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Self-modifying Cartesian Genetic Programming (SMCGP) is a general purpose, graph-based, developmental form of Genetic Programming founded on Cartesian Genetic Programming. In addition to the usual computational functions, it includes functions that can modify the program encoded in the genotype. This means that programs can be iterated to produce an infinite sequence of programs (phenotypes) from a single evolved genotype. It also allows programs to acquire more inputs and produce more outputs during this iteration. We discuss how SMCGP can be used and the results obtained in several different problem domains, including digital circuits, generation of patterns and sequences, and mathematical problems. We find that SMCGP can efficiently solve all the problems studied. In addition, we prove mathematically that evolved programs can provide general solutions to a number of problems: n-input even-parity, n-input adder, and sequence approximation to π.
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CONTRIBUTED ARTICLE
Developments in Cartesian Genetic Programming:
self-modifying CGP
Simon Harding Julian F. Miller Wolfgang Banzhaf
Received: 27 November 2009 / Revised: 26 April 2010 / Published online: 25 June 2010
Springer Science+Business Media, LLC 2010
Abstract Self-modifying Cartesian Genetic Programming (SMCGP) is a general
purpose, graph-based, developmental form of Genetic Programming founded on
Cartesian Genetic Programming. In addition to the usual computational functions, it
includes functions that can modify the program encoded in the genotype. This
means that programs can be iterated to produce an infinite sequence of programs
(phenotypes) from a single evolved genotype. It also allows programs to acquire
more inputs and produce more outputs during this iteration. We discuss how
SMCGP can be used and the results obtained in several different problem domains,
including digital circuits, generation of patterns and sequences, and mathematical
problems. We find that SMCGP can efficiently solve all the problems studied. In
addition, we prove mathematically that evolved programs can provide general
solutions to a number of problems: n-input even-parity, n-input adder, and sequence
approximation to p.
Keywords Cartesian Genetic Programming Developmental systems
S. Harding (&)W. Banzhaf
Department of Computer Science, Memorial University of Newfoundland,
St. John’s, NL A1B3X5, Canada
e-mail: simonh@mun.ca
W. Banzhaf
e-mail: banzhaf@mun.ca
J. F. Miller
Department of Electronics, University of York, York YO10 5DD, UK
e-mail: jfm7@ohm.york.ac.uk
123
Genet Program Evolvable Mach (2010) 11:397–439
DOI 10.1007/s10710-010-9114-1
1 Introduction
In genetic programming (GP), representations of programs (genotypes) are evolved
that solve computational problems. However, in such approaches the programs are
almost always static and do not change over time, or in response to environmental
inputs. In this paper, we propose a form of genetic programming in which the
programs can, over time, change, acquire new inputs or produce new outputs. The
work has been influenced by ideas expressed in the field of generative and
developmental systems, where researchers evolve genotypes that can lead to
arbitrarily large phenotypes with time dependent behaviour. Many such approaches
utilize the concept of a cell in which a fixed genotype resides. The phenotypes arise
through cell replication. The multicellular structure is finally mapped to an
appropriate computation in the application domain. We wished to be able to evolve
genotypes that could be iterated to arbitrarily sized phenotypes but in such a way
that the phenotypes were always interpretable as a program.
In biology, the phenotypes arise from genotypes through a complex interaction in
which a genotype, along with the cellular machinery and the environment gives rise
to a stage of the phenotype, which itself influences the decoding of the phenotype
for subsequent stages [5]. Regulatory mechanisms have been found to play a key
role in this transformation. In the approach described in this paper abstractions of
‘regulatory’’ mechanisms determine whether and how self-modification operations
will be applied. In computer science, this ‘‘unrolling’’ of the phenotype is often
restricted and considered analogous to self-modification or re-writing. However, in
genetic programming this notion has received only a little attention (see Sect. 2).
In the method we describe, a genotype decodes to a potentially infinite sequence
of phenotypes (which themselves are programs). Such an approach has a number of
advantages, not least, that a genotype may represent a solution to an entire class of
problems. For instance, we show later that genotypes can be evolved which encode
programs that can build parity functions with an arbitrary number of inputs, and
others that can exactly compute pin the limit of large iterations.
To accomplish this we have introduced extra functions into a GP function set that
cause modifications to the executed code itself. The technique we discuss is based
on Cartesian Genetic Programming (CGP), so we refer to it as self-modifying CGP
(SMCGP). The representation used in SMCGP is very flexible. For instance, a
genotype that has no SM functions is essentially identical to standard CGP. On the
other hand it is possible for SM functions to arise that cause the entire duplication of
the genotype. This could be seen as a kind of multi-cellular development, albeit
without the notion of a Euclidean space in which cells have to position themselves
and occupy space.
We chose to base our approach on GP, as opposed to neural networks or other
representations, as it allows the technique to be used in many different applications
requiring algorithms, e.g., mathematical regression or circuit design. The approach
we have taken is very general and in principle could encompass many types of
developmental computational systems. We also desired to devise a representation
that can work in many different domains without greatly modifying the basic
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working principle. This way we are able to test our approach on a wide variety of
problems that appear in the literature.
Another of our motivations, particularly when contrasted with developmental
systems, is that we wanted to be able to understand the evolved program. This is
especially important when demonstrating the generality of a solution. For practical
reasons, the developmental process needs to be computationally cheap. We feel that
such goals largely rule out devising a developmental GP method that emulates too
many aspects of biological systems.
SMCGP allows us not only to solve problems that cannot be solved using GP or
CGP but it allows us to find general solutions to some classes of problems. In
particular, we are able to find general solutions to problems that previously had only
been solved through a combination of CGP and human inspection. For instance, in
the first edition of the Genetic Programming and Evolvable Machines Journal,
Miller et al. [40] investigated the ‘‘digital adder problem’’ and showed a series of
evolved designs for adders that could by human inspection be generalized to
produce a design capable of adding n-bit binary numbers. In Sect. 5.4 we show how
SMCGP can obtain a general solution automatically.
The plan of the paper is as follows: We review the origins of self-modification in
computer science in Sect. 2. In Sect. 3, we review re-writing or developmental
methods, particularly those that have compared the evolution of developmental
genotypes with direct representations. We explain how SMCGP works in Sect. 4,
showing in detail how we define and use SM functions both in a self-modifying and
computational sense. We also describe how inputs and outputs are handled. The
evolutionary algorithm used and its operators and parameters are also discussed in
that section. We have devised and used a variety of different fitness function types
and primitives sets in the many experiments we have undertaken. In Sect. 4.9 we
show which function sets have been used for the various problems studied. The
experiments and results are described in Sects. 5,6and 7.
2 Review of self-modification
In this section we briefly review the existing body of work in the area of self-
modification. The first distinction we are going to make for our purposes is the
distinction between self-modified code of computers and all other kinds of self-
modification, like self-modification of organismic behaviour through learning [50]
or self-modification of brains through changes to their wiring pattern [11], or even
machines [2]. In fact, ACM/IEEE Computer Society list, under their keywords for
classification in the class Theory of Computation under point 6.1.1. ‘‘Self-modifying
machines’’ [23]. Thus, the term has a prominent place in computing, which has to do
with its history.
Early on in the design of electronic computers, it was recognized that the
distinction between data and programs really was an artificial one for the purpose of
information storage. This led von Neumann and others to the conclusion that one
should store programs and data in the same type of devices. Preceding this
development was the recognition of Turing, embodied in the famous Turing
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machine that a machine could not only modify data stored on a tape, but, if that tape
contained its program, start to modify its own program. Before Turing, Go
¨del had
realized with his numbering system a similar idea that would ultimately lead him to
the recognition of the incompleteness problem.
With von Neumann’s computer architecture came the possibility of manipulating
program code in the same way as one would manipulate data, and it was quickly
realized that one could write code that modified itself. In these early days of
computing, this was not only a possibility, but an important feature due to limited
resources available for storage. Self-modification allowed more compact code, if
less understandable one, but with some precautions, it was possible to use the
technique to optimize space requirements in memory.
One of the main uses of self-modifying code until today has been the runtime
generation of code and of code compression [4,27]. Both of these applications
allow better use of computing resources, and therefore allow run-time or memory
usage improvements over their non-modified counterparts. In the compression
domain, other aspects like security from reverse engineering also play a role.
Another line of reasoning for the usefulness of self-modification starts with
adaptation. In the context of Virtual Machines used heavily today, this has taken
hold [3]. Here, the question is again how to make intelligent use of resources.
It is only a small step from here to the idea of evolution of self-modifying models
of computation, as it was proposed repeatedly in the last decades for automata [49],
for hardware (FPGAs) [12] or computer code [44]. The latter development took
place within the field of Genetic Programming [31] which demonstrated the
evolvability of computer algorithms and paved the way for an entire new field of
algorithm development. Spector and Stoffel explicitly used self-modification in a
GP approach called ‘‘ontogenetic programming’’ [54]. Spector’s later developed the
GP language ‘‘Push’’ so that ‘‘autoconstructive evolution’’ could take place [53].
This is where evolved genotypes are responsible for the production of their own
offspring, rather than it being coded into an evolutionary algorithm explicitly. Push
allows evolved code to manipulate itself. So that programs could, in principle have
‘morphological’’ phases during which they develop into ‘‘mature’’ code which is
then executed to solve a problem. Alternatively, such programs might continue to
develop as they run, exhibiting ‘‘ontogeny’’ more in the manner of living organisms.
