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Fungal Automata

Andrew Adamatzky1, Eric Goles1,2, Genaro J. Mart´ınez1,3, Michail-Antisthenis

Tsompanas1, Martin Tegelaar4, and Han A. B. Wosten4

1Unconventional Computing Laboratory, UWE, Bristol, UK

2Faculty of Engineering and Science, University of Adolfo Ib´a˜nez, Santiago, Chile

3High School of Computer Science, National Polytechnic Institute, Mexico

4Microbiology Department, University of Utrecht, Utrecht, The Netherlands

Abstract

We study a cellular automaton (CA) model of information dynamics on a single hypha of a fungal

mycelium. Such a ﬁlament is divided in compartments (here also called cells) by septa. These

septa are invaginations of the cell wall and their pores allow for ﬂow of cytoplasm between com-

partments and hyphae. The septal pores of the fungal phylum of the Ascomycota can be closed

by organelles called Woronin bodies. Septal closure is increased when the septa become older

and when exposed to stress conditions. Thus, Woronin bodies act as informational ﬂow valves.

The one dimensional fungal automata is a binary state ternary neighbourhood CA, where every

compartment follows one of the elementary cellular automata (ECA) rules if its pores are open

and either remains in state ‘0’ (ﬁrst species of fungal automata) or its previous state (second

species of fungal automata) if its pores are closed. The Woronin bodies closing the pores are also

governed by ECA rules. We analyse a structure of the composition space of cell-state transition

and pore-state transitions rules, complexity of fungal automata with just few Woronin bodies, and

exemplify several important local events in the automaton dynamics.

Keywords: fungi, ascomycete, Woronin body, cellular automata

1 Introduction

The fungal kingdom represents organisms colonising all ecological niches [11] where they play a key role

[18, 15, 35, 12]. Fungi can consist of a single cell, can form enormous underground networks [38] and

can form microscopic fruit bodies or fruit bodies weighting up to half a ton [16]. The underground

mycelium network can be seen as a distributed communication and information processing system

linking together trees, fungi and bacteria [10]. Mechanisms and dynamics of information processing in

mycelium networks form an unexplored ﬁeld with just a handful of papers published related to space

exploration by mycelium [20, 19], patterns of electrical activity of fungi [37, 34, 1] and potential use

of fungi as living electronic and computing devices [2, 3, 4].

Filamentous fungi grow by means of hyphae that grow at their tip and that branch sub-apically.

Hyphae may be coenocytic or divided in compartments by septa. Filamentous fungi in the phylum

Ascomycota have porous septa that allow for cytoplasmic streaming [31, 24]. Woronin bodies plug

the pores of these septa after hyphal wounding to prevent excessive bleeding of cytoplasm [44, 13, 22,

42, 39, 29]. In addition, they plug septa of intact growing hyphae to maintain intra- and inter-hyphal

heterogeneity [8, 7, 40, 40, 41].

Woronin bodies can be located in diﬀerent hyphal positions (Fig. 1a). When ﬁrst formed, Woronin

bodies are generally localised to the apex [30, 43, 5]. Subsequently, Woronin bodies are either trans-

ported to the cell cortex (Neurospora crassa,Sordaria ﬁmicola) or to the septum (Aspergillus oryzae,

1

arXiv:2003.08168v1 [nlin.CG] 18 Mar 2020

Septum associated

Woronin body

~50um

septum leashin

Cytoplasmic

Woronin body

(a)

`

`

`

`

`

`

1 1 0 0 1 0 0 1 0

(b)

Figure 1: (a) A biological scheme of a fragment of a fungal hypha of an ascomycete, where we can see

septa and associated Woronin bodies. (b) A scheme representing states of Woronin bodies: ‘0’ open,

‘1’ closed.

Aspergillus nidulans,Aspergillus fumigatus,Magnaporthe grisea,Fusarium oxysporum,Zymoseptoria

tritici) where they are anchored with a leashin tether and largely immobile until they are translocated

to the septal pore due to cytoplasmic ﬂow or ATP depletion [33, 40, 39, 30, 29, 43, 45, 23, 6]. Woronin

bodies that are not anchored at the cellular cortex or the septum, are located in the cytoplasm and

are highly mobile (Aspergillus fumigatus,Aspergillus nidulans,Zymoseptoria tritici) [5, 30, 40]. Septal

pore occlusion can be induced by bulk cytoplasmic ﬂow [40] or developmental [9] and environmental

cues, like puncturing of the cell wall, high temperature, carbon and nitrogen starvation, high osmolar-

ity and low pH. Interestingly, high environmental pH reduces the proportion of occluded apical septal

pores [41].

