Geometric Landscape of Homologous Crossover for

Syntactic Trees

Alberto Moraglio & Riccardo Poli

{amoragn,rpoli}@essex.ac.uk

CEC 2005

Contents

I: Abstract Geometric Operators

II: Geometric Crossover for Syntactic Trees

III: Conclusions

I. Abstract Geometric Operators

What is crossover?

CrossoverIs there any

common

aspect ?

Is it possible to give a

representation-

independent definition

of crossover and mutation?

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100110011101000

100001100011100

Binary Strings

Permutations

Real Vectors

Syntactic Trees

Mutation & Nearness

• Mutation is naturally interpreted in terms of nearness: offspring are near the parent

• Example: Binary StringP = 0 1 0 1 1 1O = 0 1 0 1 0 1

• NEARNESS:hd(P,O)=1

Crossover & Betweenness

• Crossover is naturally interpreted in terms of betweenness: offspring are between parents

• Example: Binary StringP1 = 0 1 0|0 1 0P2 = 1 1 0|1 0 1O = 0 1 0 1 0 1hd(P1,P2)=4hd(P1,O)=3 hd(O,P2)=1

• BETWEENNES: P1---O-P2

Geometric Crossover

DEFINITION: Any crossover for which there is at least a distance (metric) such as all offspring are between parents is a geometric crossover

Geometric Crossovers across Representations

Many recombination operators for the most used representations are geometric under suitable distance:

BINARY: one-point, two-points, uniform crossovers

REAL VECTORS: line, arithmetic, discrete (non-geometric: extended line)

PERMUTATIONS: PMX, Edge Recombination, Cycle Crossover, Merge Crossover (non-geometric: order crossover)

SYNTACTIC TREES: homologous one-point & uniform crossovers (non-geometric: subtree swap crossover)

Geometric Operators Formalization

|),(|

)),((}|Pr{)|(

xB

xBzxPzUMxzfUM

}),(|{);( ryxdSyrxB

)},(),(),(|{];[ yxdyzdzxdSzyx

BALL: All points within distance r from x

SEGMENT: All points between x and y

|],[|

]),[(}2,1|Pr{),|(

yx

yxzyPxPzUXyxzfUX

UNIFORM -MUTATION: offspring z are taken uniformly within the ball of radius from the parent x

UNIFORM CROSSOVER: offspring z are taken uniformly within the segment between parents x and y

II. Geometric Crossover for Syntactic Trees

•Homologous Crossover (HC)•Hyperschema (HS)•Structural Hamming Distance (SHD)•HC is geometric under SHD via HS

One-point (Homologous) Crossover

• Alignment: align trees at the root

• Common Region: consider common topology

• Common Crossover Point: select the same crossover point for the two trees within the common region

• Subtree Swap

• Restricted: restriction of subtree swap crossover

General Homologous Crossover (HC)

• Alignment: align trees at the root

• Common Region: common trees topology

• Crossover Mask: generate crossover mask over common region

• Swap: swap nodes within the common region and swap subtrees on the boundaries of the common region

HC example - Parent Trees

Blue Parent Red Parent

All offspring under HCCommon Region: black tree structure

Crossover Mask: over common region

Within Common Region: Node swap (e.g. x2, y2)

Boundary Common Region: Subtree swap (e.g. x5. y5)

0

10

0 1 0

1 0 01

Hyperschema

Hyperschema: common region tree structure + wildcards

Wildcard “=”: different nodes same arity (replace node)

Wildcard “#”: different arity (replace subtree)

Structural Hamming Distance (SHD)

• Recursive & Bounded by 1• Trees have different root arity d=1 • Trees have same structure & all different nodes d=1• SHD is a METRIC

mqarityparityiftsdistqphdm

qarityparityif

qarityparityifqphd

TTdist

miii )()( ),(),(

1

1

)()( 1

0)()( ),(

),(

,1

21

SHD & Hyperschema

PROPERTY: SHD is function of the Hyperschema only: d(p1,p2)=g(h(p1,p2))

HC is geometric under SHD

• TO PROVE: shd(P1,O)+shd(O,P2)=shd(P1,P2)• HYPERSCHEMA: set of all offspring• WILDCARD: marginal contribution to total distance• MARGINAL BETWENNESS: for any wildcard an

offspring equals one parent or the otheroffsrping are “marginally” between parents

• WILDCARDS CONTRIBUTIONS ARE INDEPENDENT & ADDITIVE

• HENCE: offsrping are between parents also for the total distance

III. Conclusions

More Results in the paper!

• TRADITIONAL CROSSOVER: subtree swap crossover is not geometric

• SPACE STRUCTURE: SHD is connected to a “fluid” (non-graphic) neighbourhood structure

• MUTATION: SHD is connected with subtree mutation

• LANDSCAPE: when trees are interpreted as GP programs SHD gives rise to a smooth landscape hence homologous crossover is a good choice

Moral (take home message)

This result unifies syntactic trees in the context of geometric framework, together with binary strings, real vectors and permutations.

Hence, the geometric definition of crossover captures in a single formula the notion of crossover matured over last two decades of research.

As implications, the geometric unification:- simplifies and clarifies the connection between crossover and search space- gives firm fundations for a general theory of evolutinary algorithm - suggests an “automatic” way to do crossover design for new

representations

Thank you for your attention… Questions?