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Evolutionary Robotics and Open-Ended Design Automation 151

domains, such as language, culture, social organization, and technology (Basalla, 1989; Ziman,
2003), among many others. Principles of modularity, regularity, and hierarchy are, however,
nowhere as dominant as they are in engineering design. Tracing the evolution of technology over
generations of products, one can observe numerous instances of designs being encapsulated into
modules, and those modules being used as standard higher-level building blocks elsewhere.
Similarly, there is a pressure to reduce the information content in designs, by repeating or
reusing the same modules where possible, using symmetrical and regular structures, and stand-
ardizing on components and dimensions. These and other forms of regularity translate into
reduced design, fabrication, and operation costs. The organization of engineering designs,
especially as complexity increases, is typically hierarchical. The hierarchy is often organized
such that the amount of information is distributed uniformly across levels, maintaining a
‘‘manageable’’ extent of information at each stage. These principles of modularity, regularity,
and hierarchy are cornerstones of engineering design theory and practice (e.g., Suh, 1990).
Though these principles are well established, there is — like biological evolution — still a
lack of a formal understanding of how and why modular, regular, and hierarchical structures
emerge and persist, and how can we computationally emulate these successful principles in the
design automation processes.

. Functional modularity is the structural localization of function.
In order to measure functional modularity, one must have a quantitative definition of function and
structure. It is then possible to take an arbitrary chunk of a system and measure the dependency of
the system function on elements of that chunk. The more that the dependency itself depends on
elements outside the chunk, the less the function of that chunk is localized, and hence the less
modular it is. If we represent dependencies as second derivatives of function with respect to pairs of
parameters (i.e., the Hessian matrix of the fitness), then modules will be collections of parameters
that can be arranged with lighter off-diagonal elements (Wyatt and Lipson, 2003).
. Structural regularity is the compressibility of the description of the structure.
The more the structure contains repetitions, near-repetitions, symmetries, smoothness, self-
similarities, etc., the shorter its description length will be. The amount of regularity can thus be
quantified as the inverse of the description length or of its Kolmogorov complexity.
. Hierarchy of a system is the recursive composition of structure and/or function.
The amount of hierarchy can be quantified given the connectivity of functional or structural
elements (e.g., as a connectivity graph). The more the distribution of connectivities path
lengths among pairs of elements approximates a power law distribution, the more hierarchical
the system is.

4.7.1 Principles of Design
Modularity and regularity are independent principles. Principles of modularity and regularity are
often confused in the literature through the notion of reuse. Indeed, modularity has several
advantages, one of which is that modules can be used as building blocks at higher levels, and
therefore can be repeated. Nonetheless, it is easy to imagine a system that is composed of modules,
where each module appears only once. For example, opening the hood of a car reveals a system
composed of a single engine, a single carburetor, and a single transmission. Each of these units
appears only once (i.e., is not reused anywhere else in the system), but can be considered a module
as its function is localized. Its evolutionary advantage is that it can be adapted more independently,
with less impact of the adaptation on the context. A carburetor may be swapped to a newer
technology without affecting the rest of the engine system.
Similarly, there are instances of regularity without modularity. The smoothness of the hood of
the car, for example, reduces the information content of the structure but does not involve the reuse
of a particular module.