Aiming to lay a foundation of an emerging paradigm of fungal intelligence — distributed sensing

and information processing in living mycelium networks — we decided to develop a formal model of

mycelium and investigate a role of Woronin bodies in potential information dynamics in the mycelium.

The paper is structured as follows. We introduce fungal automata in Sect. 2. Properties of the

composition of cell state transition and Woronin body state transition functions are studied in Sect. 3.

Complexity of space-time conﬁguration of fungal automata, where just few cells have Woronin bodies is

studied in Sect. 4. Section 5 exempliﬁes local events, which could be useful for computation with fungal

automata, happening in fungal automata with sparsely but regularly positioned cells with Woronin

bodies. The paper concludes with Sect. 6.

2 Fungal automata M

A fungal automaton is a one-dimensional cellular automaton with binary cell states and ternary,

including central cell, cell neighbourhood, governed by two elementary cellular automata (ECA) rules,

namely the cell state transition rule fand the Woronin bodies adjustment rule g:M=hN, u, Q, f, gi.

Each cell xihas a unique index i∈N. Its state is updated from Q={0,1}in discrete time depending of

its current state xt

i, the states of its left xt

i−1and right neighbours xt

i+1 and the state of cell x’s Woronin

body w. Woronin bodies take states from Q:wt= 1 means Woronin bodies (Fig. 1) in cell xblocks

the pores and the cell has no communication with its neighbours, and wt= 0 means that Woronin

bodies in cell xdo not block the pores. Woronin bodies update their states g(·), wt+1 =g(u(x)t),

depending on the state of neighbourhood u(x)t. Cells xupdate their states by function f(·) if their

Woronin bodies do not block the pores.

Two species of mycelium automata are considered M1, where each cell updates its state as following:

xt+1 =(0 if wt= 1

f(u(x)t) otherwise

2

(a) M1, ρf= 133, ρg= 116 (b) M2, ρf= 133, ρg= 116 (c) M1, ρf= 73, ρg= 128

(d) M2, ρf= 73, ρg= 128 (e) M1, ρf= 61, ρg= 132 (f) M2, ρf= 61, ρg= 132

(g) M1, ρf= 57, ρg= 98 (h) M2, ρf= 57, ρg= 98 (i) M1, ρf= 26, ρg= 84

(j) M2, ρf= 26, ρg= 84 (k) M1, ρf= 125, ρg= 105 (l) M2, ρf= 125, ρg= 105

Figure 2: Examples of space-time dynamics of M. The automata are 103cells each. Initial conﬁgu-

ration is random with probability of a cell xto be in state ‘1’, x0= 1, equals 0.01. Each automaton

evolved for 103iterations. Binary values of ECA rules fand gare shown in sub-captions. Rule g

is applied to every iteration starting from 200th. Cells in state ‘0’ are white, in state ‘1’ are black,

cells with Woronin bodies blocking pores are red. Indexes of cells increase from the left to the right,

iterations are increasing from the to the bottom.

3

and M2where each cell updates its state as following:

xt+1 =(xtif wt= 1

f(u(x)t) otherwise

where wt=g(u(x)t).

State ‘1’ in the cells of array xsymbolises metabolites, signals exchanged between cells. Where

pores in a cell are open the cell updates its state by ECA rule f:{0,1}3→ {0,1}.

In automaton M1, when Woronin bodies block the pores in a cell, the cell does not update its

state and remains in the state ‘0’ and left and right neighbours of the cells can not detect any ‘cargo’

in this cell. In automaton, M2, where Woronin bodies block the pores in a cell, the cell does not

update its state and remains in its current state. In real living mycelium glucose and possibly other

metabolites [7] can still cross the septum even when septa are closed by Woronin bodies, but we can

ignore this fact in present abstract model.

Both species are biologically plausible and, thus, will be studied in parallel. The rules for closing

and opening Woronin bodies are also ECA rules g:{0,1}3→ {0,1}. If g(u(x)t) = 0 this means that

pores are open, if g(u(x)t) = 1 Woronin bodies block the pores. Examples of space-time conﬁgurations

of both species of Mare shown in Fig. 2.

