Information theory, predictability, and the emergence of complex life

1.8 MB·

by  Luís F Seoane

Information theory, predictability, and the emergence of complex life
Lu´ıs F Seoane1,2,3 and Ricard V. Sol´e2,3,4
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.
2ICREA-Complex Systems Lab, Universitat Pompeu Fabra (GRIB), Dr Aiguader 80, 08003 Barcelona, Spain.
3Institut de Biologia Evolutiva, CSIC-UPF, Pg Maritim de la Barceloneta 37, 08003 Barcelona, Spain.
4Santa Fe Institute, 1399 Hyde Park Road, Santa Fe NM 87501, USA.
Despite the obvious advantage of simple life forms capable of fast replication, different levels
of cognitive complexity have been achieved by living systems in terms of their potential to cope
withenvironmentaluncertainty. Againsttheinevitablecostassociatedtodetectingenvironmental
cuesandrespondingtotheminadaptiveways,weconjecturethatthepotentialforpredictingthe
environmentcanovercometheexpensesassociatedtomaintainingcostly,complexstructures. We
presentaminimalformalmodelgroundedininformationtheoryandselection,inwhichsuccessive
generationsofagentsaremappedintotransmittersandreceiversofacodedmessage. Ouragents
areguessingmachinesandtheircapacitytodealwithenvironmentsofdifferentcomplexitydefines
theconditionstosustainmorecomplexagents.
7 Keywords: Complexity,emergence,computation,evolution,predictability
1
0
2 I. INTRODUCTION automatic forecasting apparatus” (Friston, 2013; Jacob,
1998;Wagensberg,2000). Themainmessageisthatfore-
n
Simplelifeformsdominateourbiosphere(Gould,2011) seeing the future is a crucial ability to cope with un-
a
J and define a lower bound of embodied, self-replicating certainty. If the advantages of prediction overcome the
systems. But life displays an enormously broad range of problem of maintaining and replicating the costly struc-
9
complexitylevels,affectingmanydifferenttraitsofliving tures needed for inference, more complex information-

] entities, from their body size to their cognitive abilities processing mechanisms might be favored under the ap-
E
(Bonner, 1988). This creates somewhat a paradox: if propriate circumstances.
P
larger, more complex organisms are more costly to grow
.
o andmaintain,whyisnotalllifesingle-celled? Severalar- Here we aim at providing a minimal model that cap-
i guments help provide a rationale for the emergence and tures these tradeoffs. In doing so, we characterize thor-
b
persistenceofcomplexlifeforms. Asaninstance(Gould, oughly an evolutionary driver that can push towards ev-
-
q 2011) proposes that complexity is not a trait explicitly ermorecomplexlifeforms. Weadoptaninformationthe-
[ favored by evolution. A review of fossil records con- ory perspective in which agents are inference devices in-

1 vinces Gould that, across genera, phyla, and the whole teracting with a Boolean environment. For convenience,
v biosphere, we observe the expected random fluctuations this environment is represented by a tape with ones and
9 aroundthemoresuccessfuladaptationtolife. Inthisbig zeros,akintonon-emptyinputsofaTuringmachine(fig-
8 picture, bacteria are the leading life form and the com- ure1a). TheagentGlocatesitselfinagivenpositionand
3
plexity of every other living system is the product of a triestopredicteachbitofagivensequenceoflengthn–
2
random drift. Complex life would never be explicitly fa- henceitisdubbedann-guesser. Eachattempttopredict
0
vored, but a complexity wall exists right below bacteria: a bit involves some cost c, while a reward r is received
.
1 simpler forms fail to subsist. Hence, a random fluctua- for each successful prediction. 1-guessers are simple and
0
tionismorelikelytoproducemorecomplexforms,falsely assume that all bits are uncorrelated, while (n > 1)-
7
suggesting that evolution promotes complexity. guessers find correlations and can get a larger benefit
1
: Major innovations inevolutioninvolve the appearance if some structure happens to be present in the environ-
v
of new types of agents displaying cooperation while lim- ment. Awholen-bitpredictioncyclecanbedescribedas
i
X iting conflict (Maynard-Smith & Szathm´ary, 1997; Sza- a program (figure 1b). A survival function ρ depends on
thm´ary & Maynard-Smith, 1997). A specially important the number of attempts to guess bits and the number of
r
a innovation involved the rise of cognitive agents, namely correct predictions. Successful guessers have a positive
those capable of sensing their environments and reacting balance between reward and prediction cost. They get
to their changes in a highly adaptable way (Jablonka & replicated and pass on their inference abilities. Other-
Lamb, 2006). These agents were capable of dealing with wise, the agent fails to replicate and eventually dies.
morecomplex,non-geneticformsofinformation. Thead-
vantagesofsuchcognitivecomplexitybecomeclearwhen As a simple illustration of our approach, consider a 1-
considering their potential to better predict the environ- guesser living in an infinitely large environment E where
ment, thus reducing the average hazards of unexpected uncorrelated bits take value 0 with probability p and 1
fluctuations. As pointed by Francois Jacob, an organ- with probability 1 p. The average performance of a
−
ism is “a sort of machine for predicting the future – an guesser G when trying to infer bits from E is given by
2
a b c
start
new string 1.0
⇢(p,n)>0 i=1 B E
⇢
0.9
guess YES
G G wi i=i+1 i=n? ↵0.8 Extinction
NO
0.7
n
YES
0 0 1 0 1 0 1 1 1 0 1 0 0 0 E wi=bi? r=r+1 c=c+1 0.6
Survival
b ,b ,...,b NO 0.5
{ 1 2 n} 0.5 0.6 0.7 pp 0.8 0.9 1
FIG. 1 Predictive agents and environmental complexity. a An agent G interacts with an external environment E that
is modeled as a string of random bits. These bits take value 0 with probability p and value 1 otherwise. The agent tries to
guessasequenceofnbitsatsomecost,witharewardbestowedforeachcorrectlyguessedbit. Thepersistenceandreplication
of the agent can only be granted if the balance between reward and cost is positive (ρG > 0). b For a machine attempting
E
to guess n bits, an algorithmic description of its behavior is shown as a flow graph. Each loop in the computation involves
scanningarandomsubsetoftheenvironmentB =(b ,...,b )⊂E bycomparingeachb ∈B toaproposedguessw . cAmean
1 n i i
field approach to certain kind of 1-guesser (modeled in the text through equations 1, 2, and 3) in environments of infinite size
renders a boundary between survival (ρG > 0) and death (ρG < 0) as a function of the cost-reward ratio (α) and of relevant
E E
parameters for the 1-guesser model (p in this case). Note that for α<0.5 every 1-guesser survives for free.
