Categorical Approach to Swarm Computations
Jerzy Kr
, Andrew Schumann
and Krzysztof Bielas
Chair of Cognitive Science and Mathematical Modelling, University of Information Technology and
Management in Rzesz
ow, Sucharskiego 2, 35-225 Rzesz
ow, Poland
Swarm Intelligence, Category Theory, Yoneda Embedding, Emergency.
We propose the model approaching the problems of organisation, computing and emergent behavior of certain
swarms from category theory point of view. In this model the Yoneda embedding to the category of presheaves
spanned over the basic category of partial recursive functions, is activated by external stimuli and the resulting
excited domains carry collective self-organising processes. We find that the intuitionistic logic of the presheaf
topos becomes a primary logic for a way how swarms act and how they should be described algorithmically.
Swarm intelligence (Kennedy and Eberhart, 2001;
Bonabeau et al., 1999) is the phenomenon of much
concern for information science, computer science,
intelligent systems or algorithmic processes but also
for cognitive science or consciousness studies. To
understand better a more complex social feature of
swarms with much degree of complication and auton-
omy of their members, it is highly desirable to ignore
chemical and biological processes of real swarms and
to focus on their ability to solve computational tasks
in their organisation of transport networks.
There are swarms at the levels of different king-
doms. So, microbial swarms or social bacteria (the
level of Kingdom Bacteria) can be exemplified by
bacteria Pseudomonas aeruginosa which have a quo-
rum sensing allowing them to transmit signals among
the cells for their cooperation (Ben-Jacob, 2008). At
the level of Kingdom Animalia there are swarms from
the level of insects (such as ants Formicidae) to mam-
mals (such as naked mole-rats Heterocephalus glaber
and chimpanzee Pan troglodytes). Each swarm, from
bacteria to mammals, can solve logistic and transport
problems very effectively (Gordon, 2003; Michener,
Evidently that chemical and biological processes
of reacting and motoring for swarms of different lev-
els are different, too. The point is that for differ-
ent species, biologically active matters are different.
There are two main types of biologically active mat-
ters: (i) attractants (such as pheromones) which at-
tract and (ii) repellents (such as dangerous conditions)
which repel. Computationally these matters can be
presented as vertices of graph so that the swarm mo-
tion is a computational process along edges among
vertices. In this paper, we concentrate on this repre-
sentation to describe a computational nature of any
swarm at different levels.
Recently it was shown that a swarm behavior is
detected even at the level of only one cell. So,
within one cell there are different proteins (such as
microtubules and microfilaments) which are assem-
bled and disassembled under different conditions in
response on extracellular stimuli to transform the cell
and to transmit the signal. Microtubules and micro-
filaments react to external stresses to organise differ-
ent actin filament networks: unstable bunches (paral-
lel unbranched filaments), trees (branched filaments),
stable bunches (cross-linked filaments), see (Calder-
wood et al., 2000; Carlier, 1989; Carlier, 1991; For-
gacs, 1995; Hill, 1981; Mooseker and Tilney, 1975).
Hence, each filament is a ‘swarm agent’ that co-
operates with other filaments (‘agents’) in organis-
ing some emergent structures from bunches to trees.
Also, microtubules and microfilaments are responsi-
ble for changing the shape and structure of dendritic
spines of neurons so that they play a significant role
in the formation of new spines as well as stabilising
spines. Thus, due to them the signals are transmitted
through neurons (Dillon and Goda, 2005).
Thus, we observe swarms of different scale: from
a swarm behavior of some proteins (such as actin fil-
Król, J., Schumann, A. and Bielas, K.
Categorical Approach to Swarm Computations.
DOI: 10.5220/0010389502180224
In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 3: BIOINFORMATICS, pages 218-224
ISBN: 978-989-758-490-9
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
aments) to social bacteria, social insects, and mam-
mals. In this paper, we analyse a swarm behavior
of members (atoms) with emphasis on their emer-
gent effects due to their interconnections. Such a
model can abstractly describe swarms of different lev-
els and scales: from actin filaments to chimpanzee
Pan troglodytes.