3 Properties of composition f◦g

Predecessor sets

Let F={h:{0,1}3→ {0,1}} be a set of all ECA functions. Then for any composition f◦g,

where f, g ∈F, can be converted to a single function h∈F. For each h∈Fwe can construct a set

P(h) = {f◦g∈F×F|f◦g→h}. The sets P(h) for each h∈Fare available online1.

A size of P(h) for each his shown in Fig. 3c. The functions with largest size of P(h) are rule 0 in

automaton M1and rule 51 (only neighbourhood conﬁgurations (010, 011, 110, 111 are mapped into

1) in M2.

Size σof P(h) vs a number γof functions hhaving set P(h) of size σis shown for automata M1

and M2in Table 1a.

With regards to Wolfram classiﬁcation [47], sizes of P(h) for rules from Class III vary from 9 to

729 in M1(Tab. 1b). Rule 126 would be the most diﬃcult to obtain in M1by composition two ECA

rules chosen at random, it has only 9 ‘predecessor’ f◦gpairs. Rule 18 would be the easiest, for Class

III rules, to be obtained, it has 729 predecessors, in both M1(Tab. 1b) and M2(Tab. 1d). In M1,

one rule, rule 41, from the class IV has 243 f◦gpredecessors, and all other rules in that class have

81 (Tab. 1c). From Class IV rule 54 has the largest number of predecessors in M2, and thus can be

considered as most common (Tab. 1d).

Diagonals

In automaton M1for any f∈Ff◦f= 0. Assume f:{0,1}3→1 then Woronin bodies close the

pores and, thus, second application of fproduces state ‘0’. If f:{0,1}3→0 then Woronin bodes

does not close pores but yet second application of the fproduce state ‘0’.

For automaton M2a structure of diagonal mapping f◦f→h, where f, h ∈Fis shown in

Tab. 2. The set of the diagonal outputs f◦fconsists of 16 rules: (0, 1, 2, 3), (16, 17, 18, 19),

(32, 33, 34, 35), (48, 49, 40, 51). These set of rules can be reduced to the following rule. Let

C(xt) = [u(x)t= (111)] ∨[u(x)t= (111)] and B(xt) = [u(x)t= (011)] ∨[u(x)t= (010)]. Then xt= 1

if C(x)t∨C(x)t∧B(xt).

1https://figshare.com/s/b7750ee3fe6df7cbe228

4

(a) (b)

κ(ρf)

0

500

1000

1500

2000

ρf

0 20 40 60 80 100 120 140 160 180 200 220 240

(c)

Figure 3: Mapping F×F→Ffor automaton M1(a) and M2(b) is visualised as an array of pixels,

P= (p)0≤ρf≤255,0≤ρf≤255. An entry at the intersection of any ρfand ρgis a coloured as follows: red

if pρfρg=pρgρf, blue if ρg=pρgρf, green if ρf=pρgρf. (c) Sizes of P(h) sets for M1, circle, and M2,

solid discs, are shown for every function hapart of rule 0 (M1) and rule 51 (M2).

5

(a) Rules per

|P(h)|

σ γ

1 1

3 8

9 28

27 56

81 70

243 56

729 28

2187 8

6561 1

(b) M1: Class III rules

Rule σ

18 729

22, 146 243

30, 45, 60,

90, 105, 150

81

122 27

126 9

(c) M1: Class IV rules

Rule σ

41 243

54, 106, 110 81

(d) M2: Class III rules

Rule σ

18 729

22, 146 243

30, 45, 60 90,

105, 150

81

122 243

126 81

(e) M2: Class IV rules

Rule σ

41 243

54 729

106 81

110 27

Table 1: Characterisations of automaton mapping F×F→F. (a) Size σof P(h) vs a number γ

of functions hhaving set P(h) of size σ. T (b) Sizes of sets P(h) for rules from Wolfram class III.