Self-modification was also implicit in the graph re-writing system proposed by
Gruau [14], which will be described in the next section. Miller [43] also considered
a form of self-modification in his developmental method of evolving graphs and
circuits. McPhee [38] used an N-gram based GP system to produce programs that
could solve regression problems, where development was linked to an incremental
fitness function.
In a series of contributions and works, Kampis [24,25,26] pointed out that self-
modification is extremely important for living organisms, and indeed might
constitute their defining properties. Already prior to Kampis’ work, Maturana and
Varela [36] had proposed the concept of autopoiesis as a key feature of living
systems. Self-modification has also been considered in artificial organisms as they
were examined in the field of Artificial Life. Major contributions were made by the
introduction of Coreworld [46] and the TIERRA [47] and its variants [1]. In this line
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of work, the emphasis is, however, more on the observation of emergent effects in
self-modifying systems than on their use in computation.
1
Since we are interested here in moving toward more life-like behaviour of
algorithms, that exhibit adaptivity and efficiency, self-modification is taken to be
one of the key properties we want to include in our system.
3 Developmental genotypes versus direct mappings
Recently there has been an increasing interest in generative and developmental
systems (GDS) [34] and their potential benefits for evolutionary computation. Many
argue that GDS will be necessary in order to make evolutionary techniques scale up
to larger problems (see, e.g., [6]).
Kitano used a developmental method to define the architecture of an artificial
neural network (ANN). The technique used a matrix re-writing system that
manipulated adjacency matrices [29]. He claimed that he could evolve better ANNs
more quickly using the developmental approach than by direct methods, i.e., a fixed
architecture, directly encoded and evolved. However, a later paper by Siddiqi and
Lucas [52] made a more detailed study and concluded that the two approaches were
of equal quality.
Gruau also investigated an evolutionary developmental approach for ANNs. He
devised a graph re-writing method called cellular encoding [14] for local graph
transformations that control the division of cells growing into an artificial neural
network. Connection strengths (weights), threshold values and the grammar tree that
defines the graph re-writing rules were evolved using an evolutionary algorithm.
This method was shown to be effective at optimizing both the architecture and
weights at the same time, and scaled better than a direct encoding [15].
Bentley and Kumar [8] looked at a number of genotype to phenotype mappings
on a problem of creating a tessellating tile pattern. They found that what they
termed ‘‘implicit’’ developmental mapping could evolve tiling patterns much
quicker than a variety of other representations (including direct) and further, that
they could be subsequently grown (iterated) to much larger sized patterns.
Hornby and Pollack [20] evolved context-free L-systems to define three
dimensional objects (table designs). They found that their generative system could
produce fitter designs faster than direct methods. The generative approach produced
more structures with more regularity and symmetry than direct methods.
Eggenberger investigated evolving developmental and non-developmental
genetic representations on the difficult problem of optical lens design [21]. He
found that the direct method scaled very badly when compared with the
developmental approach.
Roggen and Federici [48] compared evolving direct and developmental mappings
for the task of producing specific two dimensional patterns of various sizes (the
Norwegian Flag and a pattern produced by Wolfram 1D CA rule 90). They showed
1
With the exception of recent work in AVIDA, which takes a more utilitarian approach [37].
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in both cases that as the pixel dimensions of the patterns increased the
developmental methods out performed the direct.
Gordon [13] showed that evolved developmental representations were more
scalable than direct representations for digital adders and parity functions.
Sekanina and Bidlo [51] showed how a developmental approach could be
evolved to design arbitrarily large sorting networks. Kicinger [28] investigated the
problem of design in steel structures for tall buildings and found that CA-based
generative models produced better results quicker than direct representations and
that the solutions were more compact.
Clune et al. [10] investigated the use of an indirect mapping to encode weights in
an ANN and compared with a direct ANN mapping for leg control in simulated
quadruped robots. They found the indirect mapping evolved faster to produce better
robot locomotion and it also produced much more regular gaits. The indirect
mapping used to encode ANN weights is called compositional pattern producing
networks (CPPNs). It generates neural weights between neurons in planar arrays by
evolving a mapping from the Cartesian coordinates of the two neurons to a weight
[55]. Stanley refers to such a mapping as a ‘‘novel abstraction of natural
development’’. However, unlike biological development the technique does not
involve time or iteration.
Clearly, the evidence is growing that GDS representations may be more scalable
than more direct genotype representations. Despite this, in general it is still not clear
how and whether developmental representations have advantages in a more general
computational sense. This is because, firstly, investigations have concentrated on
particular systems such as neural networks, structural design, digital circuits or
sorting networks. Secondly, in some cases the demonstrations of greater scalability
of GDS are questionable, since authors, sometimes by their own admission, have
chosen rather naive direct representations in comparison with developmental
representations.
Arguably, there has been little focus on actual computation in GDS. Instead,
much work has concentrated on pattern formation. This requires a mapping stage
where the abstract pattern is mapped to a program, design or circuit. Since the
SMCGP approach is computational from the outset, a mapping from a phenotype to
a computation is already provided. The unified representation of SMCGP includes
both developmental and non-developmental functions. As a result, comparisons of
the computational efficiencies of explicitly non-developmental (CGP) and devel-
opmental mappings (SMCGP) are more meaningful in such a setting.
4 Self-modifying CGP
4.1 Overview
SMCGP has three distinct aspects: the underlying genotype representation, the
evolutionary algorithm and the developmental process.
Algorithm 1 gives a high-level overview of the process of mapping a genotype to
a phenotype through a process of development. The first stage of the mapping is the
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modification of the genotype. This happens through the use of evolutionary
operators acting on the genotype. The developmental steps in the mapping are
outlined in lines 3–8 if the algorithm. The first step is to make an exact copy of the
genotype and call it the phenotype at iteration 0. After this the self-modification
operators are applied to produce the phenotype at the next iteration. Development
stops when either a predefined iteration limit is achieved or it turns out that the
phenotype has no self-modifications operations that are active. It is important to
note that there are various ways in which there may be no active self-modification
operations. Firstly, no self-modification operations may exist in the phenotype.
Secondly, self-modification operations are present but they are non-coding. Thirdly
the self-modification operations may not be ‘‘triggered’’ when the instructions
encoded in the phenotype are executed. These various conditions will be discussed
in the detailed description in the following sections.
Algorithm 1 Overview of genotype, phenotype and development
1: Generate genotype
2: Copy genotype to phenotype. Iteration, i=0
3: repeat
4: Apply self-modification operations to phenotype i
5: increment i
6: Calculate fitness increment, f
i
7: until (iequals number of iterations required) OR (No self-modification functions to do)
8: Evaluate phenotype fitness Ffrom fitness increments, f
i
The representation of the genotype is described in detail in Sect. 4.3. It is based
on Cartesian Genetic Programming but includes a number of new features that
support the self-modification operators. The evolutionary algorithm that operates on
this representation is simple evolutionary strategy, and is described in Sect. 4.4.
4.2 Cartesian Genetic Programming (CGP)
The term ‘‘Cartesian Genetic Programming’’ (CGP) first arose in a paper 11 years
ago on the evolution of digital circuits published in the first conference on ‘‘Genetic
and Evolutionary Computation’’ [39]. The following year, in the first edition of
Genetic Programming and Evolvable Machines, Miller et al. [40] examined how
CGP could be used to find novel digital circuits and how general digital design
principles could be deduced. Also in 2000 the method was proposed as a new and
complete method of genetic programming [42].
CGP represents programs as directed graphs. One of the benefits of this type of
representation is the implicit re-use of nodes that is characteristic of graphs. In CGP,
the genotype is a fixed-length representation where each node in the directed graph
represents a particular function and is encoded by a number of genes. One gene
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encodes the function that the node represents, and the remaining genes encode
where in the graph the node obtains its inputs from. The nodes take their inputs in a
feed-forward manner from either the output of nodes in a previous column or from a
program input (terminal). Also, the number of inputs that a node has, is dictated by
the arity of the function it represents. The program data inputs are given the absolute
data addresses 0 to n-1 where nis the number of program inputs. The data outputs
of nodes in the genotype are given sequential addresses, column by column, starting
from nto n?m-1 where mis the user-determined upper bound of the number of
nodes (equal to the number of rows multiplied by the number of columns). If the
problem requires kprogram outputs, then kintegers are added to the end of the
genotype, each one being the absolute address of the output of a node where the
program output is taken from. The two dimensional general form of a Cartesian
Genetic Program is shown in Fig. 1.
CGP uses a genotype-phenotype mapping that does not require all of the nodes to
be connected to each other. So the phenotypes can have a length from zero to the
maximum number of nodes encoded in the genotype. Thus, areas of the genotype
can be inactive and have no influence on the phenotype. This means that many
genotypes decode to exactly the same phenotype and consequently their fitnesses
are identical. This genetic redundancy (often referred to as neutrality) has been
shown to highly beneficial to the evolutionary search of CGP genotypes [41,42,56,
64].
Fig. 1 General form of two-dimensional CGP. It is a grid of nodes whose functions are chosen from a set
of primitive functions. Two parameters c, and r, respectively, define the number of columns and rows in
the grid. Each node is assumed to take as many inputs as the maximum function arity a. Every data input
and node output are labeled consecutively (starting at 0) which gives it a unique data address which
specifies where the input data or node output value can be accessed (shown in the figure on outputs of
inputs and nodes). Nodes in the same column cannot be connected to each other. The graph is directed so
that a node may only have its inputs connected to either input data or the output of a node in a previous
column. In general there may be a number of output genes (O
i
) which specify where the program outputs
are taken from. The structure of the genotype is seen below the schematic. All node function genes F
i
are
integer addresses in a look-up table of functions. All connection genes C
ij
are absolute data addresses and
are integers taking values between 0 and the address of the node at the bottom of the previous column of
nodes
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The original form of CGP did not include Automatically Defined Functions [32].