p¯G, the likelihood of emitting a correct guess: of not being able to gather enough energy to support
E
and replicate the mechanisms for inference. As will be
p¯G = pG(0)p+pG(1)(1 p), (1) shown below, it is possible to derive the critical condi-
E −
tions to survive as a function of the agent’s complexity
where pG(k) is the frequency with which the guesser
and to connect these conditions to information theory.
emits the bit value k 0,1 . A strategy that uses
As advanced above, this allows us to characterize math-
∈ { }
pG(0)=p, pG(1)=1 p (i.e. a guesser that mimics the
ematically a scenario in which a guesser’s complexity is
−
environment) makes in average
explicitly selected for.
p¯G = 2p2 2p+1 (2)
E −
II. EVOLUTION AND INFORMATION THEORY
successful predictions. Its survival function reads:
ρG =(2p2 2p+1)r c. (3) Keyaspectsofinformationtheoryrelatedeeplytofor-
E − − mulations in statistical physics (Jaynes, 1957a,b; Par-
This curve trivially dictates the average survival or ex- rondo et al., 2015) and there have been several calls
tinction of 1-guessers as a function of the cost-reward to further integrate information theory in biological re-
ratio α c/r. Note that any more complex guesser (like search (Joyce, 2002, 2012; Krakauer, 2011; Maynard-
≡
theonesdescribedbelow)wouldalwaysfareworstinthis Smith, 2000; Nurse, 2008; Walker & Davies, 2012). This
case: they would potentially pay a larger cost to infer theoryshallplayimportantrolesinpopulationorecosys-
somestructurewherenoneistobefound. Notealsothat tems dynamics, in regulatory genomics, and in chemical
the tunable parameter α codes for the severity of the signalprocessingamongothers(Adami,2012;Bergstrom
environment. & Lachmann, 2004; Dall et al., 2005; Dall & Johnstone,
The idea of autonomy and the fact that predicting 2002; Donaldson-Matasci et al., 2010; Donalson-Matasci
the future implies performing some sort of computation et al., 2008; Evans et al., 2015; Friston, 2013; Hidalgo et
suggests that a parsimonious theory of life’s complexity al.,2014;Kussell&Leibler,2005;Marzen&DeDeo,2016;
needs to incorporate reproducing individuals (and even- McNamara & Houston, 1987; Rivoire & Leibler, 2011;
tuallypopulations)andinformation(theymustbecapa- Sartori et al., 2014; Segr´e et al., 2000, 2001; Szathm´ary,
bleofpredictingfutureenvironmentalstates). Thesetwo 1989; Tkaˇcik & Bialek, 2014), but a unifying approach
components define a conflict and an evolutionary trade- is far from complete. Given its generality and power,
off. Being too simple means that the external world is information theory has also been used to address prob-
perceived as a source of noise. Unexpected fluctuations lemsthatconnectDarwinianevolutionandfarfromequi-
canbeharmfulandusefulstructurecannotbeharnessed librium thermodynamics (Drossel, 2001; England, 2013;
in your benefit. Becoming more complex (hence able Goldenfeld&Woese,2010;Nicolis&Prigogine,1977;Pe-
to infer larger structures, if they exist) implies a risk runovetal.,2014). Initsoriginalformulation,Shannon’s
3
information theory (Shannon, 1948; Shannon & Weaver,
1949)considerssymbolsbeingconveyedfromatransmit-
ter to a receiver through a channel. Shannon only deals
with the efficiency of the channel (related to its noise or
reliability) and the entropy of the source. This theory
ignores the content of the emitted symbols, despite the
limitationsofsuchassumption(Corominas-Murtraetal.,
2014; Maynard-Smith, 2000).
A satisfactory connection between natural selection
and information theory can be obtained by mapping our
survival function ρ into Shannon’s transmitter-receiver
scheme. To do so we consider replicators at an arbitrary
generationT attemptingto“send”amessageto(i.e. get-
tingreplicatedinto)alatergenerationT+1. Asuccessful
“message” thus requires the transmission of information
throughthesurvivaloftheoffspring. Theformergenera-
tionactsasatransmitter,thelaterbecomesthereceiver,
andtheenvironmentanditscontingenciesconstitutethe
channel through which the embodied message must be
conveyed (figure 2a). In other words, we can think of FIG.2 Information and evolution through natural se-
a genotype as a generative model (the instructions in an lection. a A simple diagram of the underlying evolution of
a population of bit guessers. The survival and replication of
algorithm)thatproducesthemessagethatmustbetrans-
a given agent G is indicated by branching whereas failure to
mitted. That message would be embodied by a pheno-
survive is indicated with an empty circle as and endpoint.
typeanditincludeseveryphysicalprocessandstructure
b The propagation of a successful replicator can be under-
dictated by the generative model. As discussed in (von
stood in terms of a Shannon-like transmission process from
Neumann & Burks, 1966), any replicating machine must one generation to the next.
