# Invariance of Function Complexity under Primitive Recursive Functions

Created by W.Langdon from gp-bibliography.bib Revision:1.4448

```@InProceedings{eurogp06:Woodward,
author =       "John R. Woodward",
title =        "Invariance of Function Complexity under Primitive
Recursive Functions",
editor =       "Pierre Collet and Marco Tomassini and Marc Ebner and
Steven Gustafson and Anik\'o Ek\'art",
booktitle =    "Proceedings of the 9th European Conference on Genetic
Programming",
publisher =    "Springer",
series =       "Lecture Notes in Computer Science",
volume =       "3905",
year =         "2006",
month =        "10 - 12 " # apr,
organisation = "EvoNet",
keywords =     "genetic algorithms, genetic programming",
ISBN =         "3-540-33143-3",
pages =        "310--319",
DOI =          "doi:10.1007/11729976_28",
bibsource =    "DBLP, http://dblp.uni-trier.de",
abstract =     "Genetic Programming (GP) \cite{banzhaf:1997:book}
often uses a tree form of a graph to represent
solutions. An extension to this representation,
Automatically Defined Functions (ADFs) is to allow the
ability to express modules. In Woodward we proved that
the complexity of a function is independent of the
primitive set (function set and terminal set) if the
representation has the ability to express modules. This
is essentially due to the fact that if a representation
can express modules, then it can effectively define its
own primitives at a constant cost. This is reminiscent
of the result that the complexity of a bit string is
independent of the choice of Universal Turing Machine
(UTM) (within an additive constant), the constant
depending on the UTM but not on the function.

The representations typically used in GP are not
capable of expressing recursion, however a few
researchers have introduced recursion into their
representations. These representations are then capable
of expressing a wider classes of functions, for example
the primitive recursive functions (PRFs). We prove that
given two representations which express the PRFs (and
only the PRFs), the complexity of a function with
respect to either of these representations is invariant
within an additive constant. This is in the same vein
as the proof of the invariants of Kolmogorov complexity
and the proof in Woodward.",
notes =        "Part of \cite{collet:2006:GP} EuroGP'2006 held in
conjunction with EvoCOP2006 and EvoWorkshops2006",
}

```

Genetic Programming entries for John R Woodward

Citations