However, later Walker and Miller [57,58], utilizing ideas from a technique known
as module acquisition, showed how sub-functions could be evolved and re-used in
CGP. Finally, it was shown that CGP is, under certain conditions, equivalent to a
particular form of linear GP [59].
4.3 The SMCGP representation
SMCGP’s representation, though similar to CGP, has some important differences.
SMCGP genotypes represent a linear string of nodes. That is to say, only one row of
nodes are used (in contrast to CGP which can have a rectangular grid of nodes).
Another important difference is how SMCGP represents connection genes. In CGP,
connection genes are absolute addresses, indicating where the data supplied to a
node is to be obtained, whereas SMCGP uses relative addressing. Each node obtains
its data via its connection genes by counting back from its position in the graph. As
in CGP, to prevent cycles, nodes can only connect to previous nodes (on their left).
The relative addressing allows sections of the graph to be easily moved, duplicated,
or deleted without breaking constraints of the directed graphical structure. Self-
modification also require extra genes that are used to identify parts or characteristics
of the graph that will be changed.
Another change from CGP, and previously published work on SMCGP is the way
SMCGP handles inputs and outputs. Terminals are acquired through special
functions (called INP, INPP, SKIPINP) and program outputs can be taken from a
special function called OUTPUT. This is an important change as it enables SMCGP
programs to obtain and deliver as many inputs or outputs as required by the problem
domain, during program execution. This allows the possibility of evolving general
solutions to problems.
To summarize, each node in the SMCGP graph contains a number of evolvable
elements:
The function. Represented in the genotype as an integer.
A list of (relative) connections addresses, again represented as integers.
A set of three floating point number arguments used by self-modification
functions.
There are also primitive functions that acquire or deliver inputs and outputs.
An example genotype is given in Fig. 2. The figure also shows purely
schematically some phenotypes arising at different iterations.
The actual number of inputs of a node is dictated by the arity of its function, and
in this paper there is a maximum of two inputs. If the connection gene specifies a
distance of 1 it will connect to the previous node in the list, if the gene has value 2
then the node connects 2 nodes back and so on. All the relative distances are
generated so that they are greater than zero, to avoid nodes referring directly or
indirectly to themselves.
If a gene specifies a connection pointing outside the graph, i.e., with a larger
relative address than there are nodes to connect to, then this is treated as connecting
to a null input. Terminals and outputs themselves are obtained by using special
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functions. This is described in detail in Sect. 4. This encoding is shown visually in
Sect. 4.7.
The relative addressing used allows, sub-graphs to be placed or duplicated in the
graph whilst retaining their semantic validity. This means that sub-graphs could
represent the same sub-function, but acting on different inputs. This can be done
without recalculating any node addresses thus maintaining validity of the whole
graph. So sub-graphs can be used as functions in the sense of ADFs in standard GP
(or modules in CGP [58]).
The three floating point arguments are used as arguments for the self-
modification functions, or for functions that return constant values. It is important
to note that depending on the function using them, the value may be truncated to an
integer. Section 4.8 details the available functions and their associated arguments.
Functions that are not explicitly computational (i.e., SM functions and output
functions) pass on the computations presented to them. This is discussed in
Sect. 4.5.
4.4 Evolutionary algorithm
In CGP, a (1 ?4) evolutionary strategy is often used. We do the same in this paper.
However, we begin the process by testing a population of 50 random individuals.
This helps to boot-strap the evolutionary algorithm and increases the chance of
obtaining a viable individual from which to build from. We then select the best
individual and generate four offspring by mutation. We test these new individuals,
and use the best of these to generate the next population (and if there are two or
more equally best, we pick the newer).
In the experiments in this paper, we have used a relatively high (for CGP)
mutation rate of 0.1. This means that each gene has a probability of 0.1 of being
mutated. SMCGP, like normal CGP, allows for different mutation rates to affect
different parts of the genotype (for example functions and connections could have
different mutation rates). In these experiments, for simplicity, we chose to make all
the rates the same. Mutations for the function type and relative addresses themselves
Fig. 2 The genotype maps
directly to the initial graph of the
phenotype. The genes control the
number, type and connectivity of
each of the nodes. The phenotype
graph is then iterated to perform
computation and produce
subsequent graphs. The nodes in
the phenotype that are acting as
outputs are outlined
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are unbiased; a gene can be mutated to any other valid value. Similarly, when
functions are mutated (or during the initial generation of individuals), all functions
in the set of functions chosen for a particular experiment, have the same probability
of being selected. We have yet to investigate the effect of biasing factors such as the
ratio of self-modifying to normal functions.
For the arguments, the mutation operator can choose to randomize the value
(with probability 0.1), or with probability 0.9 add noise (normally distributed, with a
standard deviation of 20).
The argument values have not been optimized and are based on our experiences
with CGP. We would expect performance increases if more suitable values were
used.
4.5 Evaluation of the SMCGP graph
The phenotype is executed in the same manner as standard CGP, so that the
computational output of the graph is obtained by recursion, starting from the output
nodes down through the functions, to the input nodes. In this way, nodes that are
unconnected are not processed and do not effect the behavior of the graph at that
stage.
For function nodes, such as ?,-and *, the output value is the result of the
mathematical operation on input values.
For graph manipulation functions (self-modifying), the graph is parsed from left
to right. The input values to nodes are found and the behavior of the node is based
on these input values. If the first input is greater or equal to the second, then the
graph manipulation function is added to a ‘‘To Do’’ list of pending modifications
and the node returns the first input. If a graph manipulation function is not added to
the ‘‘To Do’’ list it returns the value of its second input. This means that the
programs self-modifying behaviour is dependent on the particular data passing
through the graph. For Boolean functions, we add the operation to the ‘‘To Do’
regardless of the inputs.
The length of the list is usually limited as manipulations are relatively
computationally expensive to perform. In this paper we have limited the length to
just two instructions, unless stated to the contrary.
All graph manipulation functions require a number of arguments (evolved), that
determine how they operate on the graph. These are described in Sect. 4.8.
4.6 Inputs and outputs
In the classical implementation of CGP, inputs are defined as absolute addresses
that nodes can connect to. It was ensured that all node connection genes were
always a valid address. In the early versions of SMCGP [16], with the addition of
relative addressing, measures needed to be taken to deal with situations where node
connection addresses did not refer to any nodes on the graph (i.e., when addresses
went negative). To ensure that this could not happen the addresses were taken
modulo the number of inputs. This ensured that such connections always obtained a
valid program input. In this way, as the graphs grew, the addresses could reach more
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and more inputs without having to have the connections explicitly encoded (as
would be the case in CGP).
We found, however, that inputs still did not scale particularly well with problem
size, so in subsequent papers [17,18,19] we examined another strategy: Three
special input functions are now added to the function set: INP, INPP and SKIPINP.
When decoding the phenotype graph, a pointer is maintained that refers to an input.
If the first occurrence of an active input function is INP it returns the first program
input. If the first occurrence is INPP, the last program input is returned. After INP is
called it increments the input pointer. INPP decrements the input pointer. SKIPINP
allows the pointer to jump more than one input, and returns the current input before
moving the pointer by an amount given by the first argument of SKIPINP, P
0
.This
is truncated to an integer and decides whether to increment or decrement the input
pointer according to the sign of the argument. When the pointer is asked to move
beyond the last (or first) input, it simply wraps around to the other side of the input
list. This ensures that there is always an input available to be read.
2
Also in earlier work we included an extra binary gene with every node which
flagged whether the node could provide a program output [17,18,19]. However, in
the work for this paper we have removed output genes and instead introduced a
primitive function called OUTPUT that provides a program output. This was partly
done because we thought, like the introduction of input functions, it would improve
the ability of the approach to scale to larger problems with different numbers of
outputs and also to make the input-output mechanisms more consistent.
A number of measures need to be taken when the number of OUTPUT nodes
does not match the number of program outputs.
If there are no OUTPUT nodes in the graph, then the last nnodes in the graph
are used.
If there are more OUTPUT nodes than are required, then the right-most
OUTPUT nodes are used until the required number of outputs is reached.
If the graph has fewer OUTPUT nodes than are required graph, then nodes are
chosen as outputs by moving forwards from the right-most node flagged as an
output.
If there is a condition where not enough nodes can be used as outputs (as there
are not enough nodes in the graph), the individual is labeled as corrupt and is
given a bad fitness score to prevent selection.
4.7 Examples
The figures used throughout this paper are generated automatically by the SMCGP
program. When reading the graphs there are several important things to note:
Each function has a different colour (or shade of grey), however, the same
colour may be used differently in different function sets.
2
In this model, if a node wishes to connect to a negative address, a default value is returned. For binary
problems this value is FALSE, for numeric problems the default value is 0. INP, INPP and SKIPINP are
all terminal functions with no inputs. So connection genes are ignored
408 Genet Program Evolvable Mach (2010) 11:397–439
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The pictured graphs cannot be converted back into phenotypes. For clarity many
details have been omitted (such as the values for parameters).
Unconnected nodes are drawn with a smaller square and without their function
type as a label.
In figures with more than one phenotype, each graph represents one iteration.
Figure 3shows a phenotype with one output node and two INP (input) nodes.