pass on a physically embodied copy of its instructions –
hencethephenotypemustalsoincludeaphysicalrealiza-
tion of the algorithm encoded by the genotype1. Finally,
anyevolutionarypressure(includingtheinteractionwith A. Messages, channels, and bit guessers
otherreplicatingsignals)canbeincludedascontrivances
of the channel. Let us first introduce our implementation of environ-
ments (channels), messages, and two classes of agents:
Followingasimilarideaofmessagesbeingpassedfrom transmittersandreceivers. Ourreplicatingagentswillbe
one generation to the next one, (Maynard-Smith, 2000) dubbedbit-guessersbecauseefficienttransmissionwillbe
proposes that the replicated genetic message carries equivalent to accurately predicting channel conditions.
meaningful information that must be protected against The notation may seem arid, so it is good to retain a
the channel contingencies. Let us instead depart from a central picture (figure 3): Guessers G posses a gener-
replicating message devoid of meaning. We realize that ative model ΓG that must produce messages that fare
the channel itself would convey more reliably those mes- well in an environment E. Both these messages and the
sagesembodiedbyaphenotypethatbetterdealswiththe environments are modeled as bit strings. What follows
environmental (i.e. channel) conditions. Dysfunctional is a rigorous mathematical characterization of how the
messages are removed due to natural selection. Efficient different bit sequences are produced.
signals get more space in successive generations (figure We consider m-environments, strings made up of m
2b). Through this process meaningful bits of environ- random bits. Bits within m-environments are sorted, so
mental information are pumped into the replicating sig- that it matters what bits follow each other. We might
nals, such that the information in future messages will consider one given m-environment – i.e. one realization
anticipate those channel contingencies. Meaningful in- (E) of m sorted random bits (e E, i = 1,...,m, e
i i
∈ ∈
formationisnotprotectedagainstthechannelconditions 0,1 ). Alternatively, we might work with the ensem-
{ }
(including noise), but emerges naturally from them. ble (E ) of m-environments – i.e. all possible environ-
m
ments of the same size ((e E ) E , l = 1,...,2m,
i,l l m
∈ ∈
i = 1,...,m), or with a sample (Eˆ ) of this ensemble
m
((E Eˆ ) E , l=1,..., Eˆ ). We might evaluate
l m m m
1 Note that many of the phenotypic structures build in order to thep∈erforma⊂nceofreplicating||sign||alsandbitguessersin
getreplicatedarelaterdismissed(think,e.g.,aboutthegermvs.
singlem-environmentsorinthecorrespondingensemble.
somaticcelllines). Wepresentacleardivisionbetweengenotype
Weintroducem-environmentstomodeltheconditions
andphenotypeforsakeofillustration. Weareawareofthemurky
frontierbetweentheseconcepts. of the channel through which messages propagate. To
4
attempt to transmit an n-bit message (W, usually with For every bit that attempts to traverse the channel a
n < m) we extract an n-sized word (B E) from the costcispaid. Arewardr =c/α iscashedinonlyitthat
⊂
corresponding environment by picking up the bit at a bit is successfully received. α is an external parameter
random position in E and the successive n 1 bits. The that controls the payoff. The survival function reads:
−
b B arecomparedtothew W (withb ,w 0,1 )
i i i i
an∈d each w gets through if w∈= b . Hence att∈em{ptin}g ρG(α) = (p¯G α)r. (9)
i i i E E −
to get messages through an environment effectively be-
comesaninferencetask: ifaguessercananticipatewhat Asaruleofthumb,ifp¯GE >αthegivenguesserfareswell
bit follows, it has a greater chance of sending messages enough in the proposed environment.
through, and hence moving into a later generation. It is useful to quantify the entropy per bit of the mes-
In this paper the generative model ΓG of the guessers sages produced by G:
is endowed with a minimal ability to react to the envi-
n
ronment on the spot (this mechanism will be explained H(G) = 1 (cid:88)(cid:88)pG(k;i)log(cid:0)pG(k;i)(cid:1), (10)
below). Insteadofconveyingafixedstring,ΓG buildsW −n
i=1 k
as a function of the broadcast history:
and the mutual information between the messages and
wi =wi(w1,...,wi−1;b1,...,bi−1). the environment:
Hence, the evaluation of a generative model is rather 1 (cid:88)n (cid:88)
I(G:E) = p (k,k(cid:48);i)
based on the ensemble of messages that it can produce. G,E
n ×
There is a compromise worth investigating between the i=1k,k(cid:48)
fidelity of the message that an agent tries to convey and (cid:18) p (k,k(cid:48);i) (cid:19)
G,E
log . (11)
its ability to react to environmental conditions in real pG(k;i)p (k(cid:48);i)
E
time. Exploring this tradeoff is left for future work. The
reaction capabilities of our bit-guessers is kept to a min- To evaluate the performance of a guesser over an en-
imum. semble Eˆ of environments (instead of over single envi-
m
Usually,aguesserattemptstoemitann-bitwordmany ronments)weattemptN broadcastsovereachofN dif-
g e
(Ng) times through the same channel. For each one of ferent environments (El Eˆm, l = 1,...,Ne Eˆm )
these broadcasts, a new n-sized word Bj E (with of a given size. For sim∈plicity, instead of lab≡elin||g bj||,