It is by now a rather well-established fact that cer-
tain biological systems can be considered as comput-
ing devices and biological computers based on them
are not merely a dream, e.g. (Adamatzky, 2010;
Erokhin et al., 2012; Roquet et al., 2016). It is a
less known, but rather expected fact, that some such
systems exemplify, or interpret, non-classical logic
which has been claimed to exist in computing bi-
ological systems recently (Schumann, 2019; Schu-
mann, 2017; Ohya and Volovich, 2011). The au-
thors of (Adamatzky and Siccardi, 2015) showed that
quantum gates can be interpreted in the actin fila-
ment swarms. There were also previous attempts
toward modelling an emerging consciousness from
quantum computations taking place in some of the
systems e.g. (Hameroff, 1998). A special attention
has been paid to biological swarms like ant and bee
colonies, fish schooling, birds flocking, horse herd-
ing, bacterial colonies, multinucleated giant amoebae
or the actin filaments. In this context there emerged
the new branch of computer science which can be
called swarm intelligence and is devoted to analysing
the collective and decentralised behavior of swarms
as reacting on external factors. Defining the factors
as stimuli, i.e. attractants and repellents acting on
swarms (like low or high temperature, light or dark-
ness), one observes the reaction produced, or calcu-
lated, by a swarm which is not just a simple additive
reaction of members of the swarms but rather an ‘in-
telligent’ emergent reaction of big domains of them
without any centralised control. The process can be
further used to solve certain mathematical hard prob-
lems like Travelling Salesman Problem, the Steiner
Tree Problem, the Generalised Assignment Problem
and some other (Schumann, 2019). Even though the
instances of these tasks are solved only case by case
and no general algorithm is certainly known or deriv-
able from the cases, the collective decentralised reac-
tion of the swarms members makes the job.
Hence, one of the goals of the present paper is
to analyse to what extent non-classical logic is an
inherent feature of certain biological swarms. Con-
nected with this is the attempt to characterise what
emergent collective behavior of a swarm could be in
order to take part in swarm’s computations. Given
the possibility to interpret as non-classical (many-
valued, quantum) logical gates in the set of the al-
lowed swarm’s reactions on external factors, is this
merely the interpretation in the fundamentally classi-
cal swarms? Or, maybe, swarms are fundamentally
non-classical, say, many-valued or even quantum.
Stating differently, the swarm’s motility and self-
organisation are driven by inherently non-classical
logical circuits, or they are classical where non-
classicality is just an external option due to an inter-
pretation. So the fundamental question in this context
would be whether one merely interprets non-classical
gates and the following logic within inherently clas-
sical biological systems or we are really facing true
non-classical realm of the systems which is not re-
ducible to classical one.
We think that this is a highly nontrivial problem to
decide the above reducibility at the fundamental level
which could help understanding the swarm’s intelli-
gence and using it in developing swarm-designed al-
gorithms and software. We motivate our interest in
this issue by analogous problem of existence of hid-
den variables in quantum theory which has marked
significantly the development of quantum mechanics
and science in general, for years. Another motivation
is the special fundamental role of intuitionistic logic
in physical world in general which could also have its
impact on understanding of the swarm activity con-
sidered here, see (Isham, 2000; Kr
ol, 2006; Lands-
man, 2007)
We do not solve the general reducibility problem
here, although, we propose a model shedding sub-
stantial light on the problem. The method we em-
ploy is categorical which means that it relies on find-
ing suitable category K where a swarm W would be
embedded. All categorical constructions referred to
in this paper, like category of sheaves or Yoneda em-
bedding into it, are elementary and a reader can find
them in any of many textbooks from category the-
ory, however, we work here with (MacLane and Mo-
erdijk, 1994) which is certainly more than enough.
Next, our task is to identify categorical construction
responsible for the collective behavior spreading over
regions in W . This is achieved by taking the cate-
gory of presheaves on K, SET
, and Yoneda em-
bedding y : W SET
. The model is simplis-
tic since it treats the excited regions of W as very
regular, however, we think that filtering of this cat-
egorical setup of regular domains through biological
realisations should approach the real behavior. An-
other simplification comes from considering swarms
as pseudo atomistic structure where there is a net of
atoms-nodes connected by actin fibers. Nevertheless
such a presentation is rather generally accepted e.g.
(Galkin et al., 2015; Adamatzky, 2018).
The choice of K is already important and should
Categorical Approach to Swarm Computations
reflect the internal processes of a swarm. For rela-
tively simple swarms like actin filaments or multinu-
cleated giant amoebae one can distinguish nodes con-
nected by links which together constitute the structure
of the swarm. The nodes reaction on stimuli is par-
tially responsible for what is called the intelligent be-
havior of the swarm or its computational power. Thus,
we adopt here the point of view that there are nested
processes already at nodes (or in between two nodes)
which determine the group behavior of swarms with-
out necessity to increase the number of connections.