The output of the program will be the Binary XOR (BXOR) of the two input nodes.
INP returns the next available input. If the program only had one input, both of the
INP nodes would return the same value. If there are two inputs, the INP node on the
left will return the first input, the next INP node will return the second input. If there
were more than two inputs, the additional inputs would be ignored.
Figure 4introduces the INPP node. Suppose that the program has two inputs x
0
and x
1
. The leftmost INP function returns x
0
, the next INP function returns x
1
. The
INPP returns x
1
also, this is because the second INP function would have left the
input pointer pointing to x
0
(due to the list of inputs being a circular list). Thus the
first BXOR function returns x0x1and the second BXOR function returns x1
x0x1:Since the OUTPUT function is directly connected to this it returns the
same.
Figure 5shows a simple phenotype with two outputs. The first output value is
equal to the first input. The second output is the exclusive-OR of the first two input
values.
Figure 6also shows a phenotype with two outputs (the nodes used as outputs are
drawn with a box around them). However, here there is only one OUTPUT node in
the phenotype. The SMCGP interpreter attempts to find other nodes to use as
outputs. Here the next node has been selected to be used as an output.
Figure 7demonstrates the use of the DUP (duplicate) operator. The duplication
operator here duplicates a section of the graph made of three nodes BXOR, INP, and
BXOR. It inserts them next to the DUP node. The source of these three nodes is the
BXOR node used in the first iteration and it’s neighbouring two nodes to the right.
This demonstrates two things. The first is how the duplication operator can make
Fig. 3 Example showing the a simple phenotype with 1 output and 2 input nodes
Fig. 4 Example showing the a simple phenotype with 1 output and 3 input nodes
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programs grow by inserting copies of other parts of the phenotype into the next
iteration. The other observation is that SMCGP can move nodes that were
previously unconnected and connect them to form part of the program.
Phenotypes in SMCGP can also shrink. Figure 8shows an example of the DEL
(deletion) operator in use. In the first iteration, the program uses three inputs. In the
next iteration, the DEL node has removed the first node. The left most BXOR no
longer connects to an input, but instead receives copies of the ‘‘default’’ value. The
INP (input nodes) in the first iteration use inputs 0, 1 and 2. In the next iteration the
nodes would use inputs 0 and 1. In a further iteration, the first node (BXOR) is
removed. Each time the DEL occurs the program’s functionality is also changed.
4.8 Self-modification functions
The way self-modifying functions act is defined by four variables. Three of them are
the argument genes that are double precision numbers. We denote them P
0
,P
1
, and
P
2
. The other variable is the integer position in the phenotype of the self-modifying
node (i.e., the leftmost node is position 0). We denote this x. In the definitions of the
SM functions we often need to refer to the values taken by node connection genes
(which are all relative addresses). We denote the jth connection gene on node at
position i,byc
ij
.
There are several rules that decide how addresses and arguments are treated:
When P
i
are added to the x, the result is treated as an integer.
Fig. 5 Example showing multiple outputs
Fig. 6 Example showing a phenotype with multiple outputs, but only one OUTPUT node
Fig. 7 Example showing the use of DUP (duplicate operator)
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Address indexes are corrected if they are not within bounds. Addresses below 0
are treated as 0. Addresses that reach beyond the end of the graph are truncated
to the graph length.
Start and end indexes are sorted into ascending order (if appropriate).
Operations that are redundant (e.g., copying 0 nodes) are ignored, however, they
are taken into account in the ‘‘To Do’’ list length.
The exact rules obeyed by various graph manipulation functions are shown in
Table 1.
The list of self-modification functions in Table 1is large and some are quite
complex, however, it is at this stage unclear what the minimal useful set of self-
modifications should be. To some extent this question may only be answerable
through extensive experimentation using evolution.
4.9 Function sets for experiments
There are a number of different function sets used in these experiments, so they
have been grouped into sets as in Table 5. Table 6contains the set of functions in
each of these groups. The choice of function sets is determined by the problem type,
and by any previous work that we wish to compare with. Tables 2,3and 4detail the
various individual functions.
5 Experiments: digital circuits
5.1 Fitness function for parity and adder
In this section we describe how to use SMCGP to evolve programs that generate
adder and parity circuits having an arbitrary number of inputs. The aim is to evolve
Fig. 8 Example showing the use of DEL (deletion operator)
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a program that on each iteration, produces the next larger circuit by taking more
inputs and performing the appropriate function (even-parity or bitwise addition).
Even parity circuits consist of ninputs, and a single output that is true when there
are an even number of True bits in the input. For the adder, the circuits take two
n-bit, binary encoded integers and return one n?1-bit number that is the numerical
sum of the two inputs.
Digital circuits have often been studied in genetic programming [30,32], and
some systems have been used to produce ‘‘general’’ solutions [22,60,61,63]. A
general solution is a program that can output a digital circuit for an arbitrary number
of inputs, for example it may generate a parity circuit of any size.
Table 1 Definition of the self-modification functions
Basic
Delete (DEL) Delete the nodes between (P
0
?x) and (P
0
?x?P
1
)
Add (ADD) Add P
1
new random nodes after (P
0
?x)
Move (MOV) Move the nodes between (P
0
?x) and (P
0
?x?P
1
) and insert after
(P
0
?x?P
2
)
Duplication
Overwrite (OVR) Copy the nodes between (P
0
?x) and (P
0
?x?P
1
) to position
(P
0
?x?P
2
), replacing existing nodes in the target position
Duplication (DUP) Copy the nodes between (P
0
?x) and (P
0
?x?P
1
) and insert after
(P
0
?x?P
2
)
Duplicate preserving
connections (DU2)
Copy the nodes between (P
0
) and (P
0
?P
1
) and insert after (P
0
?P
2
)
Duplicate preserving
connections (DU3)
Copy the nodes between (P
0
?x) and (P
0
?x?P
1
) and insert after
(P
0
?x?P
2
). When copying, this function modifies the c
ij
of the copied
nodes so that they continue to point to the original nodes
Duplicate and scale
addresses (DU4)
Starting from position (P
0
?x) copy (P
1
) nodes and insert after the node at
position (P
0
?x?P
1
). During the copy, c
ij
of copied nodes are
multiplied by P
2
Copy to stop
(COPYTOSTOP)
Copy from xto the next ‘‘COPYTOSTOP’’ or ‘‘STOP’’ function node, or
the end of the graph. Nodes are inserted at the position the operator
stops at
Stop marker (STOP) Marks the end of a COPYTOSTOP section
Connection modification
Shift connections
(SHIFTCONNECTION)
Starting at node index (P
0
?x), add P
2
to the values of the c
ij
of next P
1
nodes
Shift connections 2
(MULTCONNECTION)
Starting at node index (P
0
?x), multiply the c
ij
of the next P
1
nodes by P
2
Change connection (CHC) Change the (P
1
mod 3)th connection of node P
0
to P
2
Function modification
Change function (CHF) Change the function of node P
0
to the function associated with P
1
Change parameter (CHP) Change the (P
1
mod 3)th parameter of node P
0
to P
2
Miscellaneous
Flush (FLR) Clears the contents of the ‘‘To Do’’ list
P
i
are the evolved arguments of the self-modification functions. xrepresents the absolute position of the node
in the graph, where the leftmost node has position 0. c
ij
is the jth connection gene on node at position i
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In the case of parity, to begin with, we evolve for two input bits. When a
successful solution is found, the fitness function requires that the program produces
a two bit circuit, followed by a three bit circuit. Then when a genotype has been
found that solves the two bit problem and on iteration solves the three bit problem,
the fitness function changes so that now in addition to the previous behaviour the
genotype, when iterated twice, produces a phenotype that solves the four-bit
problem. The process continues in this way until we obtain a phenotype that
correctly implements the required function with 20 inputs. We refer to the
application of each parity or adder as a test case.
Fitness is computed as the number of correctly predicted bits over all test cases. If
the candidate solution fails to find a totally correct solution for a given input size, it
is not tested on other input sizes. We evolve for 19 test cases (2 inputs to 20 inputs).
The fitness function is designed to force the SMCGP to find a solution that grows
through each test case to the next. In this way, the chance of a general solution is
maximised. The fitness function pseudo code is shown in Algorithm 2.
