Mathematical Models of Artificial Genetic Representations with - - PowerPoint PPT Presentation

Mathematical Models of Artificial Genetic Representations with Neutrality

Carlos M. Fonseca1, Vida Vukaˇ sinovi´ c2 and Nino Baˇ si´ c3

1 CISUC, Department of Informatics Engineering,

University of Coimbra, Portugal

cmfonsec@dei.uc.pt

2 Computer Systems, Joˇ

zef Stefan Institute, Ljubljana, Slovenia

vida.vukasinovic@ijs.si

3 Faculty of Mathematics, Natural Sciences and Information Technologies,

University of Primorska, Koper, Slovenia

nino.basic@famnit.upr.si

Dagstuhl Seminar 17191, 7-12 May 2017

Acknowledgements

• COST Action CA15140 on Improving Applicability of Nature-Inspired Op-

timisation by Joining Theory and Practice (ImAppNIO) for a Short-Term Scientific Mission grant

COST is supported by the EU Framework Programme Horizon 2020

• National funds through Portuguese Foundation for Science and Technology

(FCT)

• European Regional Development Fund (FEDER) through COMPETE 2020

Operational Program for Competitiveness and Internationalisation (POCI)

Outline

• Background
• Neutrality in natural evolution
• Neutrality in artificial evolution
• Uniformly neutral representations
• Mathematical formulation
• Concluding remarks

1. Background

1.1. Neutrality in natural evolution

• Most mutations at the genotypic level are not expressed in the phenotype

(Kimura, 1968)

• Genotypes connected by neutral mutations form (large) neutral networks
• Random genetic drift instead of natural selection
• Accumulation of neutral mutations may lead to beneficial mutations later
• Neutrality is believed to account for improved search space exploration
• Massive redundancy and neutrality

• RNA fitness landscapes (Schuster et al. 1994)
• Genotype is a sequence of nucleotides (bases)
• Phenotype is a shape (secondary structure, represented as a graph)
• Shape space considerably smaller than sequence space (redundancy)
• Few common shapes, many rare ones (non-uniform redundancy)
• Many single-base (and even two-base) mutations are neutral
• Such genotype-phenotype mappings are defined by the physical laws gov-

erning the folding process (and may have themselves evolved)

1.2. Neutrality in artificial evolution

• Genotype-phenotype mappings are referred to as representations
• Good representations and operators are crucial to evolutionary algorithm

performance

• The influence of genotypic redundancy and neutrality on search perfor-

mance is (still) not well understood

• Larger search spaces make the problem harder (?)
• Larger neighbourhoods induced by neutral networks (may) make the

problem easier (???)

• There have been attempts to identify representation properties that influence

the performance of evolutionary algorithms (Rothlauf, 2006)

• Several contradicting results in the literature (Galv´

an-L´

• pez et al, 2011)

• Many artificial redundant representations have been proposed
• Emphasis on very high redundancy
• Not amenable to analysis, typically evaluated experimentally
• Will focus on a family of representations based on error control codes

(Fonseca and Correia, 2005)

• Emphasis on low redundancy (is high redundancy really justified?)
• Various degrees of uniform redundancy, neutrality, connectivity, locality,

and synonymity can be obtained

• Have allowed the influence of the above properties on optimisation per-

formance to be studied experimentally (Correia, 2013)

• Not trying to model natural representations!

1.3. Uniformly neutral representations

Split redundant genotypic space into 2ℓ−k interspersed classes

Uniformly neutral representations

Map genotypes in each class so as to form neutral networks

In the binary case, block error-control codes define suitable genotypic classes

• one main class (C0, the code itself)
• 2ℓ−k −1 cosets (Cj)
• minimum distance is at least 2

Decoding

• Determine genotype class (polynomial division)
• Map genotype to main class (add a constant)
• Decode to obtain phenotype (truncation)

⇒ Neutral networks can be explicitly designed by specifying the representation

• f the zero phenotype, z j, in each class Cj

• Exhaustive enumeration of all representations based on a given main class

has only been achieved for ℓ−k = 3 bits of redundancy

• Higher redundancy leads to extremely large numbers of different represen-

tations

• MDS depiction of some neutral network shapes (ℓ = 7, k = 4)

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

2. Mathematical formulation

2.1. Binary representations

• The genotypic space is a vector space G = Zℓ

2, where addition of two vec-

tors, ⊕, is defined as the componentwise XOR operation, and scalar multi- plication is the multiplication of a vector by a constant from Z2.