Such an attitude can be seen as the step supporting the
orchestrated objective reduction hypothesis in cogni-
tive sciences and neural networks stating that our con-
sciousness is the result of deep processes taking place
in neurons rather than due to the myriad of connec-
tions (Schumann, 2019). This hypothesis was formu-
lated in (Hameroff and Penrose, 2014) and it is con-
firmed by our reasoning above that the same computa-
tional power of swarms is observed at different levels
and scales: from actin filament to mammals. In or-
der to fulfil this requirement we are choosing as K the
Turing category of partial computations (Cockett and
Hofstra, 2008). Another reason for the choice of such
K is the attempt to consider classically computations
executed between nodes as morphisms of K. Even
though they are classically defined the entire internal
structure of SET
is intuitionistic based on Heyting
(not Boolean) algebras.
Following the pseudo-atomic model of actin filaments
(Galkin et al., 2015; Adamatzky, 2018), it is repre-
sented by a graph hV, E, Q, f i where V is a set of ver-
tices, E is a set of edges, Q is a set of allowed states,
and f is a transition function switching the nodes
states. f : Q × [0, 1] Q calculates the new state de-
pending on the fraction of excited nodes in around the
given node. Here we adopt the point of view that the
change of the state is accompanied by the change of
a node which mathematically corresponds to a com-
putation process realised in principle by a classical
Turing machine. We do not distinguish here the one
computation over another, this will be left as option
and implemented at further stage via introducing phe-
nomenological parameters. Thus, the state space Q
is eventually extended for the infinite many allowed
states such that they correspond to recursively com-
putable functions (partial recursive) f : N N (or
more generally f : N
The category K representing a swarm W is thus
K = Comp(N) the Turing category (Cockett and Hof-
stra, 2008) which objects are k-tuple products N
morphisms partial recursive functions f : N
Any Turing category is equipped with a Turing ob-
ject (which here is N) and realises a notion of com-
putability which, in a general Turing category, can be
not necessary SET-based. The basic categorical con-
struct behind Turing categories is a partial category
which realises partial concepts of Cartesian closed-
ness or powers (Mulry, 1994). Such a partial category
is Turing if there exists a Turing object in it.
There are defining features of Comp(N) relevant
to the process of representing W by it. Let us discuss
that point more closely. The particular purpose of
the categorification of the simplest biological swarms
(like actin filaments) is
1. considering W as computing system composed of
computing nodes,
2. the computations by nodes are elementary it
is represented by partially recursive functions f :
resulting in changing the states of
3. the partiality enables to take into account algo-
rithms which do not halt on some data and thus
giving no definite result,
4. different algorithms at different nodes can be
5. the collective computations in W leads to the
emergent behavior and swarms computability and
The space of natural numbers N is the common do-
main for all programs and all data on which programs
compute. The reason is G
odel numbering which en-
codes programs as natural numbers and data as natu-
ral numbers and the result of a computation is again
a code which is a natural number. We can effec-
tively enumerate all partial recursive functions (PR).
PR functions are generated by three operations: con-
stant (assigning a constant value a to a set of n vari-
ables), successor (assigning the value x + 1 to a vari-
able x and projection on i-th variable of a set of k vari-
ables). The point is that all Turing computable func-
tions are represented by PR functions. Hence, coding
PR, we code all Turing programs executed on natural
numbers. We do not need to leave the realm of nat-
ural numbers when talking about all possible Turing
computations on all Turing machines.
Let e be a code for some Turing machine (a natural
number). There exists the enumeration of PR func-
tions, {φ
, such that (Cockett and Hofstra, 2008;
Baez, 2019)
T1. The functions φ
, x
, ..., x
) =
(e, x
, x
, ..., x
), n > 0 are PR.
Paradigms-Methods-Approaches 2021 - Workshop on Novel Computational Paradigms, Methods and Approaches in Bioinformatics
T2. There exists a set of PR functions S
, n, m N
such that
(e, x
, ..., x
, v
, ..., v
) =
(e, x
, ..., x
), v
, ..., v
for any Turing machine e and any natural m, n > 0.
T1. is known as an universality property and in fact it
states that N is a universal Turing machine. T2. is the
so -called parameterisation theorem.
A swarm W is now interpreted such that its mem-
bers correspond to objects while morphisms to com-
putations resulting by executing algorithms on data
in nodes which the process computes the change of
a state in the nearby node. The rules for composi-
tion of morphisms in Comp(N) makes the computa-
tions spreading over the entire swarm. This makes
that the point 1. of the categorification procedure is
fulfilled. The point 2. is fulfilled as well. Regard-
ing 3., the partiality of computations reflects the fact
that not all computations halt we do not have gen-
eral algorithms to predict this fact. Thus, a nature
of computability refers to unpredictable halting which
on the level of category theory is encapsulated in the
notion of partial functions and is defined within re-
striction category. One example of a restriction cat-
egory crucial for this paper is our category of N
with partial recursive functions. So, already PR func-
tions contain the halting indeterminacy which reflects
the fundamental fact that primitive recursive functions
are nontrivially extended by PR ones and PR repre-
sent all Turing computations. Thus, 3. is realised by
the model based on Comp(N) which is also true for
4. Namely, different algorithms and Turing machines
are indeed involved in the computations realised in a
swarm represented by Comp(N), since it obviously
holds (Cockett and Hofstra, 2008)
Lemma 1. T1. and T2. hold true in Comp(N).