Table 2 Binary functions
Function Operation
BAND a AND b
BOR a OR b
BNAND NOT (a AND b)
BXOR a XOR B
BNOR NOT (a OR b)
BNOT NOT a
BIAND (NOT a) AND b
BF0 FALSE
BF1 (a AND b)
BF2 a AND (NOT b)
BF3 (a AND (NOT b)) or (a AND b)
BF4 (NOT a) AND b
BF5 ((NOT a) AND b) OR (a AND b)
BF6 ((NOT a) AND b) OR (a AND (NOT b))
BF7 ((NOT a) AND b) OR (a AND NOT(b)) OR (a AND b)
BF8 ((NOT a) AND (NOT b))
BF9 ((NOT a) AND (NOT b)) OR (a AND b)
BF10 ((NOT a) AND (NOT b)) OR (a AND NOT (b))
BF11 ((NOT a) AND (NOT b) OR a AND (NOT b) OR a AND b)
BF12 ((NOT a) AND (NOT b) OR (NOT a) AND b)
BF13 ((NOT a) AND (NOT b) OR (NOT a) AND b OR a AND b)
BF14 ((NOT a) AND (NOT b) OR (NOT a) AND b OR a AND (NOT b))
BF15 ((NOT a) AND (NOT b) OR (NOT a) AND b OR a AND (NOT b) OR a AND b)
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Table 3 Mathematical
functions Function Operation
NOP No operation
DADD a ?b
DSUB a -b
DMULT a * b
DDIV a / b
CONST a constant, defined by P
0
AVG (a ?b) / 2
DSQRT Square root of a
DRCP 1 / square root of a
DABS Absolute value of a
TANH tanh(a)
TANHNN tanh(a ?b)
FACT Factorial
POW a
b
COS cosine(a)
SIN sine(a)
MIN min(a, b)
MAX max(a, b)
IFLTE If (a\0) return b, else 0
INDX Current node index
INCOUNT Number of inputs
Table 4 Input and output functions
Function Operation
INP Return input pointed to by current_input, increment current_input
INPP Return input pointed to by current_input, decrement current_input
SKIPINP Return input pointed to by current_input, current_input =current_input ?P
0
OUTPUT Return data provided
P
0
is the first argument gene
Table 5 The function set used
in each experiment
See Table 6for the definition of
each group
Experiment Function set
Adder A
Parity (full) B
Parity (reduced) C
Fibonacci D
Pi (patterns) E
Approximating Pi F
Power regression F
Classification F
Squares D
Sum of numbers D
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Algorithm 2 Fitness function
1: Fitness, F=0
2: Copy genotype to phenotype. Iteration, i=0
3: repeat
4: BREAK =FALSE
5: Apply self-modification operations to phenotype i
6: increment i
7: Calculate fitness on test case, f
i
, by counting number of incorrect bits
8: if f
i
=0then
9: BREAK =TRUE
10: end if
11: F=F?f
i
12: until i=LIMIT OR BREAK
For the parity case LIMIT = 19, so the largest parity function tested has 20 inputs.
However, we test the solutions for generality by testing to 24 bits. We chose 24 bits
for two reasons. First, the largest evolved parity circuit we found in the literature
was 22 bits [45]. It should be noted that Poli and Page used all 16 two input Boolean
functions in their function set, whereas we just use AND, NAND, OR and NOR.
Secondly, this is the largest circuit we can test within reasonable time and
reasonable memory requirements.
Essentially, the fitness function used for evolving both types of circuit is the
same, except that for the adder fitness function, the number of inputs and outputs
grow each time. So the genotype should add two binary inputs, the phenotype at the
first iteration, should add two two-bit binary numbers, and so on. The LIMIT =6,
but we tested evolved solutions further in order to check for generalization. Due to
the demand on computational resources, we stop exhaustively evaluating the
circuits at 10 bits (i.e., the addition of two ten bit numbers). For larger sizes, we
sample the input space by testing with 10,000 random numbers having up to 1,000
inputs.
For the adder, function set A (shown in Table 6) is used. For the parity
problem, two different function sets were compared. One contains all two-input
Boolean functions, set B in Table 6and the other which contains a reduced set of
Boolean functions, set C. Both sets of functions have been used in work by other
authors.
5.2 Parity results
Table 7shows the average number of evaluations required to evolve for a given
number of inputs. The success rate was 100%. Results are based on 50 trials per
function set. Although, Poli and Page evolved solutions to even-parity up to 22
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inputs, they only gave numbers of evaluations for a single evolutionary run when
the number of inputs was 13, 15, 17, 20 and 22 [45].
In Table 8, the results are compared to previous CGP representations and
Koza’s figures for GP with ADFs [32]. The SMCGP results are clearly highly
competitive. We have included Koza’s figures for reference. Koza calculated the
computational effort for a 99% success rate and so represent the number of
evaluations assuming the most favourable number of runs and numbers of
generations. More detailed comparisons between CGP and other methods have
been previously published in [58]. There it was seen that Embedded Cartesian
Genetic Programming (ECGP) was highly competitive with other GP methods. It
Table 6 Function sets per experiment
Function
name
ABCDEF
OUTPUT X X X X X X
INP XXXXXX
INPP X X X X X
SM XXXXXX
NOP XX
DADD X X X
DSUB XX
DMULT X X
DDIV XX
CONST XX
DSQRT XX
POW XX
COS XX
SIN XX
MIN XX
MAX XX
AVG XX
DRCP XX
DABS XX
TANH XX
LOG XX
LN XX
INCOUNT X
BAND X X
BNAND X X
BXOR X
BNOR X X
NOR X X
BIAND X
BF0 to BF15 X X
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should be noted that SMCGP is solving a different problem than CGP and ECGP.
SMCGP were evolved to solve not just one instance of the parity problem, but a
sequence of parity problems.
After evolution, solutions were tested for inputs of up to 24 bits. It was found that
all solutions generalized to problems of this size. This is an improvement over our
previous work, where it was found that 96% of solutions generalised to all tested
problems.
For both sets of functions, most solutions apparently had generalized when even-
5 parity is reached. Since, when the genotype can solve 3, 4 and 5 input problems it
Table 7 Average evaluations
required to find a program that
will solve parity up to a given
number of bits (50 runs)
Results are for both function sets
used
Input bits Reduced Full
3 247,753 37,276
4 275,663 41,697
5 278,635 43,016
6 298,104 43,593
7 318,376 150,719
8 322,843 150,721
9 322,843 150,722
10 322,843 150,722
11 322,851 150,722
12 322,851 150,722
13 322,866 150,722
14 322,866 150,722
15 322,866 150,722
16 322,866 150,722
17 322,870 150,722
18 322,870 150,722
19 322,874 150,722
20 322,874 150,722
Table 8 Comparison with previous work on evolving parity
Input bits Reduced Full SMCGP 2007 CGP ECGP GP
4 275,663 41,697 28,811 81,728 65,296 176,000
5 278,635 43,016 58,194 293,572 181,920 464,000
6 298,104 43,593 199,256 972,420 287,764 1,344,000
7 318,376 150,719 410,128 3,499,532 311,940 1,440,000
8 322,843 150,721 1,080,656 10,949,256 540,224
GP is Koza’s tree GP (with ADFs), ECGP is embedded CGP and CGP is conventional CGP. With the
exception of the figures of Koza, the figures show average evaluations required to find a given sized parity
circuit. Results for higher numbers of inputs are not available for CGP or ECGP. The figures for Koza
represent computational effort so they represent a minimum number of evaluations required to achieve
99% success. The minimum is selected from the ‘‘ideal’’ number of runs and number of generations. The
blank symbol indicates that figures are unavailable
Genet Program Evolvable Mach (2010) 11:397–439 417
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very often was able to solve all other tested parity sizes. This is reflected in the
results in Table 7, where the number of evaluations required to solve a problem
stops increasing, because once a solution is found to generalize, no further evolution
is required.
Spector also examined the even-parity problem, however, he used a different
function set [53]. He found better scaling behaviour than Koza on even-parity
functions up to six inputs.
Other approaches have looked at finding general solutions. Huelsbergen [22]
evolved machine language level programs that could iterate over the bits in a string
and from this parity could be easily determined. The solutions would be suitable for
any length bit string. Recursion has also been successfully used to solve the parity
problem [61,62,63]. These approaches produced programs rather than circuits to
solve the problem. They also used high level programming constructs rather than
purely boolean logical primitives.
5.3 A general solution to parity
The genotype in Fig. 9was evolved with a ‘‘To Do’’ list length of 1. The 20 node
genotype only has 7 active nodes. The inactive nodes are shown as unconnected
smaller squares. Nodes INPP at positions 0 and 2, obtain inputs x
1
and x
0
,
respectively. Three Boolean functions BNOR, BAND and BOR appear at positions
4, 5, and 6. The OUTPUT function obtains the single output from the BOR node.
The only active SM node is DUP at position 1. It carries arguments which cause it to
copy eight nodes beginning at the node on its left (INPP) and insert them
immediately after itself. The action of the DUP node is shown using the curved line
with an arrow emanating from the box. Since the genotype has no connections that
are left of the first node, when DUP copies it disconnects the first two nodes in the
generated phenotype. These appear at the beginning (left) of the new phenotype
(iteration 1) at positions 0 and 1. It is important to note that in this phenotype a
previously inactive node (BNAND) at position 9 becomes active.
We can see that the genotype computes even-2 parity as follows. Denote the
outputs of node, iby z
i
. Note denotes the exclusive-OR operation. When two or
more Boolean arguments are side by side (as if being multiplied), it is assumed that
the Boolean AND (BAND) operation is applied to the arguments (e.g., xy is
equivalent to BAND (x,y)). Overbar represents inversion.
z0¼x1
z1¼x1
z2¼x0
z4¼BNORðz2;z1Þ¼x0þx1¼x0x1¼ð1x0Þð1x1Þ
z5¼BANDðz1;z2Þ¼x1x0
z6¼BORðz5;z4Þ¼z5þz4¼z5z4z4z5
ð1Þ
Substituting for z
5
and z
4
, expanding and then canceling terms we obtain
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Fig. 9 An evolved genotype iterated twice that computes even-parity
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123
Fig. 10 The inductive hypothesis in diagrammatic form
420 Genet Program Evolvable Mach (2010) 11:397–439
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z6¼x1x0ð1x0Þð1x1Þx1x0ð1x0Þð1x1Þ
z6¼x1x01x1x0x1x0x1x0ð1x1x0x1x0Þ
z6¼x1x01ð2Þ
Thus z
16
=z
6
is the even-3 parity function. When the eight duplicated nodes are
inserted into the genotype just before position 2 they cause the activation of the
BNAND node at position 9 in the new phenotype. This inverts the function
computed by the eight duplicated nodes in the genotype. So the output of this block
of nodes (denoted A), is x2x1since the INPP functions return the inputs in
descending order.
