Categorical Approach to Swarm Computations

Jerzy Kr

´

ol

a

, Andrew Schumann

b

and Krzysztof Bielas

c

Chair of Cognitive Science and Mathematical Modelling, University of Information Technology and

Management in Rzesz

´

ow, Sucharskiego 2, 35-225 Rzesz

´

ow, Poland

Keywords:

Swarm Intelligence, Category Theory, Yoneda Embedding, Emergency.

Abstract:

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.

1 INTRODUCTION

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,

1969).

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.

a

https://orcid.org/0000-0002-7296-7355

b

https://orcid.org/0000-0002-9944-8627

c

https://orcid.org/0000-0003-3259-7676

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-

218

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

Copyright

c

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

K

op

, and Yoneda em-

bedding y : W → SET

K

op

. 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

219

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

K

op

is intuitionistic based on Heyting

(not Boolean) algebras.

2 SWARM AS A CATEGORY

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

k

→ N

m

).

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

k

and

morphisms partial recursive functions f : N

k

→ N

m

.

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 :

N

k

→ N

m

– resulting in changing the states of

nodes,

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

used,

5. the collective computations in W leads to the

emergent behavior and swarms computability and

‘intelligence’.

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

i

}

i∈N

, such that (Cockett and Hofstra, 2008;

Baez, 2019)

T1. The functions φ

e

(x

1

, x

2

, ..., x

n

) =

:

Φ

(n)

(e, x

1

, x

2

, ..., x

n

), n > 0 are PR.

Paradigms-Methods-Approaches 2021 - Workshop on Novel Computational Paradigms, Methods and Approaches in Bioinformatics

220

T2. There exists a set of PR functions S

n

m

, n, m ∈ N

such that

Φ

(n+m)

(e, x

1

, ..., x

m

, v

1

, ..., v

n

) =

Φ

(m)

(S

n

m

(e, x

1

, ..., x

m

), v

1

, ..., v

n

).

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

k

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.

3 EMERGENT PHENOMENA IN

THE CATEGORICAL SWARMS

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

K

op

, where SET is the

category of sets and functions between them and K

op

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

K

op

are thus contravariant func-

tors F : K → SET and morphisms are natural transfor-

mations between F’s. 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

K

op

, the Yoneda embedding

y : K → SET

K

op

.

Let a

s

∈ Ob(K), s ∈ I be objects and h

p

∈

Hom(K), p ∈ J be morphisms of K. An object a

in K is sent to the functor F

a

∈ SET

K

op

such that

F

a

(b) = Hom(b, a) – the set of all arrows (morphisms)

from b ending at a. We also say that the functor F

a

is

at the stage b ∈ K meaning that we consider the set

F

a

b. Thus, by changing the stage we have different

sets of arrows within a single presheaf F

a

.

Given two presheaves P, F in SET

K

op

and a mor-

phism η : F → P in SET

K

op

the sets on stages a

s

in

both functors are related as in the graph in Fig. 1

Fa

1

η

a1

//

F(h)

Pa

1

P(h)

Fa

2

η

a2

//

Pa

2

Figure 1: The natural transformation η on stages a.

Hence, the natural transformation η between 2

functors F and P in SET

K

op

is a family of maps be-

tween sets on every stage a: {η

a

}

a∈K

: Fa → Pa such

that for every morphism h : a

1

→ a

2

in K the square

above commutes, i.e. η

a2

◦ F(h) = P(h) ◦ η

a1

. Given

two natural transformations η

1

: F → P, η

2

: P → G

there emerges the composition η

2

◦η

1

: F → G which

Categorical Approach to Swarm Computations

221

is also a natural transformation as a morphism in

SET

K

op

. Thus, we have indeed SET

K

op

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

P

to ev-

ery pair (F, P) of objects (functors or presheaves) in

presheaves category SET

K

op

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

K

op

. 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

Hom

W

(b, a) ⊂ W – the sets of connections in W .

Hom

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

a

: K → SET in SET

K

op

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

a

such that the entire collection of F

a

, 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

K

op

. 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

y

→ SET

K

op

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

K

op

. Let F, P be two presheaves in SET

K

op

and

their corresponding two excited domains at the stage

c ∈ K (for a and the representable F = R

a

this is the

set of arrows from c ending at a), Fc ⊂ R

a

c, P

a

⊂ R

a

c.

R

a

is the representable functor which on stage c reads

R

a

c = Hom

K

(c, a). Then the intersection (pullback)

of F and P, F ∩ P, exists in SET

K

op

, i.e. the pullback

square below commutes

Now, given excited domain D ⊂ W such that

R

a

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

a

and F. It holds

Theorem 1 (The Yoneda lemma). For an arbitrary

functor F : K

op

→ SET and the representable functor

F ∩ P

//

P

m2

F

m2

//

R

a

Figure 2: Pullback of F and P in SET

K

op

as subobject of

R

a

.

R

a

: K

op

→ SET , there exists a bijective correspon-

dence at every a (an object in K)

Θ

F,R

: Nat. trans. (R

a

, F)

'

→ Fa

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

K

op

. Given two excited domains,

D

1

, D

2

by certain external stimuli s

1

, s

2

, the domains

can interact, e.g. they collide, dissipate, inhibit etc.

which leads to further deformations and generating a

resulting domain D

3

such that the logical functions

are deﬁned on D

i

, i = 1, 2, 3. For example, one can de-

ﬁne conjunction of D

1

and D

2

which is the pullback

D

1

∩ D

2

. 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

K

op

is usually much more com-

plete than K itself and, e.g., the pullbacks for sure ex-

ist in SET

K

op

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

K

op

is also richer. As

a consequence, the general conclusion is that when-

ever external stimuli cause the excitations of D

1

, D

2

such that they have components in SET

K

op

, the logic

of such collective processes is the internal logic of

SET

K

op

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.

4 CONCLUSION

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

222

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-

gence.

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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