Mathematical Models of Artificial Genetic Representations with Neutrality c 2 and Nino Baˇ Carlos M. Fonseca 1 , Vida Vukaˇ c 3 sinovi´ si´ 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´ opez 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 2 ℓ − 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 ( C 0 , the code itself) • 2 ℓ − k − 1 cosets ( C j ) • 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 of the zero phenotype, z j , in each class C j
• 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) 5 5 8 6 1 6 4 4 2 3 5 4 3 2 7 8 2 8 3 6 1 6 8 1 1 7 2 4 7 3 5 7
2. Mathematical formulation 2.1. Binary representations • The genotypic space is a vector space G = Z ℓ 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 Z 2 . • The phenotypic space is P = Z k 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 m i ( g ) = m ( g , e i ) , where e i is a vector of length ℓ with a 1 on the i -th component and zeros elsewhere. • A mutation m i is neutral if r ( g ) = r ( m i ( g )) . • M ⊆ G is a neutral network if for each g 1 , g 2 ∈ M there exists a sequence of genotypes h 1 = g 1 , h 2 ,..., h µ = g 2 , where h j ∈ M for all j = 1 ,..., µ , and neutral mutations m i j for j = 1 ,..., µ − 1 such that h j + 1 = m i j ( h j ) .
2.3. Representation properties • Uniform redundancy: r is uniformly redundant if | r − 1 ( p 1 ) | = | r − 1 ( p 2 ) | for all p 1 , p 2 ∈ P • Connectivity: c r = 1 � r ( N ( r − 1 ( p ))) � � | P | ∑ � p ∈ P where N ( r − 1 ( p )) = { g ∈ G \ r − 1 ( p ) |∃ h ∈ r − 1 ( p ) : d G ( g , h ) = 1 } • Synonymity: s r = 1 1 | P | ∑ ∑ d G ( g , h ) � | r − 1 ( p ) | � p ∈ P { g , h }⊆ r − 1 ( p ) 2 • Locality: 2 ∑ l r = d P ( r ( g 1 ) , r ( g 2 )) ℓ | G | { g 1 , g 2 }⊆ G : d G ( g 1 , g 2 )= 1
2.4. Uniformly neutral representations • Let ν be an inclusion of P into G. G g g ∼ h ν h 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 = { z 0 , z 1 ,..., z τ − 1 } , where τ = | G / ν ( G ) | and z 0 ∈ 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 ( z 0 ) = 0 P 3. for each g ∈ ν ( P ) , r ( z 0 ⊕ g ) = r ( z i ⊕ 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 ( · , z 0 )) . • 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, c r ( p 1 ) = c r ( p 2 ) s r ( p 1 ) = s r ( p 2 ) and for every p 1 , p 2 ∈ P. Moreover, if r is fully compatible with ν and Z, then l r ( p 1 ) = l r ( p 2 ) .
• 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, c r ′ = c r s r ′ = s r . and Moreover, if r , r ′ are fully compatible with ν and Z, Z ⊕ c, respectively, then l r ′ = l r . • Theorem 3 Let ν be a linear inclusion of P into G. Let π be a permutation of 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, c r ′ = c r s r ′ = s r . and
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
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