Now we turn out attention to the second iteration. When DUP inserts nodes 2 to 9
after itself, the nodes 4 to 9 are shifted right (becoming nodes 12–17) in the second
iteration phenotype (enclosed in a box labeled B in the Fig. 9). We now prove that
this set of nodes carries out the exclusive OR of its input (emanating from the
NAND node, which we call y) with the input variable (in this case x
1
) (Fig. 10).
z12 ¼x1
z14 ¼BNORðx12;yÞ¼BNORðx1;yÞ¼ð1x1Þð1yÞ
z15 ¼BANDðy;z12Þ¼BANDðy;x1Þ¼x1y
z16 ¼BORðz15;z14 Þ¼z15 þz14 ¼z15 z14 z14z15
z17 ¼BNANDðz16;z16Þ¼z16 1
ð3Þ
Substituting for z
14
and z
15
in z
16
and then noting that in the last term when x
1
y
multiplies ð1x1Þwe obtain ðx1yx1yÞwhich is zero, thus we can simplify
z16 ¼x1yð1x1Þð1yÞx1yð1x1Þð1yÞ
z16 ¼x1yð1x1Þð1yÞ¼x1y1x1yx1y
z16 ¼1x1y
z17 ¼x1y
ð4Þ
Since we have seen that the nodes in section C compute the odd parity of the
supplied input yand the acquired input (by INPP) we find that at iteration two the
phenotype computes y1x01¼yx1x01¼x3x2x1x01:This
is the even-4 parity function. To construct a proof by induction we will assume that
for ninputs the phenotype computes even-nparity.
Table 9 Evaluations required
to evolve to each size
An ninput adder adds two n-bit
numbers
No. of bits in each pair Average Evals % Success
1 2,415 100.0
2 952,965 94.0
3 1,043,732 88.0
4 1,083,890 86.0
5 1,237,723 86.0
6 1,439,856 86.0
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The upper diagram shows the even-nparity function E
n
. This is the inductive
hypothesis. We have already seen that the function enclosed in box A produces at
the next iteration the two disconnected nodes and the function in A, y¼xnxn1
followed immediately by the function in box B, yn2¼yxn2;thus the new
phenotype, E
n?1
generates the function,
Enþ1¼yn2Enxn1xn2
Enþ1¼yxn2Enxn1xn2
Enþ1¼xnxn1xn2Enxn1xn2
Enþ1¼xnEn
ð5Þ
Thus the inductive hypothesis also applies for the n?1th iteration. We have
already seen that at iteration two, the form of the phenotype obeys the inductive
hypothesis. Hence the general case is proved.
5.4 Adder results
Table 9shows the average number of evaluations required to evolve a program that
could grow to an adder of a given size (via intermediate sizes). We analysed when
the adder solutions began to generalise (i.e., could solve up to 6 bits addition, even
though they were evolved to solve a smaller problem).
After evolving to a 6 ?6 bit adder, the successful solutions were tested on larger
problems. Adders were tested up to 1,000 ?1,000 bits, and remarkably, all were
found to successfully generalise. Again, we had to sample the input space for larger
problem sizes. We took 1,000 different input patterns for every input size from 10 to
1000 bits.
5.5 A general solution to n-bit binary addition
We prove formally that an evolved genotype produces a general adder. The
genotype was evolved with a ‘‘To Do’’ list of length equal to 1. It is 20 nodes long
but only 7 nodes are active. This can be seen in the first row of Fig. 11. There are
two input obtaining nodes INP at positions 0 and 1. There are three Boolean
functions used BF6 (Exclusive-OR), BF11 and BF1 appearing at positions 3, 4, and
6. The OUTPUT function receives the output from the BXOR node at position 3.
The only active SM node is DUP at position 2. Its arguments cause it to copy 14
nodes beginning at the node on its left (BF6) and insert them (at the next iteration)
immediately before the node at 17. The action of the DUP node is shown using an
arrow emanating from the box containing the 14 nodes that will be re-inserted. After
insertion in the phenotype after node 16 the formerly inactive nodes, 7, 10, 11, 14,
15, 16 (see box B) are activated by nodes on the right (in box A). After the copying
operation the nodes in box A in the genotype, beginning with node 3, become the
nodes in the new phenotype beginning at node 17 (box A). As a result, new inputs
(inside box B) are obtained through the action of INP nodes at positions 10 and 11
and two more SM functions DU2 and DUP, located at positions 7 and 14,
respectively, become active. However, because the length of the ‘‘To Do’’ list is 1
422 Genet Program Evolvable Mach (2010) 11:397–439
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x1y
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
x1y
1y
2
x2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
Copy here
A
BA
B
ADD(x1, y1, c0)
Fig. 11 A schematic of an evolved genotype and two subsequent phenotypes which represent a general binary adder
Genet Program Evolvable Mach (2010) 11:397–439 423
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these instructions have no self-modifying role. Instead they have the passive
computational role of passing their second input. When the phenotype at the first
iteration is executed the DUP function (at position 2) cause fourteen more nodes to
be copied after node 16. This is shown in box B in the second iteration (Table 10).
To establish that these operations when repeated will construct an adder of an
arbitrary size we need to consider the way the phenotypes can be divided into
recognizable modules (i.e., a series of simple adders). In Fig. 12 we show how the
phenotypes can be decomposed into a series of one-bit adder modules. Initially the
genotype implements a one bit adder without carry, ADD(x0, y0). A one bit adder
with carry-in, C
in
, and carry-out, C
out
is defined by the truth table shown in
Table 11.
The equations for the sum and carry bits, Sand C
out
are given below.
S¼xyCin
Cout ¼xy CinðxyÞð6Þ
Let us examine the nodes in the box labeled ADD(x0,y) in Fig. 12. First note that it
uses functions BF
6
,BF
11
and BF
1
(see 2). BF
6
is the exclusive-OR function, BF
1
is
the AND function and BF11ða;bÞ¼1bab:The equations below show the
outputs z
i
of all the active nodes with labels i.
z0¼x0
z1¼y0
z2¼y0
z3¼x0y0
z4¼1x0y0x0ðx0y0Þ¼1y0x0y0
z5¼x0y0
z6¼y0ð1y0x0y0Þ¼x0y0
ð7Þ
When C
in
= 0 in Eq. 6and the resulting equations compared with Eq. 7one easily
sees that z
5
and z
6
are identical with Sand C
out
, respectively. Thus the active nodes
in the genotype implement a one-bit adder with no carry-in. The next thing we need
to show is that the nodes in Fig. 12 in the box labeled ADD(x1, y1, c0) do indeed
implement a one bit adder with carry-in and carry-out (Eq. 6). Let us examine the
Table 10 Percentage of
solutions that generalise to
various, un-evolved input
lengths
No. Of inputs % Success
10 80
50 76
100 76
250 76
500 76
750 76
1000 76
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x0y0
x0y
0x1y
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
x0y
0x1y
1y
2
x2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
ADD(x0, y0)
ADD(x0, y0)
ADD(x0, y0)
ADD(x0, y0, c0)
ADD(x1, y1, c0) ADD(x2, y2, c1)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
Fig. 12 After each duplication (box B in Fig. 11) a one-bit adder with carry ADD(x, y, c) is created which receives new inputs through two activated INP functions and
the carry from the previous stage through a passive DU2 node. It in turn passes the computed sum to an OUTPUT node and the new carry to a DU2 node
Genet Program Evolvable Mach (2010) 11:397–439 425
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nodes 10–20. One more additional Boolean function, BF7ða;bÞ¼abab is
used. Once again we write equations for the output of all nodes.
z10 ¼x1
z11 ¼y1
z14 ¼c0
z15 ¼BF7ðy1;x1Þ¼x1y1x1y1
z16 ¼x1y1
z17 ¼x1y1c0
z18 ¼BF11ðc0;x1y1c0Þ¼1ðx1y1c0Þc0ðx1y1c0Þ
¼1ðx1y1Þc0ðx1y1Þ
z19 ¼x1y1c0
z20 ¼BF1ðx1y1x1y1;1ðx1y1Þc0ðx1y1ÞÞ
¼ððx1y1Þx1y1Þð1ðx1y1Þc0ðx1y1ÞÞ
¼ððx1y1Þðx1y1Þc0ðx1y1Þx1y1x1y1ðx1y1Þc0x1y1c0x1y1
¼x1y1c0ðx1y1Þð8Þ
Comparing with Eq. 6we see that the output Sis correctly obtained from z
19
and the
carry-out is obtained from z
20
. In the phenotype at iteration 2 the carry out is passed
to the next module by the SM node DU2 (node 21).
So to summarize, at the first iteration. the duplicated section of the phenotype
activates previously inactive nodes which form the front section of a one-bit adder
and also activates a SM node which passes the carry of the previous adder (via a
DUP node) into the newly formed adder. In the second iteration the phenotype
consists of three adders connected in the manner of a ripple carry adder. The first
adder block originated in the original genotype (first phenotype). It passes out the
sum bit computed to an OUTPUT node and passes out its carry (ADD(x0, y0)) to
the next adder block, which is a full one bit adder (with carry-in and carry-out). This
is shown as ADD(x1, y1, c0). This in turn passes its carry (c1) to the final adder
block, ADD(x2, y2, c1) which passes out the two most significant sum bits. The
process is entirely regular and it can be easily seen that carrying out further
iterations adds a new adder block, ADD(x, y, c) into the existing adder. Thus we can
see that the iterated genotype represents an arbitrary bit adder.
