Auditoría de mercadotecnia Análisis ambiental

We will consider here evolvability under the class of evolutionary mechanisms Mthat contains all those mechanisms that use a mutation functionM usuch that M u(l)∈ w((lx, ly)).12 Specifically, we will consider qual-evolvability as

defined in Section 6.2.1. Our distance function dis(l, l0) equals 0 if lx = lx0

and ly =l0y, and 1 otherwise.13

In the following when we refer to Manhattan distance between two nodes we refer to the distance of the shortest strictly horizontal and/or vertical path between the two nodes. Note that m−1 is the maximum Manhattan distance between two nodes on the grid. The Manhattan distance between two nodes (x, y) and (x0, y0) lower bounds the number of timeslots needed for an entity to come alive on node (x0, y0) as a result of life being propagated among neighboring nodes starting from an entity that resides on node (x, y). This is a direct result of the fact that M u(l)∈w((lx, ly)).

A simplified analysis. Here we will examine the qual-evolvability of all entity classes C such that C ∈ Lm. In contrast with the next section, we

will not be concerned with how the size of the entity classes affects their probability of evolution, or how the latter behaves in relation to m, i.e., the parameter that determines the size of the grid and consequently the possible distances among nodes.

As we mentioned above, we will consider two cases concerning the food- process. In both cases |Ft_{| =k, for some positive integer k ≤m}2_{. That is,}

11_{We remind the reader that S}t_{.Lis the set of entities in the system during timeslott.}

12_{Here M u(l) stands forS.M.M u(l, S).} 13_{Given that L}

m contains no multisets and our distance function is discrete, qual-

the density q of food on the grid is constant and equal to k/m2. In case (i) we will have thatFt _=Ft+1_{, for allt≥0. In case (ii) we will have thatF}t_is

a subset of Gm chosen uniformly at random among all subsets of Gm of size k.14

For both cases (i) and (ii) the following holds. For c = 1, i.e., if the entities need to eat at every timeslot, no entity class C such that |C|> k is evolvable, since there is not enough food to sustain as many entities as the size of C. More generally, if C is evolvable then |C| ≤ck.

Let us focus on case (i) now. We have that if C is evolvable then for all l ∈ C there exists l0 ∈ L0 such that Q(F0, l, l0).15 What is more, we can also easily find a mechanism that makes the above statement work the other way around too. One such mechanism is the mechanism where b(l) is a positive constant, i.e., there exists pb ∈ (0,1] such that St.M.b(l) = pb for

all timeslotst, andM u(l) chooses uniformly randomly among the entities on the four neighboring nodes to (lx, ly).

Let us demonstrate why this mechanism works as promised, i.e., if for all

l ∈C there exists l0 ∈L0 _{such that Q(F}0_{, l, l}0_{) thenC is evolvable. Suppose}

that for all l∈C it holds that there existsl0 ∈L0 _{such thatQ(F}0_{, l, l}0_{), then}

the above mechanism has as a result that alll ∈Chave a positive probability to evolve, and that probability is independent of any accuracy parameter . Moreover, because of the nature of Ft _{in case (i), if an entity evolves it will}

be sustained forever. Hence, C is evolvable, according to Definition 6.2.3. Taking m into account. Here we take into account how the probability of evolution is affected by the size of the class to evolve. The size of the classes C could be equal tom or larger16.

In line with Remark 6.2.2, we address this concern by parameterizing all of our components with m and focusing on how things turn out in relation to m. Hence, instead of C ⊂Gm, we consider Cm ⊂Gm.17 We will consider

only the most interesting of the two cases described above, i.e., case (ii) where for some density q ∈ [0,1] we will have that Ft

m is a subset of Gm chosen

uniformly at random among all subsets of Gm of size km =dqm2e.

Let us start by demonstrating that|Cm| ≤ckm is not enough to guarantee 14_{For case (i) the environment S.E would need to contain (as information) both kand}

F0, while for case (ii) onlykwill be enough.

15_{We remind the reader that L}0_{is the initial set of entities.} 16_{The size ofC can be as large asm}2_.