In the next section we present analysis of the point
5. related to the collective computability.
The reaction of a swarm to external stimuli can be a
source for collective and possibly emergent behavior.
We want to understand this from the category theory
point of view. Let A be an atractant and R a repel-
lent. They lead to the appearance of certain excita-
tions of the swarm which spread over its domains. It
seems natural to think about these excitations as orig-
inated in certain morphisms/computations/change of
the state of a node in W . However, such stimuli are not
in W so there is also another possibility to describe the
following excitations of W as being categorically ex-
ternal to Comp(N). We follow the relation of presheaf
category to the base category in this respect.
Given a category, say K, one can create the cate-
gory of presheaves on it, SET
, where SET is the
category of sets and functions between them and K
is the category opposite to K, i.e. morphisms are
taken with opposite direction than in K and objects
are the same as in K (MacLane and Moerdijk, 1994).
The objects of SET
are thus contravariant func-
tors F : K SET and morphisms are natural transfor-
mations between Fs. Understanding the presheaves
level of swarms refers to the concept of varying sets
and to the internal logic of toposes (MacLane and Mo-
erdijk, 1994).
Given the swarm W spanned on objects-nodes of
Comp(N) and an external stimuli which leads to the
excitations restricted to certain node(s) and connec-
tions, the excited region is travelling within W with
the effect of eventual increasing, deforming or in-
hibitting the excitation area. The regions can then
meet themselves, interact, dissipate etc. (Adamatzky,
2018). We want to understand this process categori-
cally, especially its computational aspect.
There is a natural embedding of K = Comp(N)
into SET
, the Yoneda embedding
y : K SET
Let a
Ob(K), s I be objects and h
Hom(K), p J be morphisms of K. An object a
in K is sent to the functor F
such that
(b) = Hom(b, a) the set of all arrows (morphisms)
from b ending at a. We also say that the functor F
at the stage b K meaning that we consider the set
b. Thus, by changing the stage we have different
sets of arrows within a single presheaf F
Given two presheaves P, F in SET
and a mor-
phism η : F P in SET
the sets on stages a
both functors are related as in the graph in Fig. 1
Figure 1: The natural transformation η on stages a.
Hence, the natural transformation η between 2
functors F and P in SET
is a family of maps be-
tween sets on every stage a: {η
: Fa Pa such
that for every morphism h : a
in K the square
above commutes, i.e. η
F(h) = P(h) η
. Given
two natural transformations η
: F P, η
: P G
there emerges the composition η
: F G which
Categorical Approach to Swarm Computations
is also a natural transformation as a morphism in
. Thus, we have indeed SET
as a cate-
gory which, however, is very rich from the categor-
ical point of view, namely it is a topos. In particu-
lar there always exist exponential objects F
to ev-
ery pair (F, P) of objects (functors or presheaves) in
presheaves category SET
and there always exists
the object of all subobjects (monics) of F, P (F), (here
P means the operation of taking the power object).
Both, exponential and power objects are presheaves,
hence functors, in SET
. Moreover, finite products
and coproducts exist as does the subobject classifier
(MacLane and Moerdijk, 1994).
The sets Hom(b, a), when restricted to W , are
(b, a) W the sets of connections in W .
(b, a) W is not just a set of physical connec-
tions in actin filaments originated in the node a it is
rather a set of possible signals (computations) which
can be sent to a from b through the physical connec-
tions. The physical connections constitute a kind of
skeleton for categorified W .