Table 11 Truth table of a one-bit adder with carry-in and carry-out
C
in
xyC
out
SC
in
xyC
out
S
0 000 00 010 1
0 100 10 111 0
1 000 11 011 0
1 101 01 111 1
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We have many different solutions for adders which appear to be general and it is
highly likely that some of these are very innovative ways of building general adders.
Such analysis remains for the future.
6 Experiments: patterns and sequences
6.1 Digits of p
The task here is to find a program that on each iteration will output the next digit of
p, i.e., the first time the program is executed it outputs 3. Then after self-
modification is applied, the program encoded in the phenotype again should output
1. Then 4, 1, 5, and so on.
The program inputs are the iteration, i, and the previous output value.
An integer data type was used. Again, function set E was used (see Table 6). The
fitness function terminated iteration when an incorrect digit was given as output.
Fitness is defined as the total number of correct digits that were output before
making a mistake. Evolution was allowed to continue for 10,000,000 evaluations (or
100 digits of p).
6.2 Digits of presults
The experiment was repeated 310 times. The longest sequence found was 31 digits.
The shortest was 5 digits, and the average number of correct digits was 14. The best
evolved solution produced 31 digits of p, before outputting an incorrect digit. The
evolved output sequence and the expected output sequence are shown in Table 12.
Figure 13 shows the development stages of the first ten iterations of this program.
Each iteration outputs the next digit of p, starting with 3 in the first step. The
program consistently uses the final node, a copy to stop (CTS) function, as its
output. This is because no OUTPUT nodes were used, so the graph runner defaulted
to using the last node in the graph as output.
The program unfolds as follows. In iteration 0, CTS (Copy To Stop) returns the
constant 3.29, but because the program is interpreted as integers, the value is
truncated to 3. In iteration 1, INP returns the first input, which is the current iteration
(1). The CTS node returns the DSQRT (square root) of 1. In iteration 2, CTS returns
the truncated value of the constant, i.e., 4. In iteration 3, the CTS node returns the
square root of the square root of the current iteration (3). As a truncated integer, this
Table 12 Output from the pgenerator compared to actual digits
Evolved output Correct digits
3141592653 5897932384 6264338327
9 653334444 4444444444 4444444444
4444555555 5555555555 5555555555
5555555555
3141592653 5897932384 6264338327
9 502884197 1693993751 0582097494
4592307816 4062862089 9862803482
5342117068
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Fig. 13 The first ten developmental steps of a program that produces a pdigits sequence
428 Genet Program Evolvable Mach (2010) 11:397–439
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is 1. In iteration 4, again, the CTS node returns a value (5) from a constant. In
iteration 5, the output comes from adding two constants 4 and 5, to return 9. In
iteration 6, here the CTS node connects to a MOV (Move) function which is
returning the square root of 6 (as integer), i.e., 2. In iteration 7, the output (6) is the
sum (DADD) of 4 and 2 (which is the integer square root of 7). In iteration 8, the
output comes from the average (AVG) of 4 and 6 (via the same calculations as
iteration 7), to get 5. In iteration 9, the input value is 9, so the square root function
now outputs 3. The left most CTS function returns 3 (via the MOV node connected
to the top input). This is because of the order of modification nodes has reached a
limit on the ‘‘To Do’’ list, and the operation has failed—changing which of the
inputs is selected as the output. The program continues in this fashion for the first 31
digits.
6.3 Squares
In this task, we ask that evolution finds a program that generates a sequences of
squares 0, 1, 4, 9, 16, 25, ... without using multiplication or division operations. As
Spector [54] who first devised this problem points out, this task can only be
successfully performed if the program can modify itself—as it needs to add new
functionality in the form of additions to produce the next integer in the sequence.
Hence, genetic programming, without iteration, will be unable to find a general
solution to this problem.
6.4 Squares results
Programs were evolved using a very restricted function set. Programs have one
input: the current iteration. The initial graph size was set to 20 nodes. Mutation rate
0.1. Function set D from Table 6was used.
Out of 50 trials and a maximum of 1,000,000 evaluations, all successfully
evolved the first ten outputs correctly. The average number of evaluations needed
was 35,224. The minimum and maximum number of evaluations are 135 and
249,969. With a standard deviation of 53,609. Of these, 84% were found to
successfully generalise to the first 100 values.
6.5 Fibonacci
In a task similar to the squares problem (see Sect. 6.3), we evolve a program that
when iterated produces the Fibonacci sequence. Again, we limit the function set to
force evolution to find a solution that requires self-modification.
We evolve for both the first nand mnumbers in the sequence and test for
generality to 42 numbers (after which the value exceeds a long int). We have
previously noted that the programs produced by evolution generalize although they
have an irregular pattern to begin with (see Sect. 6.3). We were intrigued to see the
behaviour when starting the base case for Fibonacci at either the arbitrary start of 1,
1 or at the next step of that sequence 1, 2.
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6.6 Fibonacci results
Programs were evolved using a very restricted function set. Programs have one
input, the numeric constant 1. The initial graph size was set to 20 nodes. Mutation
rate 0.1. Function set D was used (see Table 6).
Out of 50 trials and a maximum of 1,000,000 evaluations, all successfully
evolved the first ten outputs correctly. The average number of evaluations needed
was 71,812. The minimum and maximum number of evaluations are 630 and
924,333. With a standard deviation of 162,998. Of these, 46% were found to
successfully generalise to the first 100 values.
7 Experiments: mathematical
7.1 Approximating p
There exist several iterative approaches to approximating p[9]. For example, the
Gregory-Leibniz series calculates:
p¼4
14
3þ4
54
7þ4
9...
This series is simple, but requires a large number of iterations to reach good
accuracy. Another method
3
uses recursion to find an approximation:
p¼nðtanðp
=
nÞtan3ðp
=
nÞ
3þtan5ðp
=
nÞ
5tan7ðp
=
nÞ
7...Þ
for n=1, 2, . Curiously, there has been little work on evolving approximators to
p, despite it being a well defined problem with many human designed solutions to
compare against.
Krohn et al. [33] employed an artificial developmental system based on fractal
proteins [7] to produce approximations to pusing two different approaches. The first
approach was to generate the digits as a binary sequence. The second, and more
successful, approach was to use the output of the developmental system to provide
values for the equation:
X
I
i¼1PN
n¼2Bn;i
Qi
t¼1B1;i
where Iis the number of developmental iterations, Nis the number of behavioural
genes (an output of the evolved program) and B
n,i
is the output of the nth
behavioural gene at iteration i.
Our fitness function was designed to produce a program where subsequent
iterations of the program would produce more accurate approximation to p.
Programs were allowed to iterate for a maximum of ten iterations. If the output after
an iteration did not approximate pmore closely, evaluation was stopped and a large
3
http://www.ams.org/featurecolumn/archive/pi-calc.html.
430 Genet Program Evolvable Mach (2010) 11:397–439
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fitness penalty applied. Note that it is possible that after the ten iterations the output
value diverges from p, and the quality of the result would therefore worsen.
The fitness score of an individual is defined as the absolute error of the last
output. In addition, the string representation of the output was checked to ensure that
all digits matched correctly. We were limited to a precision of 14 decimal places
(i.e., 3.14159265358979), due to using double precision representation. Four
variants of the fitness function were tested, each with different configurations of
inputs given to the programs, these are described in Table 13.
7.2 Approximating presults
The statistical results for these experiments are shown in Table 14. Each experiment
was conducted approximately 150 times. The standard deviations are large and
overlap, which means that the algorithms appear to perform similarly.
7.3 Example pgenerator
Figure 14 shows the output of an evolved SMCGP program that accurately
converges to p. Table 15 shows the output of the program at each time step. As the
program is relatively short, it was possible to extract the evolved generating
function:
fðiÞ¼ cosðsinðcosðsinð0ÞÞÞÞ i¼0
fði1Þþsinðfði1ÞÞ i[0
ð9Þ
Equation 9is a nonlinear recurrence relation. However, it can be shown that it
converges rapidly to p. When i=10, the output matches the first 2,048 digits of p.
4
We can note that the value of pis a fixed point of Eq. 9since x=x?sin(x)is
obeyed when x=p. It is stable since f0(x)=1?cos(x) = 0 when x=p. How rapidly
it converges to pcan be seen from the following argument. Suppose at some
Table 13 Variants of the fitness function with different input strategies
Config. Inputs Description
A One input: the current iteration Functions can be built using the current iteration counter as
a parameter
B One input: numeric constant (1) The program has no real input, and therefore has to build a
structure that performs the iterative process
C Two inputs: the current iteration
and last outputted value
This form can, in some sense, be viewed as recurrent, as
programs can depend on the previous output
D Two inputs: numeric constant (1)
and last outputted value
This form can also be viewed as recurrent, as programs can
depend on the previous output
4
Tested using the mpmath library for Python: http://www.code.google.com/p/mpmath/.
Genet Program Evolvable Mach (2010) 11:397–439 431
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iteration m,f(m) is close to p. Then we can write f(m)=p-d, where dis a small
quantity. Then from Eq. 9fðmþ1Þ¼pdþsinðpdÞpd3
3!:
7.4 Summing numbers
Here we wished to evolve a program that could sum an arbitrarily long list of
numbers. At the n-th iteration, the evolved program should be able to take ninputs
and compute the sum of all the inputs. We devised this problem because we thought
it would be difficult for genetic programming, but relatively easy for a technique
such as neural networks. The premise being, that neural networks appear to perform
well when combining input values and genetic programming seems to work well
using feature selection on the inputs.