(b) Sizes of sets P(h) for rules from Wolfram class IV.

f◦f f

0 0, 1, 2, 3, 16, 17, 18, 19, 32, 33, 34, 35, 48, 49, 50, 51

1 128, 129, 130, 131, 144, 145, 146, 147, 160, 161, 162, 163, 176, 177, 178, 179

2 64, 65, 66, 67, 80, 81, 82, 83, 96, 97, 98, 99, 112, 113, 114, 115

3 192, 193, 194, 195, 208, 209, 210, 211, 224, 225, 226, 227, 240, 241, 242, 243

16 8, 9, 10, 11, 24, 25, 26, 27, 40, 41, 42, 43, 56, 57, 58, 59

17 136, 137, 138, 139, 152, 153, 154, 155, 168, 169, 170, 171, 184, 185, 186, 187

18 72, 73, 74, 75, 88, 89, 90, 91, 104, 105, 106, 107, 120, 121, 122, 123

19 200, 201, 202, 203, 216, 217, 218, 219, 232, 233, 234, 235, 248, 249, 250, 251

32 4, 5, 6, 7, 20, 21, 22, 23, 36, 37, 38, 39, 52, 53, 54, 55

33 132, 133, 134, 135, 148, 149, 150, 151, 164, 165, 166, 167, 180, 181, 182, 183

34 68, 69, 70, 71, 84, 85, 86, 87, 100, 101, 102, 103, 116, 117, 118, 119

35 196, 197, 198, 199, 212, 213, 214, 215, 228, 229, 230, 231, 244, 245, 246, 247

48 12, 13, 14, 15, 28, 29, 30, 31, 44, 45, 46, 47, 60, 61, 62, 63

49 140, 141, 142, 143, 156, 157, 158, 159, 172, 173, 174, 175, 188, 189, 190, 191

50 76, 77, 78, 79, 92, 93, 94, 95, 108, 109, 110, 111, 124, 125, 126, 127

51 204, 205, 206, 207, 220, 221, 222, 223, 236, 237, 238, 239, 252, 253, 254, 255

Table 2: Diagonals of automaton M2.

6

Commutativity

In automaton M1, for any f, g ∈Ff◦g6=g◦fonly if f6=g. In automaton M2there are 32768

pairs of function which ◦is commutative, their distribution visualised in red in Fig. 3b.

Identities and zeros

In automaton M1there are no left or right identities, neither right zeros in hF,F,◦i. The only left

zero is the rule 0. In automaton M2there are no identities or zeros at all.

Associativity

In automaton M1there 456976 triples hf, g, hion which operation ◦is associative: (f◦g)◦h=

f◦(g◦h). The ratio of associative triples to the total number of triples is 0.027237892. There are

104976 associative triples in M2, a ratio of 0.006257057.

4 Tuning Complexity: Rule 110

To evaluate on how introduction of Woronin bodies could aﬀect complexity of automaton evolution,

we undertook two series of experiments. In the ﬁrst series we used fungal automaton where just one

cell has a Woronin body (Fig. 5). In the second series we employed fungal automaton where regularly

positioned cells (but not all cells of the array) have Woronin bodies.

State transition functions gof Woronin bodies were varied across the whole diapason but the state

transition function fof a cell was Rule 110, ρf= 110. We have chosen Rule 110 because the rule is

proven to be computationally universal [25, 14], P-complete [32], the rules belong to Wolfram class

IV renown for exhibiting complex and non-trivial interactions between travelling localisation [46], rich

families of gliders can be produce in collisions with other gliders [26, 27, 28].

We wanted to check how an introduction of Woronin bodies aﬀect dynamics of most complex

space-time developed of Rule 110 automaton. Thus, we evolved the automata from all possible initial

conﬁgurations of 8 cells placed near the end of n= 1000 cells array of resting cells and allowing

to evolve for 950 iterations. Lempel–Ziv complexity (compressibility) LZ was evaluated via sizes of

space-time conﬁgurations saved as PNG ﬁles. This is suﬃcient because the ’deﬂation’ algorithm used

in PNG lossless compression [36, 21, 17] is a variation of the classical Lempel–Ziv 1977 algorithm [48].

Estimates of LZ complexity for each of 8-cell initial conﬁgurations are shown in Fig. 4a. The initial

conﬁgurations with highest estimated LZ complexity are 10110001 (decimal 177), 11010001 (209),

10000011 (131), 11111011 (253), see example of space-time dynamics in Fig. 4b.