bji ∈ Bj for j = 1,...,Ng and i = 1,...,n) i⊂s extracted we stack together all Ng Ne n-sized words Wj ani,ld
from the same environment – i.e., as in real life, channel Bj. This way bj Bj an×d wj Wj for i = 1,...,n
conditions vary. The manufactured message ensemble and j = 1,...,iN∈N . We hiav∈e pG(k;i), p (k(cid:48);i),
becomes Wj with wj Wj. g e Eˆm
i ∈ p (k,k(cid:48);i), pG (i), and p¯G defined just as before,
We can calculate different frequencies with which the G,Eˆm Eˆm Eˆm
guessers or the environments present bits with value only with j running through j = 1,...,NgNe. Also
k,k(cid:48) 0,1 : as before, we average the payoff across environments to
∈{ } determine whether a guesser’s messages get successfully
1 (cid:88)Ng transmitted or not given the length of the environment
pG(k;i) = δ(wj,k), (4) ensemble (m) and α:
N i
g
j=1
ρG (α) = (p¯G α)r. (12)
1 (cid:88)Ng Eˆm Eˆm −
p (k(cid:48);i) = δ(bj,k(cid:48)), (5)
E N i Note that
g
j=1
pG,E(k,k(cid:48);i) = N1 (cid:88)Ng δ(wij,k)δ(bji,k(cid:48)), (6) I(G:Eˆm) = n1 (cid:88)n (cid:88)pG,Eˆm(k,k(cid:48);i)×
g i=1k,k(cid:48)
j=1
(cid:32) (cid:33)
pGE(i) = N1g (cid:88)Ng δ(wij,bji)⇒ (7) log pGp(Gk,E;ˆim)p(kEˆ,mk((cid:48)k;(cid:48)i;)i) (13)
j=1
1 (cid:88)n is different from
p¯G = pG(i); (8)
⇒ E n E
i=1 1 (cid:88)Ne
I(G:E) = I(G:E ). (14)
withδ(x,y)beingDirac’sdelta. NotethatpG(k;i)hasa (cid:104) (cid:105)Eˆm Ne l=1 l
subtle dependency on the environment (because G may
react to it) and that p¯GE indicates the average probabil- We use (cid:104)·(cid:105)Eˆm to indicate averages across environments
ity that guesser G successfully transmits a bit through of an ensemble Eˆ .
m
channel E.
5
frequency. Hence:
ΓG(1) g =max p (k(cid:48);1) ; (15)
E ≡ 1 k(cid:48) { E }
Ifboth0and1appearequallyoftenwechoose1without
loss of generality. If the agent succeeds in its first guess,
its safest next bet is to emit the bit (0 or 1) that more
frequently follows g in the environment. We proceed
1
similarlyifthefirsttwobitshavebeencorrectlyguessed,
ifthefirstthreebitshavebeencorrectlyguessed,etc. We
define p (k;i) as the probability of finding k = 0,1
B|Γ
{ }
at the i-th position of the Bj word extracted from the
environment, provided that the guess so far is correct:
m i−1
1 (cid:88) (cid:89)
FIG.3 Fromagenerativemodeltoinferenceaboutthe pB|Γ(k(cid:48);i) = Z(i) δ(bji,k(cid:48)) δ(bji(cid:48),gi(cid:48)). (16)
j=1 i(cid:48)=1
world. A diagrammatic representation of the algorithmic
logic of the bit guessing machine. Our n-guesser contains The index j, in this case, labels all n-sized words within
a generative model (represented by a pool of words) from the environment (bj Bj) E and Z(i) is a normal-
which it draws guesses about the environment. If a bit is i ∈ ⊂
izationconstantthatdependsonhowmanywordsinthe
successfullyinferred,thechosenconjectureispursuedfurther
environment match ΓG up to the (i 1)th bit:
by comparing a new bit. Otherwise, the inference is reset. E −
m i−1
(cid:88) (cid:89)
Z(i) = δ(bji(cid:48),gi(cid:48)). (17)
j=1i(cid:48)=1
The message-generation process ΓG embodied by our
n-guessers could be implemented in different ways, in- It follows:
cluding Artificial Neural Networks (ANNs) (Hopfield,
1988),spikingneurons(Maass&Bishop,2001),Bayesian ΓG(i=2,...,n) g = max(cid:8)p (k(cid:48);i)(cid:9). (18)
E ≡ i k(cid:48) B|Γ
networks (Jensen, 1996; Pearl, 1985), Turing machines
(Turing, 1936), Markovian chains (Markov, 1971), (cid:15)- Note that the pool of bits in ΓG consists of an n-sized
machines(Crutchfield&Young,1989),RandomBoolean E
word, which is what they try to emit through (i.e. it
Networks (RBNs) (Kauffman, 1993), among others.
constitutes the guess about) the channel. If a guesser
These devices elaborate their guesses through a series of
would not be able to react to environmental conditions,
algorithms (e.g. back-propagation, message passing, or
the word W that is actually generated at every emission
Hebbianlearning)providedtheyhaveaccesstoasample wouldbethesameineverycaseandwj =g always; but
of their environment. i i
we also allow our guessers a minimal reaction if one of
Intherealworld,trialanderrorandevolutionthrough thebitsfailstogetthroughthechannel(i.e. ifoneofthe
naturalselectionwouldbethealgorithmwiringthemes- guesses is not correct). This minimal reaction capacity
sage generation processes (i.e. the genotype) into our by our guessers results in:
agents. Thiswouldestablishsomedynamicsoutofwhich
the relevant inference would emerge, which is a very in- wj = ΓG(i l)=g , (19)
teresting process. However, in this paper we aim at un- i E − i−l
derstandingthelimitsimposedbyachannel’scomplexity where l is the largest i at which wj = bj. This means
andthecostofinference;speciallywhentheperformance that a guesser restarts the broadcasit(cid:54)of ΓiG whenever it
E
of the guessers is almost optimal. Therefore, we will as- makes a mistake2.
sumethatouragentsperformanalmostperfectinference All together, our guesser consists of a generative
given the environment where they live. This best infer- model ΓG that contains a pool of bits and a simple
encewillbehard-wiredintheguesser’sgenerativemodel conditional instruction. This is reflected in the flow
of their environment ΓG as explained right ahead.
