abstract = "In this paper we introduce, formalize, and
experimentally validate a novel concept of functional
modularity for Genetic Programming (GP). We rely on
module definition that is most natural for GP: a piece
of program code (subtree). However, as opposed to
syntax-based approaches that abstract from the actual
computation performed by a module, we analyze also its
semantic using a set of fitness cases. In particular,
the central notion of this approach is subgoal, an
entity that embodies module's desired semantic and is
used to evaluate module candidates. As the cardinality
of the space of all subgoals is exponential with
respect to the number of fitness cases, we introduce
monotonicity to assess subgoals' potential utility for
searching for good modules. For a given subgoal and a
sample of modules, monotonicity measures the
correlation of subgoal's distance from module's
semantics and the fitness of the solution the module is
part of. In the experimental part we demonstrate how
these concepts may be used to describe and quantify the
modularity of two simple problems of Boolean function
synthesis. In particular, we conclude that monotonicity
usefully differentiates two problems with different
nature of modularity, allows us to tell apart the
useful subgoals from the other ones, and may be
potentially used for problem decomposition and enhance
the efficiency of evolutionary search.",
notes = "GECCO-2009 A joint meeting of the eighteenth
international conference on genetic algorithms
(ICGA-2009) and the fourteenth annual genetic
programming conference (GP-2009).