1. Geometric Landscape of Homologous Crossover for Syntactic Trees Alberto Moraglio & Riccardo Poli {amoragn,rpoli}@essex.ac.uk CEC 2005

2. Contents I: Abstract Geometric Operators II: Geometric Crossover for Syntactic Trees III: Conclusions

3. I. Abstract Geometric Operators

4. What is crossover? Crossover Is there any common aspect ? Is it possible to give a representation- independent definition of crossover and mutation? 100000011101000 100111100011100 100110011101000 100001100011100 Binary Strings Permutations Real Vectors Syntactic Trees

5. Mutation & Nearness Mutation is naturally interpreted in terms of nearness: offspring are near the parent Example: Binary String P = 0 1 0 1 1 1 O = 0 1 0 1 0 1 NEARNESS:hd(P,O)=1

6. Crossover & Betweenness Crossover is naturally interpreted in terms of betweenness: offspring are between parents Example: Binary String P1 = 0 1 0|0 1 0 P2 = 1 1 0|1 0 1 O = 0 1 0 1 0 1 hd(P1,P2)=4 hd(P1,O)=3 hd(O,P2)=1 BETWEENNES: P1---O-P2

7. 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

8. 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)

9. Geometric Operators Formalization BALL: All points within distance r from x SEGMENT: All points between x and y 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

10. II. Geometric Crossover for Syntactic Trees Homologous Crossover (HC) Hyperschema (HS) Structural Hamming Distance (SHD) HC is geometric under SHD via HS

11. 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

12. 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

13. HC example - Parent Trees Blue Parent Red Parent

14. All offspring under HC Common 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 1 0 0 10 1 00 1

15. Hyperschema Hyperschema: common region tree structure + wildcards Wildcard “=”: different nodes same arity (replace node) Wildcard “#”: different arity (replace subtree)

16. 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

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

18. 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 other  offsrping are “marginally” between parents WILDCARDS CONTRIBUTIONS ARE INDEPENDENT & ADDITIVE HENCE: offsrping are between parents also for the total distance

19. III. Conclusions

20. 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

21. 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

22. Thank you for your attention… Questions?