We assumed that a cell in the position n−100 has a Woronin body which can be activated (Fig. 5),

i.e. start updating its state by rule f, after 100th iteration of the automaton evolution. We then

run 950 iteration of automaton evolution for each of 256 Woronin rules and estimated LZ complexity.

In experiments with M1we found that 128 rules, with even decimal representations, do not aﬀect

space time dynamics of evolution and 128 rules, with even decimal representations, reduce complexity

of the space-time conﬁguration. The key reasons for the complexity reduction (compare Fig. 4b and

c) are cancellation of three gliders at c. 300th iteration and simpliﬁcation of the behaviour of glider

guns positioned at the tail of the propagating wave-front. In experiments with M2128 rules, with

even decimal representations, do not change the space-time conﬁguration of the author. Other 128

rules reduce complexity and modify space-time conﬁguration by re-arranging the structures of glider

guns and establishing one oscillators at the site surrounding position of the cell with Woronin body

(Fig. 4d).

In second series of experiments we regularly positioned cells with Woronin bodies along the 1D

array: every 50th cell has a Woronin body. Then we evolved fungal automata M1and M2from

exactly the same initial random conﬁguration with density of ‘1’ equal to 0.3. Space-time conﬁguration

of the automaton without Woronin bodies is shown in Fig. 6a. Exemplar of space-time conﬁgurations

7

Size, kB

20

30

40

50

60

70

Initial conﬁguration, decimal encoding

0 50 100 150 200 250

(a)

(b) (c)

(d)

Figure 4: (a) Estimates of LZ complexity of space-time conﬁgurations of ECA Rule 110 without

Woronin bodies. (b) A space-time conﬁguration of ECA Rule 110 evolving from initial conﬁguration

10110001 (177), no Woronin bodies are activated. (c) A space-time conﬁguration of M1Rule 110

evolving from initial conﬁguration 10110001 (177), Woronin body is governed by rule 43; red lines

indicate time when the body was activated and position of the cell with the body. In (bcd), a pixel in

position (i, t) is black if xt

i= 1.

8

`

`

`

`

`

`

1 1 0 0 1 0 0 1 0

Figure 5: Only one cell has Woronin body.

of automata with Woronin bodies are shown in Fig. 6b–h. As seen in Fig. 7 both species of fungal

automata show similar dynamics of complexity along the Woronin transition functions ordered by their

decimal values. The automaton M1has average LZ complexity 82.2 (σ= 24.6) and the automaton

M278.4 (σ= 22.1). Woronin rules gwhich generate most LZ complex space-time conﬁgurations are

ρg= 133 in M)1(Fig. 6b) and ρg= 193 in M)2(Fig. 6e). The space-time dynamics of the automaton

is characterised by a substantial number of gliders guns and gliders (Fig. 6b). Functions being in the

middle of the descending hierarchy of LZ complexity produce space-time conﬁgurations with declined

number of travelling localisation and growing domains of homogeneous states (Fig. 6cg). Automata

with Woronin functions at the bottom of the complexity hierarchy quickly (i.e. after 200-300 iterations)

evolve towards stable, equilibrium states (Fig. 6dh).

5 Local events

Let us consider some local events happening in the fungal automata discussed in Sect. 4: every 50th

cell of an array has a Woronin body.

Retaining gliders. A glider can be stopped and converted into a station localisation by a cell

with Woronin body. As exempliﬁed in Fig. 8a, the localisation travelling left was stopped from further

propagation by a cell with Woronin body yet the localisation did not annihilate but remained stationary.

Register memory. Diﬀerent substrings of input string (initial conﬁguration) might lead to diﬀer-

ent equilibrium conﬁgurations achieved in the domains of the array separated by cells with Woronin

bodies. When there is just two types of equilibrium conﬁgurations they be seen as ‘bit up’ and ‘bit

down’ and therefore such fungal automaton can be used a memory register (Fig. 8b).

Reﬂectors. In many cases cells with Woronin bodies induce local domains of stationary, sometimes

time oscillations, inhomogeneities which might act as reﬂectors for travelling localisations. An example

is shown in Fig. 8c where several localisations are repeatedly bouncing between two cells with Woronin

bodies.