A guesser’s generative model usually depends on the
environment where it is deployed, so we note ΓG ΓG.
≡ E
This ΓGE will consist of a pool of bits gi ∈ ΓGE (figure 2 Note that more elaborated guessers would not only reset their
3) and a series of rules dictating how to emit those bits: guess. They might browse through a tree with conditional in-
either in a predetermined order or as a response to the structionsateverypoint. Besidesanextendedmemorytostore
thegrowingnumberofbranches,theywouldalsorequirenested
channel’s changing conditions. Whenever we pick up an
if-else instructions. On the other hand, ANNs or Bayesian net-
environment E = e ,i = 1,...,m , the best first guess
i worksmightimplementsuchtree-browsingwithoutexcessiveif-
{ }
possible will be the bit (0 or 1) that shows up with more elsecosts.
6
chart in figure 3. Coming back to biology, and as noted 1 0.75
above, the generative model is akin to a genotype and
the resulting bits emitted as guesses about the channel 0.9
would make up a phenotype.
0.65
0.8
We have made a series of choices regarding how to im-
plement environmental conditions. These choices about 0.550 10 20 30
implementation affect the way some randomness enters 0.7
the model (reflected in the fact that, given an environ-
ment E, a guesser might come across different words 0.6
B E) and also how we implement our guessers (in-
⊂
cluding their minimal adaptability to wrong bits). We
0.5
came up with a scheme that codes guessers, the en- 0 10 20 30
Envinronment size (m)
vironment (or channel), and the messages transmitted
as bit strings. This allows us a direct measurement of FIG.4 Probability of correctly guessing a bit in envi-
information-theoretical features which are suitable for ronment ensembles of constant size. p¯n,averageprob-
m
our discussion, but the conclusions at which we arrive abilitythatn-guesserscorrectlyguess1bitinm-environments
should be general: survival will depend on a replicator’s for different n values. Here p¯1m can be computed analytically
abilitytoextractusefulinformationfromthechannel(en- (solidlineinthemainplot)andmarksanaverage,lowerpre-
codedinthemessagethatitconveysfromonegeneration dictability boundary for all guessers. In the inset, the data
hasbeensmoothedandcomparedtoagivenvalueofα(repre-
to the next one), and on the cost-efficiency tradeoff re-
sentedbyahorizontalline). Attheintersectionbetweenthis
lated to how meaningful bits are.
line and p¯n we find m¯n(α), the environment size at which
m
Because of the minimal implementation discussed, all n-sizedagentsguessjustenoughbitstosurvivegivenα. No-
guessers of the same size are equal. Environmental en- ticethatn-guessersareevaluatedonlyinenvironmentsofsize
sembles of a given size are considered equivalent as well. m≥n.
Hence,thenotationisnotaffectedifweidentifyguessers
and environments by their sizes. Accordingly, in the fol-
lowing we substitute the labels G and E by the more
informative ones n and m respectively. Hence ρG (α) A. Guessers isolated in environments of fixed size
Em
becomes ρn(α), p¯G becomes p¯n, etc.
m E m Figure 4 shows p¯n, the average probability that n-
m
guessers correctly guess 1 bit in m-environments. The 1-
guesser (that lives off maximally decorrelated bits given
the environment) establishes a lower bound. More com-
III. LIFE COMPLEXITY IS TUNED TO A plex machines will guess more bits in average, except
PREDICTABILITY-REPLICATION TRADEOFF
for infinite environment size m , at which point all
DICTATED BY THE ENVIRONMENT → ∞
guessers have equivalent predictive power.
Asmgrows,environments(whichconsistofcollections
We report a series of numerical experiments. Some of random bits) get less and less predictable. Impor-
of them deal with guessers in environment ensembles of tantly, the predictability of shorter words decays faster
fixed size, others allow guessers to switch between envi- than that of larger ones, thus enabling guessers with
ronmentsizestofindaplacewheretothrive. Theimpor- larger n to survive where others would perish. There
tant message is that the complexity of the guessers that are 2n possible n-words, of which m are realized in each
canpopulateagivenenvironment(i.e. thecomplexityof m-environment. When m >> 2n, the environment im-
the most successful messages traversing the correspond- plements an efficient, ergodic sampling of all n-words
ingchannel)istunedbythepredictabilityoftheenviron- – thus making them maximally unpredictable. When
ment. Environment sizes exist at which simple guessers n < m < 2n the sampling of n-sized words is far from
dieoffbutinwhichmorecomplexlifeflourishes–thusof- erg∼odicandanon-trivialstructureisinducedintheenvi-
fering an explanation to real-life excursions beyond bare ronmentbecausethesymmetrybetweenn-sizedwordsis
replicativesuccess. Ourschemeallowsustoquantifythe broken–theycannotbeequallyrepresentedduetofinite
different aspects involved in the tradeoff, and to charac- size sampling effects.
terize this evolutionary pressure behind complex replica-
Thisallowsthatcomplexguessers(thosewiththeabil-
tors.
ity to contemplate larger words, keep them in memory,
Finally,someofthesimulationsengagen-guesserswith andmakechoicesregardinginformationencodedinlarger
different n in direct competition with each other. Yet strings) can guess more bits, in average, than simpler
other experiments explore what may happen if resources agents. In terms of messages crossing the channel, while
could get exhausted. This is suggestive of how the cur- shorterwordsaremeaninglessandbasicallygettransmit-
rent model will be explored in future research. ted (i.e. are correctly guessed) by chance alone, larger
7
words might contain meaningful, non-trivial information a 0.3
thatgetsuccessfullytransmittedbecausetheycopewith 0.1
the environment in the adequate way.
