. Kriegeskorte, N., Mur, M., Ruff, D. A., Kiani, R., Bodurka, J., Esteky, H., Tanaka, K., &
Bandettini, P. A. (2008). Matching categorical object representations in inferior temporal
cortex of man and monkey. Neuron, 60, 6, p. 1126–1141.
14 U. Priss
12. Kumar, C. A., Ishwarya, M. S., & Loo, C. K. (2015). Formal concept analysis approach to
cognitive functionalities of bidirectional associative memory. Biologically Inspired Cognitive
Architectures, 12, p. 20-33.
13. Kuznetsov, S. O. (2004). Machine learning and formal concept analysis. In International
Conference on Formal Concept Analysis, Springer, p. 287-312.
14. Ludwig, V,; Adachi, I; Matsuzawa, T. (2011). Visuoauditory mappings between high lumi-
nance and high pitch are shared by chimpanzees (Pan troglodytes) and humans. Proceedings
of the National Academy of Sciences, 108, 51, p. 20661-20665.
15. Nadig, A., & Shaw, H. (2012). Acoustic and perceptual measurement of expressive prosody
in high-functioning autism: Increased pitch range and what it means to listeners. Journal of
Autism and Developmental Disorders, 42, 4, p. 499-511.
16. Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our
capacity for processing information. Psychological review, 63, 2, p. 81.
17. Parish-Morris, J., Sariyanidi, E., Zampella, C., Bartley, G. K., Ferguson, E., Pallathra, A. A.,
... & Brodkin, E. S. (2018). Oral-Motor and Lexical Diversity During Naturalistic Conver-
sations in Adults with Autism Spectrum Disorder. In Proceedings of the Fifth Workshop on
Computational Linguistics and Clinical Psychology: From Keyboard to Clinic, p. 147-157.
18. Smith, E. G., & Bennetto, L. (2007). Audiovisual speech integration and lipreading in
autism. Journal of Child Psychology and Psychiatry, 48, 8, p. 813-821.
19. Varela, F.; Thompson, E.; & Rosch, E. (1992). The Embodied Mind: Cognitive Science and
Human Experience, MIT Press.
20. Wille, Rudolf (1984). Liniendiagramme hierarchischer Begriffssysteme. Studien zur Klassi-
fikation. Indeks Verlag.