Modiﬁers. Cells with Woronin bodies can act as modiﬁers of states of gliders reﬂected from them

and of outcomes of collision between travelling localizations. In Fig. 8d we can see how a travelling

localisation is reﬂected from the vicinity of Woronin bodies three times: every time the state of the

localisation changes. On the third reﬂection the localisation becomes stationary. In the fragment

(Fig. 8e) of space-time conﬁguration of automaton with Woronin bodies governed by ρg= 201 of the

fragment we can see how two localisations got into contact with each in the vicinity of the Woronin

body and an advanced structure is formed two breathing stationary localisations act as mirror, and

there are streams of travelling localisations between them. A multi-step chain reaction can be observed

in Fig. 8f: there are two stationary, breathing, localisations at the sites of the cells with Woronin bodies.

A glider is formed on the left stationary localisation. The glider travel to the right and collide into

right breather. In the result of the collision the breath undergoes structural transitions, emits a glider

travelling left and transforms itself into a pair of stationary breathers. Meantime the newly born glider

collided into left breather and changes its state.

6 Discussion

As a ﬁrst step towards formalisation of fungal intelligence we introduced one-dimensional fungal au-

tomata operated by two local transition function: one, g, governs states of Woronin bodies (pores are

open or closed), another, f, governs cells states: ‘0’ and ‘1’. We provided a detailed analysis of the

9

(a) (b) M1,ρg= 133 (c) M1,ρg= 29 (d) M1,ρg= 49

(e) M2,ρg= 193 (f) M2,ρg= 5 (g) M2,ρg= 221 (h) M2,ρg= 174

Figure 6: (a) ECA Rule 110, no Woronin bodies. Space-time evolution of M∞(bcd) and M∈(e–h)

for Woronin rules shown in subcaption. LZ complexity of space-time conﬁgurations decreases from (b)

to (d) and from (e) to (h). Every 50th cell has a Woronin body.

10

Size, kB

40

60

80

100

120

140

Woronin rule ρg, decimal encoding

0 50 100 150 200 250

Figure 7: Estimations of LZ complexity of space-time, 500 cells by 500 iterations, conﬁgurations of

M1, discs, and M1, circles, for all Woronin functions g.

(a) M1,

ρg= 2

(b) M1,ρg= 15 (c) M1,

ρg= 21

(d) M1,

ρg= 31

(e) M1,ρg=

29

(f) M1,

ρg= 201

Figure 8: (a) Localisation travelling left was stopped by the Woronin body. (b) Analog of a memory

register. (c) Reﬂections of travelling localisations from cells with Woronin bodies. (d) Modiﬁcation of

glider state in the vicitinity of Woronin bodies. (e) A fragment of conﬁguration of automaton with

ρg= 29, left cell states, right Woronin bodies states. (f) Enlarged sub-fragment of the fragment (d)

where Wonorin body tunes the outcome of the collision. For both automata ρf= 110.

11

Figure 9: An example of 5-inputs-7-outputs collision in M2,ρf= 110, ρg= 40. Every 50th cell has a

Woronin body. Cells state transitions are shown on the left, Woronin bodies state transitions on the

right. A pixel in position (i, t) is black if xt

i= 1, left, or wt

i= 1, right.

12

magma hf, g, ◦i, results of which might be useful for future designs of computational and language

recognition structures with fungal automata. The magma as a whole does not satisfy any other prop-

erty but closure. Chances are high that there are subsets of the magma which might satisfy conditions

of other algebraic structures. A search for such subsets could be one of the topics of further studies.

Another topic could be an implementation of computational circuits in fungal automata. For

certain combination of fand gwe can ﬁnd quite sophisticated families of stationary and travelling

localisations and many outcomes of the collisions and interactions between these localisations, an

illustration is shown in Fig. 9. Thus the target could be, for example, to construct a n-binary full

adder which is as compact in space and time as possible.

The theoretical results reported show that by controlling just a few cells with Woronin bodies it is

possible to drastically change dynamics of the automaton array. Third direction of future studies could

be in implemented information processing in a single hypha. In such a hypothetical experimental setup

input strings will be represented by arrays of illumination and outputs could be patterns of electrical

activity recorded from the mycelium hypha resting on an electrode array.

Acknowledgement

AA, MT, HABW have received funding from the European Union’s Horizon 2020 research and in-

novation programme FET OPEN “Challenging current thinking” under grant agreement No 858132.

EG residency in UWE has been supported by funding from the Leverhulme Trust under the Visiting

Research Professorship grant VP2-2018-001.

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