0.08
Notethatthissymmetrybreakingtofavorpredictabil-
0.06
ity of larger words is just a mechanism that allows us
0.2
to introduce correlations in a controlled and measurable 0.04
way. In the real world this mechanism might correspond
0.02
to asymmetries between dynamical systems in temporal
or spatial scales. Although our implementation is rather 0
0 10 20 30
ad hoc (suitable to our computational and conceptual 0.1
needs), we propose that similar mechanisms might play
important roles in shaping life and endowing the uni-
verse with meaningful information. Indeed, it might be
extremely rare to find a kind of environment in which
0
wordsofallsizesbecomenon-informativesimultaneously. 0 10 20 30
b
Back to our experiments, the mutual information be-
tween a guesser’s response and the environment (i.e. be-
tween broadcast messages and channel conditions) fur- 0.6
ther characterizes the advantages of more complex repli-
cators. Figure 5a shows I(G : E ) and I(G:E) .
m (cid:104) (cid:105)Em
As we noted above, these quantities are not the same.
0.4
Let us focus on 1-guesser for a moment to clarify what
these quantities encode.
Given an environment, 1-guessers have got just one
bit that they try to emit repeatedly. They do not react 0.2
to the environment – there is not room for any reaction
within one bit, so their guess is persistently the same.
Themutualinformationbetweentheemittedbitandthe
arbitrary words B E that 1-guessers come across is 0
⊂ 0 10 20 30
precisely zero, as shown in the inset of figure 5a. Hence,
Envinronment size (m)
I(G:E) capturesthemutualinformationduetothe
(cid:104) (cid:105)Em
slightreactioncapabilitiesofguesserstotheenvironmen- FIG.5 Mutualinformationandentropy. Guesserswith
tal conditions. n=1 (crosses), n=2 (squares), n=5 (pluses), and n=10
(triangles)arepresented. aI(G:E )and(cid:104)I(G:E)(cid:105) (in-
While the bits emitted by 1-guessers do not corre- m Em
set) quantify the different sources of information that allow
late with B E, they do correlate with each given E
more complex guessers to thrive in environments in which
⊂
since they represent the most frequent bit in the envi- simplerlifeisnotpossible. bTheentropyofaguesser’smes-
ronment. Accordingly, the mutual information between sage given its environment seems roughly constant in these
a 1-guesser and the aggregated environments (reflected experiments despite the growing environment size. This sug-
by I(G : E )) is different from zero (figure 5a). To gests an intrinsic measure of complexity for guessers. Larger
m
this quantity contribute both the reaction capability of guessers look more random even if they might carry more
guessers and the fact that they have hard-wired a near- meaningful information about their environment. The thick
optimal guess in ΓG, as explained in section II.A. blacklinerepresentstheaverageentropyoftheenvironments
E (which approaches log(2)) against which the entropy of the
We take the size of a guesser n as a crude characteri- guessers can be compared.
zation of its complexity. This is justified because larger
guessers can store more complex patterns. H(G)
(cid:104) (cid:105)Em
indicates that more complex guessers look more entropic
thanlesscomplexones(figure5b). Largerguesserscome B. Evolutionary drivers
closertotheentropyleveloftheenvironment(blackthick
lineinfigure5b), whichitselftendsrapidlytolog(2)per The key question is whether the payoff may be favor-
bit. This is remarkable: better performing guessers ap- ableformorecomplexguessersprovidedthattheyneeda
pearmoredisorderedtoanexternalobserverevenifthey more costly machinery in order to get successfully repro-
are better predictors when considered within their con- duced. As discussed above, the cost of units of Artificial
text. Notethat H(G) isbuiltbasedonthebitsactu- NeuralNetworksorthenumberofstatesinBayesiannet-
(cid:104) (cid:105)Em
ally emitted by the guessers, meaning that this quantity works would enter this evolutionary game if such imple-
correlates with the complexity of the phenotype. For mentations of bit guessers were chosen. To keep the dis-
guessers of fixed size n, we observe a slight decay of cussionsimple,andwithoutlossofgenerality,bitguessers
H(G) as we proceed to larger environments. incur only in a cost proportional to the number of bits
(cid:104) (cid:105)Em
8
that they try to transmit. Equation 12 captures all the a 0.15
forces involved: the cost of transmitting larger messages
versustherewardofasuccessfultransmissionthatcomes
aftermorecomplexenvironmentscouldbeapprehended.
Guessers of a given size survive in an environment en-
0.1
semble if, in average, they can guess enough bits of the
environmentor,putotherwise,iftheycanconveyenough
bits through the channel (in any case, they survive if
p¯n > α, which implies ρn > 0). Setting fix a value
m m
of α we find out graphically m¯n(α), the largest envi- 0.05
ronment at which n-guessers can survive (figure 4, in-
set). Because m-environments look more predictable to
more complex guessers we have that m¯n(α) > m¯n(cid:48)(α) if
n > n(cid:48). This guarantees that for α > 0.5 there always
0
exist m-environments from which simple life is banned 0 10 20 30
while more complex life can thrive – i.e. situations in Envinronment size (m)
which environment complexity is an explicit driver to- b
wards more complex life forms. For α 0.5 any guesser 120
≤
survives in average. Under that circumstance a guesser’s
80
complexity becomes a neutral trait and we may expect
that complexity does emerge out of sheer fluctuations.