• The phenotypic space is P = Zk

2, with the same operations.

• A binary representation is a surjective mapping r : G → P. If ℓ > k, the

representation is redundant.

2.2. Mutations and neutral networks

• A mutation is a bilinear mapping m : G × G → G, where m(g,e) = g ⊕ e

for each g,e ∈ G. The single point mutation of the i-th component of g is denoted by mi(g) = m(g,ei), where ei is a vector of length ℓ with a 1 on the i-th component and zeros elsewhere.

• A mutation mi is neutral if r(g) = r(mi(g)).
• M ⊆ G is a neutral network if for each g1,g2 ∈ M there exists a sequence of

genotypes h1 = g1,h2,...,hµ = g2, where hj ∈ M for all j = 1,...,µ, and neutral mutations mi j for j = 1,...,µ −1 such that hj+1 = mi j(hj).

2.3. Representation properties

• Uniform redundancy: r is uniformly redundant if

|r−1(p1)| = |r−1(p2)| for all p1, p2 ∈ P

• Connectivity:

cr = 1 |P| ∑

p∈P

• r(N(r−1(p)))
• where N(r−1(p)) = {g ∈ G\r−1(p)|∃h ∈ r−1(p) : dG(g,h) = 1}
• Synonymity:

sr = 1 |P| ∑

p∈P

1 |r−1(p)|

2

• ∑

{g,h}⊆r−1(p)

dG(g,h)

• Locality:

lr = 2 ℓ|G|

∑

{g1,g2}⊆G:dG(g1,g2)=1

dP(r(g1),r(g2))

2.4. Uniformly neutral representations

• Let ν be an inclusion of P into G.

ν g h g ⊕ h G P g ∼ h r

• Let ∼ be a relation on G defined as g ∼ h ⇔ g⊕h ∈ ν(P).

Note that ∼ is an equivalence relation. The corresponding equivalence classes are called cosets.

• An equivalent definition of coset of ν(P) in G is g ⊕ ν(P) = {g ⊕ ν(p) :

p ∈ P}, where g is a coset representative.

• Let Z = {z0,z1,...,zτ−1}, where τ = |G/ν(G)| and z0 ∈ ν(P), denote the

set of representatives of all cosets of ν(P) in G, i.e. Z is a transversal.

• Definition 1 Let ν be an inclusion of phenotype space P in genotype space

G and Z be a transversal of all cosets of ν(P) in G. A representation r is compatible with ν and Z if the following conditions hold:

• 1. Z forms a neutral network in G
• 2. r(z0) = 0P
• 3. for each g ∈ ν(P), r(z0 ⊕g) = r(zi ⊕g) for every i = 0,...,τ −1.

• Definition 2 A representation r is said to be fully compatible with inclu-

sion ν and transversal Z if r is compatible with ν and Z, and r|ν(P) = ν−1(m(·,z0)).

• Theorem 1 Let ν be a linear inclusion of P into G. Let r be a representation

which is compatible with inclusion ν and transversal Z. Then, cr(p1) = cr(p2) and sr(p1) = sr(p2) for every p1, p2 ∈ P. Moreover, if r is fully compatible with ν and Z, then lr(p1) = lr(p2).

• Theorem 2 Let ν be a linear inclusion of P into G. Let r,r′ be represen-

tations which are compatible with inclusion ν and transversals Z, Z ⊕ c,

• respectively. Then,

cr′ = cr and sr′ = sr . Moreover, if r,r′ are fully compatible with ν and Z, Z ⊕c, respectively, then lr′ = lr .

• Theorem 3 Let ν be a linear inclusion of P into G. Let π be a permutation
• f the components of g ∈ G such that π(ν(P)) = ν(P). Let r,r′ be repre-

sentations which are compatible with inclusion ν and transversals Z, π(Z),

• respectively. Then,

cr′ = cr and sr′ = sr .

3. Concluding remarks

• Developed a mathematical framework for the study and characterisation of

artificial genetic representations

• Formalised a class of uniform neutral representations from the literature
• Equivalence classes of representations related to:
• Translations
• Permutations (automorphisms of ν(P))
• Restrict enumeration to representatives of such equivalence classes
• Now targeting the enumeration of representations with 4 bits of redundancy
• Will allow the effect of locality on search performance to be studied under

fixed values of the other properties