The functor R
: K SET in SET
sending an
object b to the set in SET of arrows from b ending
at a, Hom(b,a), is particularly important both in cat-
egory theory and for swarms. It is the representable
functor and in fact for any a in K there is such repre-
sentable F
such that the entire collection of F
, a K
defines the Yoneda embedding. Further this repre-
sentability allows for considering excited regions in
W K as sets of morphisms ending at a which could
define the representable subfunctor in SET
. Then
the logic of emergent collective behavior of W fol-
lows the dynamics of excited domains in W which
are grouped in subpresheaves on stages. Stated differ-
ently, the Yoneda embedding
y : W K
defines the representable (sub)functors as governing
excited domains of W . In particular, the intersec-
tions of two colliding domains is given by pullback
in SET
. Let F, P be two presheaves in SET
their corresponding two excited domains at the stage
c K (for a and the representable F = R
this is the
set of arrows from c ending at a), Fc R
c, P
is the representable functor which on stage c reads
c = Hom
(c, a). Then the intersection (pullback)
of F and P, F P, exists in SET
, i.e. the pullback
square below commutes
Now, given excited domain D W such that
b, b D in W (which is the set of excited arrows
in W ending at a in D) we have bijection of Fa with
the set of natural transformations of R
and F. It holds
Theorem 1 (The Yoneda lemma). For an arbitrary
functor F : K
SET and the representable functor
Figure 2: Pullback of F and P in SET
as subobject of
: K
SET , there exists a bijective correspon-
dence at every a (an object in K)
: Nat. trans. (R
, F)
where Fa is a set which is assigned to a by functor F.
So, we can approach the emergent logic of excited
domains in W based on Yoneda embedding and in-
ternal logic of SET
. Given two excited domains,
, D
by certain external stimuli s
, s
, the domains
can interact, e.g. they collide, dissipate, inhibit etc.
which leads to further deformations and generating a
resulting domain D
such that the logical functions
are defined on D
, i = 1, 2, 3. For example, one can de-
fine conjunction of D
and D
which is the pullback
. We consider the domains as sets of arrows
which are precisely subfunctors in the presheaves cat-
egory evaluated at certain stages a, b. The point is
that the category SET
is usually much more com-
plete than K itself and, e.g., the pullbacks for sure ex-
ist in SET
as there exist exponents and finite lim-
its and colimits. Moreover, usually the sheaf cate-
gory contains also richer partiality structure that the
base category K (Mulry, 1994) which means that the
computability space of SET
is also richer. As
a consequence, the general conclusion is that when-
ever external stimuli cause the excitations of D
, D
such that they have components in SET
, the logic
of such collective processes is the internal logic of
which is intuitionistic, i.e. neither using set
theoretic axiom of choice nor the logical principle
of the excluded middle are allowed. The model pre-
sented here indicates also that the emergent collective
computability of swarms can differ from the set based
computability in K. Both the properties above will be
more thoroughly analysed in a separate publication.
There are many algorithms developed for explicating
the swarm behavior from the swarm motility of birds
and horses within the Particle Swarm Optimization
(Kennedy and Eberhart, 1995; Wang et al., 2011) to
coworking of ants within the Ant Colony Optimiza-
tion (Dorigo and Stutzle, 2004), bees within the Ar-
tificial Bee Colony Algorithm (Karaboga and Akay,
Paradigms-Methods-Approaches 2021 - Workshop on Novel Computational Paradigms, Methods and Approaches in Bioinformatics
2009) and within the Bees Algorithm for General-
ized Assignment Problem (Ozbakir et al., 2010), and
many others. There are defined different logic circuits
on the basis of different swarms: ants (Coello Coello
et al., 2000), bees (Mollabakhshi and Eshghi, 2013),
slime mould (Adamatzky, 2010; Schumann, 2019),
etc. Nevertheless, there is no general theory of
swarm computation which would summarise all the
approaches. In other words, there is no ‘metamath-
ematics’ or ‘foundations’ of swarm intelligence. In
the research programme of (Aczel et al., 2013) in the
foundations of mathematics, there was proposed ho-
motopic type theory as ultimate mathematical foun-
dations. In our approach, we assume that within this
programme we can also identify isomorphic compu-
tational structures to define types and their hierar-
chies of different chemical and biological systems as
substrates of swarm computing. In order to fulfill
this task, we have started with defining categories on
swarms. Presumably in the course of defining suit-
able mathematical structures behind various phenom-
ena realised by intelligent swarms we need certain
modifications of toposes, e.g. (Asselmeyer-Maluga
and Kr
ol, 2019). It is our preliminary result and rather
draft in developing ‘foundations’ of swarm intelli-
The proposed categorical model for swarm com-
putability and collective behavior indicates that the in-
trinsic logic of such swarm phenomena has to be in-
tuitionistic. The particular case of intuitionistic logic
is Boolean logic encompassing multivalued (also in-
finite many) Boolean logic, since Heyting algebras
on which toposes are built on are generalisations of
Boolean algebras. Deciding up to what extent the
appearance of the intuitionistic logic is generic for
swarm intelligence in general, requires further stud-
ies which would contain also the detail development
of the scenario proposed here.
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