17_{Note here that the subscriptmonC}

mdoes not signify the complexity of the entities

in Cmas ndoes in Definition 6.2.4 or as in the example of Section??. Heremis just the

the evolvability of C.18 Let rt be a predicate such that rt((x, y)) is true iff at timeslot t an entity l such that (lx, ly) = (x, y) is reproduced (clearly,

reproduced by an entity that resides on a node neighboring (x, y)). We have that Pr[rt((x, y))]≤1−(3/4)4, where 1−(3/4)4 is the probability for

rt_{((x, y)) to happen given that there are entities on all neighboring nodes of}

(x, y).

Assuming that the densityq 6= 1, then we have for allt ≥1 that Pr[Cm evolves at timeslot t] ≤Pr[∀l ∈Cm :rt((lx, ly))∨(¬rt((lx, ly))∧(lx, ly)∈ [ t0_{∈{t−c,...,t−1}} Ft0)] ≤ Y l∈Cm Pr[rt((lx, ly))∨(¬rt((lx, ly))∧(lx, ly)∈ [ t0_{∈{t−c,...,t−1}} Ft0)] ≤((1−(3/4)4) + (3/4)4(1−(1−q)c))|Cm| ≤α|Cm|_, for some α∈(0,1).

Subsequently, assuming thatL0 _6=C

m, Pr[Cm evolves in g(m,1/) timeslots] =Pr[ _ t∈{1,...,g(m,1/)} Cm evolves at timeslot t] ≤α|Cm|_g(m,1/)

As such, if|Cm| ≥m then the probability of success decreases exponen-

tially withm. Let us summarize by saying that if the elements of a sequence {Cm} are evolvable with a probability that is decreasing at most polynomi-

ally with m then |Cm| ∈ O(logm). This result is a consequence of the cost

that we have to pay (in terms of probability) in order to keep the entities fed. It is still possible though for Cm of size O(logm) to be evolvable.

Specifically, given the above result the strongest thing that we may pos- sibly prove is that |Cm| ∈ O(logm) and |Cm| ≤ ckm is enough to render C

evolvable. However, a simple consideration will indicate this to be unlikely. Consider c = 1, m = 1024, |Cm| = log2m = 10, and say that we have just

enough food, i.e., km = 10. Thus, we ask whether Cm can evolve in a place

where we have 106 _{nodes and there is only food for 10 of them. A good guess}

18_{Note that this restriction, which practically says that there is at least the bare mini-} mum of food on the grid in order to sustain as many entities as those inCm, was enough

would be that we cannot get evolvability in this setting; we would instead need to boost km, and probably have it in Ω(m2).

In order to address this questions we made some simulations of the sys- tem.19 Those indicate the following. It turns out that the system has a threshold behavior. Specifically, there is a threshold value, say q∗, forq that determines whether the probability that everyone dies is negligible or it is close to 1. For example, for c= 1 this threshold value of q lies in the interval [0.59,0.6]. Naturally, for larger values ofc the value of the thresholdq∗ gets smaller and smaller.

Simulations show that the density of alive entities in L0 does not make much difference. Moreover, the density of alive entities, say qA, for a specific q converges as m gets larger.20 _{The fact that F}t

m is uniformly random and St.L⊆F_mt for all timeslotst, indicates that the system reaches a state where, in the long run, every configuration of density roughly equal to qA is pretty

much as probable as any other. Or, more precisely, there exists some small real numberηsuch that the great majority of the configurations have density in [qA−η, qA+η]. This is the key point in our analysis here.

We have that asmgets larger ηgets smaller and the number of nodes on the grid gets larger. As such, we may regard the system to have a behavior where at each timeslot, each node carries an alive entity with probability

qA, independent from the other nodes.21 And in this setting the probability

that a certain entity class |Cm|evolves is (qA)|Cm|. Since |Cm| ∈O(logm) we

have that the probability of evolution decreases at most polynomially with

m. All in all, the above indicate that for allCm such that |Cm| ∈ O(logm),

if km > m2q∗ then Cm is evolvable.