Wecansimulatesomedynamicsthatgravitatearound 80
40
m¯n(α). After evaluating an n-guesser N N times in
g e
·
an arbitrary m-environment, the guesser is promoted to
m+1 if ρˆn(α,N ,N ) > 0, where ρˆ represents an em-
m g e
piricallyaccumulatedrewardinsteadoftheensembleav- 40 00.55 0.6 0.65 0.7
erage. If ρˆn(α,N ,N ) < 0, the guesser is demoted to
m g e
m 1. The steady state of this process is character-
−
ized by a distribution Pn(m;α,N ,N ) Pn(m,α), i.e.
g e
≡
the frequency with which n-guessers are found in envi-
0
ronments of a given size (figure 6a). The overlap and 0.5 0.6 0.7 0.8 0.9 1
gaps between Pn(m,α) for different n suggest that: i)
some guessers would engage in harsh competition if they
FIG. 6 Dynamics around m¯n(α). Again, guessers with
neededtoshareenvironmentsofagivenkindandii)there
n = 1 (solid line), n = 2 (dashed line), n = 5 (dotted line),
is room for different guessers to get segregated into envi-
and n=10 (dot-dashed line). a Pn(m,α) tells us how often
ronments of increasing complexity. The average
do we find n-guessers in m-environments when they are al-
(cid:88) lowedtoroamconstrainedonlybytheirsurvivalfunctionρn.
mˆn(α) = mPn(m,α) (20) The central value mˆn of Pn(m,α) must converge to m¯n(mα)
m and oscillations around it depend (through N and N ) on
g e
how often do we evaluate the guessers in each environment.
should converge to mˆn(α) m¯n(α) under the appro-
(cid:39) b Average mˆn for n = 1,2,5,10 and standard deviation of
priate limit – i.e. if we evaluate the guessers numerically
Pn(m,α) for n = 1,10. Deviations are not presented for
enoughtimesastoapproachthesemeanfieldvalues. Fig-
n = 2,5 for clarity. The inset represents a zoom in into the
ure 6b shows dynamically-derived averages mˆn(α) and main plot.
some deviations around them as a function of α.
It is easily justified that guessers drop to simpler envi-
ronments if they cannot cope with the complexity. It is
less clear why they should seek more complicated envi-
vironments might appear only if actual (n=0) guessers
ronmentsiftheythriveinagivenone. Thismighthappen
(cid:54)
die, as we explain below. We tracked the population
spontaneously if simpler environments get crowded or if
using P (n,t), the proportion of 0-, 1-, 2-, 3-, and 4-
resources get exhausted, as we study right ahead. m
guessers through time. (These experiments where the
To simulate a competition dynamics, n-guessers with
more computationally demanding, that is why we took
n = 0, 1, 2, 3, and 4 were randomly distributed occu-
n = 1,2,3,4 instead of the values n = 1,2,5,10 used
pying 100 environments of fixed m and were assigned an
throughoutthepaper. Theinsightsgainedfromthesim-
initial ρˆ(t = 0) = nρ with i = 1,...,100 labeling the
i 0
ulations do not depend on the actual values of n.)
guesser at each environment. Larger guessers start out
with larger ρˆ(t = 0) representing that they come into At each iteration, a guesser (say the i-th one) was
i
being with a larger metabolic load satisfied. A 0-guesser chosen randomly and evaluated with respect to its en-
represents an unoccupied environment. New empty en- vironment. Then the wasted environment was replaced
9
a 100 b c 1 d 0.7
75 0.875 0.675
50 0.75 0.65
25 0.625 0.625
0 0.5 0.6
0 15 30 0 15 30 10 20 30 40 10 20 30 40
Envinronment size (m) Envinronment size (m) Envinronment size (m) Envinronment size (m)
FIG. 7 Evolutionary drivers: competition. Coexisting replicators will affect each other’s environments in non-trivial
wayswhichmayoftenresultincompetition. Weimplementadynamicsinwhich1-,2-,3-,and4-guessersexclusivelyoccupya
finitenumberofenvironmentsofagivensize(fixedm). The100availableslotsarerandomlyoccupiedatt=0andgrantedto
thebestreplicatorsasthedynamicsproceed. WeshowP (n,t=10000)form=5,...,39andα=0.6(a),α=0.65(b). The
m
mostabundantguesseratt=10000isshownforα∈(0.5,1)(c)andα∈(0.6,0.7)(d). Oncemisfixed,thereisanuppervalue
of α above which no guesser survives and all 100 available slots remain empty. Competition and the replication-predictability
tradeoff segregate guessers according to the complexity of the environment – i.e. of the transmission channel. Coexistence of
differentguessersseemspossible(e.g. m=15inb),butitcannotbeguaranteedthatthedynamicshaveconvergedtoasteady
distribution.
by a new, random one with the same size. We ensured resources they depend on). In figure 8b given some α
that every guesser attempts to guess the same amount and β = 1, 1- and 2-guessers can only survive within
of bits in average. This means, e.g., that 1-guessers some under and upper limits (figure 8b). As announced
are tested twice as often as 2-guessers, etc. If after the above,thismightestablishevolutionaryincentivestoex-
evaluationwefoundthatρˆ(t+∆t)<0,thentheguesser ploremorecomplexscenariosevenifsomeguesserhasits
i
died and it was substituted by a new one. The n of needs already satisfied in a simple environment.
the new guesser was chosen randomly after the current
distribution P (n,t). If ρˆ(t+∆t) > 2nρ , the guesser
m i 0
got replicated and shared its ρˆ with its daughter,
i IV. DISCUSSION
who overrode another randomly chosen guesser. This
replication at 2nρ represents that parents must satisfy
0 In this paper we have considered a fundamental ques-
a metabolic load that grows with the size n of the
tionrelatedtotheemergenceofcomplexityinlivingsys-
guessers. There is a range (0<ρˆ <2nρ ) within which
i 0 tems. The problem being addressed here is how more
guessers are alive but do not replicate. Figure 7a and
complex organisms can overcome the cost of their com-
b show P (n,t = 10000) with α = 0.6 and 0.65. Note
m plexity by developing a higher potential to predict the
that for large environments all guessers combined do not
external environment. As suggested by several authors
addupto100–i.e. mostlyemptyslotsremainandmost
(Friston, 2013; Jacob, 1998; Wagensberg, 2000) the be-
guessers get extinguished. The most abundant guesser
havioral plasticity provided by the exploratory behavior
after 10000 iterations is shown in figure 7c as a function
of living systems can be understood it terms of their po-
of m and α.
tential for dealing with environmental information (Ger-
hart and Kirschner, 1997).
An alternative evolutionary pressure is introduced if
Ourmodelsmakeanexplicitapproachbyconsideringa
the bits in the environment represent resources that
replication-predictability tradeoff under very general as-
might get exhausted. Thinking from the message broad-
sumptions,namely: (a)Morecomplexenvironmentslook
casting perspective, a spot on the channel might appear
moreunpredictabletosimplerreplicatorsand(b)Agents
crowdedifitisengagedinasuccessfultransmission. As-
thatcankeepalargermemoryandmakeinferencesbased
sume that every time that a bit is correctly guessed, it
onmoreelaboratedinformationcanextractenoughvalu-
gets exhausted (or gets crowded) with an efficiency β so
able bits from the environment as to survive in those
that in average each bit cannot contribute any reward
more challenging situations. Despite the inevitable cost
β(p¯n/m) of the time. The average reward extracted by
m inherent to the cognitive machinery, a selection process
a guesser from an ensemble (figure 8) becomes:
towards more complex life is shown to occur. This offers
(cid:18) p¯n (cid:19) an explicit evolutionary pressure towards more complex
r˜mn = 1−β mm p¯nmr. (21) life forms.
Inourstudyweidentifyatransmitter(replicatorsata
Smaller guessers living in very small environments givengeneration),areceiver(replicatorsatthenextgen-
quickly crowd their channels (alternatively, exhaust the eration), and a channel (any environmental conditions)
10
a 0.65 lier picture introduced in (Maynard-Smith, 2000). That
metaphor assigned to the DNA some external meaning
thathadtobepreservedagainsttheenvironmentalnoise.
Contrary to this, we propose that, as messages attempt
0.6
to travel from a generation to the next one, all channel
conditions (including noise) pump relevant bits into the
transmitted strings – hence there is no need to protect
0.55
meaningagainstthechannel. Thewaythatweintroduce
correlationsinourscheme(throughasymmetrybreaking
betweentheinformationbornebyshortandlargerwords
0.5 due to finite size effects) is compatible with this view.
However, interestingly, it also suggests that meaningful
informationmightarisenaturallyeveninhighlyunstruc-
tured environments when different spatial and temporal
0.45
0 10 20 30 scales play a relevant role.
b 0.61
This way of integrating information theory and Dar-
winiand evolution is convenient to analyze the ques-
tions at hand that concern the emergence of complex
life forms. But it also opens further research questions.
0.59 As discussed at the beginning of the paper, guessers and
theirtransmissiblemessagesmightandshouldshapethe
transmission channel (e.g., by crowding it, as explored
briefly in section III.B). What possible co-evolutionary
dynamics between guessers and channels can be estab-
0.57
lished? Are there stable ones, others leading to extinc-
tion, etc? Do some of them, perhaps, imply open-ended
evolution? Which ones? Also, a guesser’s transmitted
message might be considered an environment in itself,
0.55 opening the door to ecosystem modeling based on bare
0 10 20 30
Envinronment size (m) information theory. It is also suggested the exploration
of different symbiotic relationships from this perspective
FIG. 8 Evolutionary drivers: exhausted resources. and how they might affect coevolution.
Rather than monopolizing channel slots (as in figure 5), we Finally, animportantquestionwasleftasidethatcon-
can also conceive individual bits as valuable, finite resources
cernsthememoryvsadaptabilitytradeoffofbitguessers.
thatgetexhaustedwhenevertheyarecorrectlyguessed. Then
Here we studied guessers with a minimal adaptability to
asuccessfulreplicatorcanspoilitsownenvironmentandnew
focus on the emerging hierarchy of complexity. Adapt-
conditions might apply to where life is possible. a Average
ability at faster (say, at behavioral) temporal scales is
rewardobtainedby1-,2-,5-,and10-guessersinenvironments
ofdifferentsizeswhenbitsgetexhaustedwithefficiencyβ =1 linked to more complex inferences with richer dynamics.
whenever they are correctly guessed. b Given α=0.575 and This brings in new dilemmas as to how to weight the
α = 0.59, 1- and 2-guessers can survive within upper and different building blocks of complex inference – e.g. how
lower environment sizes. If the environment is too small, re- dowecomparememoryandif-elseorwhileinstructions?
sourcesgetconsumedquicklyandcannotsustainthereplica- These and other questions are left for exploration in fu-
tors. In message transmission language, the guessers crowd ture research.
their own channel. If the environment is too large, unpre-
dictability takes over for these simple replicators and they
perish.
Acknowledgments
The authors thank the members of the Complex Sys-
throughwhichamessage(ideallyinstructionsabouthow tems Lab, as well as Fernando Maestre and his team,
to build newer replicators) is passed on. Darwinian evo- and Max Tegmark, Jeremy Owen, Henry Lin, Jordan
lution follows naturally as effective replicators transit a Horowitz,andDavidWolpertforveryusefuldiscussions.
channel faster and more reliably thus getting more and This study was supported by an European Research
more space in successive generations. The inference task Council Advanced Grant (SYNCOM), the Botin Foun-
is implicit as the environment itself codes for meaningful dation, by Banco Santander through its Santander Uni-
bits of information that, if picked up by the replicators, versities Global Division and by the Santa Fe Institute.
boostthefitnessofthephenotypesembodiedbythesuc- This work has also been supported by the Secretaria
cessful messages. d’Universitats i Recerca del Departament d’Economia i
This view is directly inspired by a qualitative ear- Coneixement de la Generalitat de Catalunya.