Input vectors consist of random sequences of integers. The fitness is defined as
the absolute cumulative error between the output of the program and the expected
sum of the values. We evolved programs which were evaluated on input sequences
of 2 to 10 numbers. The function sets consists of the self-modifying functions and
just the ADD function.
Table 14 Finding pusing various inputs to the evolved programs
Config. % Success Avg. Evals Min SD Min., Avg. iterations
A 96.7 8,952,441 6,731 17,766,584 3, 5.95
B 99.4 2,905,673 526 8,826,492 3, 5.49
C 96.2 4,953,518 869 13,674,006 3, 6.78
D 98.7 2,146,348 642 9,515,520 3, 5.90
Experiment types: A, One input, the current iteration; B, One input, numeric constant (1); C, Two inputs,
the current iteration and last outputted value; D, Two inputs, numeric constant (1) and the last outputted
value. Iterations refers to the number of times the program has to be iterated before it reaches pto 14
decimal places
Fig. 14 SMCGP program that produces an approximation to p. Each row is a different time step
432 Genet Program Evolvable Mach (2010) 11:397–439
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7.5 Summing numbers results
All experiments were found to be successful, in that they evolved programs that
could sum between 2 and 10 numbers (depending on the number of iterations the
program is iterated). Table 16 shows the number of evaluations required to reach
the n-th sum (where nruns from 2 to 10).
After evolution, the best individual for each run was tested to see how well it
generalized. This test involved summing a sequence of 100 numbers. It was found
that most solutions generalized, however, in 1% of cases, they did not.
We also tested the ability of conventional CGP to sum a set of numbers. Here
CGP could only be evolved for a given size of set of input numbers. The results
(based on 200 runs) are also shown in Table 16. This experiment revealed that CGP
is able to solve this problem only for a smaller sets of numbers. This shows a clear
benefit of the self-modification approach in comparison with the direct encoding.
7.6 Regression
We devised a new problem that tests the ability of SMCGP to learn a ‘‘modular’
regression problem. The task is to evolve a program that, depending on the iteration,
approximates the expression x
n
where nis the iteration number. The fitness function
applies xas integers from 0 to 20. The fitness is defined as the number of wrong
outputs (i.e., lower is better). Function set F was used, as detailed in Table 6.As
with the squares problem, without self-modification, it would be impossible for GP
to produce a general solution to this problem.
We evolved to n=10 and then tested for generality up to n=20. As with other
experiments, we evolved incrementally. We first required the programs to solve
n=1. When that was successful, we evolved for n=1 and n=2. Next for
n=1, 2, 3 and so on.
Table 15 Output from program shown in Fig. 14
Iteration Output error Output Correct digits
0 3.14159265358979 0 0
1 2.47522590819691 0.666366745392881 0
2 0.897795232223359 2.24379742136643 0
3 0.115840730988874 3.02575192260092 0
4 0.000258905467357184 3.14133374812244 3
5 2.89235302375346E-12 3.1415926535869 11
6 0 3.14159265358979 14
7 0 3.14159265358979 14
8 0 3.14159265358979 14
9 0 3.14159265358979 14
Output is error is the difference between p(.Net’s Math.PI) and the output from the program. Correct
digits is the count of the correctly matching digits after the decimal point. Errors will appear to be 0 when
the actual error is very small
Genet Program Evolvable Mach (2010) 11:397–439 433
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7.7 Regression results
Table 17 shows the results summary for the powers regression problem. All runs
were successful. In this instance, we see an interesting difference between the two
starting conditions. If the fitness function starts with n=1, ..., 5, we find that fewer
evaluations are required to reach n=10. However, this leads to reduced
generalization. Using a Kolmogorov-Smirnov test, we find that the difference in
the evaluations required is statistically significant (P\0.01).
7.8 Classification
In this experiment we wanted to investigated the behaviour of SMCGP on a problem
in which there appeared to be no clear benefit for self-modification. The problem we
chose is a protein classification problem—as described in [35]. The task is to predict
the location of a protein in a cell, from the amino acids in the particular protein. We
used the entire dataset as training set. The set consisted of 2,427 entries, with 19
variables each and 1 output. The function set for SMCGP includes all the
mathematical operators in addition to the self-modifying command set. The CGP
function set contained just the mathematical operators.
We allowed the phenotype to iterate the number of times specified in the
genotype before we tested the program on the training set. Function set F (see
Table 6) was used for the SMCGP function set, and for CGP the same set was used
but without the self-modification functions.
Table 16 Evaluations required to evolve a SMCGP program that can add a set of numbers of a given
size
Size of set Average Minimum Maximum SD % CGP
2 50 50 50 0 100
3 1,208 54 6,764 987 80
4 2,338 62 19,307 2,025 95.8
5 3,120 87 23,149 2,549 48
6 4,026 126 42,168 4,068 38.1
7 5,010 204 48,824 5,447 0
8 5,931 213 68,201 7,033 0
9 6,788 231 87,949 8,348 0
10 7,434 246 87,976 8,779 0
Hundred percent of SMCGP experiments were successful. The % success rate for conventional CGP is
also shown
Table 17 Summary of results
for the powers regression
problem
Number of initial
test sets
Average
evaluations
SD Percentage
generalize
1 687156 869699 60.4
5 527334 600800 55.6
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7.9 Classification results
Table 18 shows the summary of results for the protein localization problem. We see
under these conditions, that both CGP and SMCGP perform similarly.
This is encouraging as it suggests that using SMCGP does not worsen
performance (compared with CGP) on a problem where there is no clear need for
self-modification. Of course, further work is needed to confirm this on a wider
collection of problems.
8 Conclusions and further work
It is 10 years since the birth of Cartesian Genetic Programming and it is fitting that
we should now be reporting on an improved form of it called Self-Modifying CGP.
The new technique has borrowed some concepts from developmental biology. It
extends the capability of CGP. Unlike CGP, SMCGP allows us to evolve solutions
to whole classes of problems rather than specific instances with a fixed number of
inputs and outputs.
We have also shown that it is suitable for a wide range of computational
problems, and that it can out perform previous approaches on many of these
problems. Given the remarkable success of the technique, it might be worth
investigating whether other methods of GP couldn’t be extended following a similar
route. We speculate that in both linear GP an in tree-based GP, it should be possible
to implement analogous operations, perhaps with similar effects.
There remains much to be investigated. For example, it is unclear what the best
parameter configuration should be. Here we used arbitrary parameters, and tried to
maintain consistency throughout experiments, however, it is likely that these were
sub optimal. Parameter settings seem to be very different from CGP. For example,
here we use small genotypes and large mutation rates, whereas CGP appears to work
best with large genotypes and small mutation rates.
The available function set is also an area that needs investigating. The current set
of self-modifying functions is unlikely to be optimal. However, it is very hard to
predict what functions are actually most useful. When examining the evolved
Table 18 Results summary for the bioinformatics classification problem
- CGP SMCGP
Average fitness (training) 66.81 66.83
SD fitness (training) 6.35 6.45
Average fitness (validation) 66.10 66.18
SD fitness (validation) 5.96 6.46
Avg. evaluations to best fitness (training) 7,679 7,071
SD evaluations to best fitness (training) 2,452 2,644
Avg. evaluations to best fitness (validation) 7,357 7,161
SD evaluations to best fitness (validation) 2,386 2,597
Genet Program Evolvable Mach (2010) 11:397–439 435
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programs, our intuition has sometimes been proved wrong about whether certain
functions would be beneficial to evolution.
The methods used to activate self-modifying functions depending on the values
of numerical inputs need to be investigated further to ascertain how useful they are.
At present they are designed with bi-arity functions in mind. It would be interesting
to consider activation mechanisms for self-modifying functions inspired by
epigenetics in biology.
Although SMCGP is a relatively complex technique, conceptually it is simple,
and it is straight forward to implement. SMCGP execution can be implemented very
efficiently, and graphs containing thousands of nodes are easily handled. We have
yet to explore the possibilities that this brings.
A significant aspect of SMCGP is that it can produce mathematically provable
solutions to general classes of problems. It seems possible that it could produce
hitherto unknown general solutions to some problems. These solutions may have
utility in a number of research domains. We intend to continue to investigate such
areas. Utilizing a theorem prover in the fitness function, rather than carrying out
post-hoc proofs though desirable, is likely to be computationally intractable as
theorem provers, even if they could be used, have poor time complexity.
The form of SMCGP we have described here uses a one-dimensional graph
representation, however, we have also been investigating a form of SMCGP in
which programs can be developed that are a two-dimensional sheet of nodes. Early
experiments indicate that this allows solutions to problems to be evolved even faster
than the one-dimensional form.
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... CGP was initially proposed to be a solution nding algorithm for an electronic circuit in [1] but was later expanded to other problem [2]. Variations of CGP such as the SMCGP [3] and Embedded CGP [4] were later proposed showing faster solution nding on benchmark problems. Due to the capability of CGP to be able to generate complex function, it has been applied to generate Articial Neural Networks (ANN) [5] and other classication functions using generated rule sets. ...
... (1) to(3). Eqs.(4) to(6)shows the segment s of the v chromosome decodes into the 3 dimensional coordinate, (x m , y m , z m ) where m = 1, 2, . . . ...
... Self-modifying CGP introduced another type of node, a self-modifying (SM) node, one that refers to the code itself [26,28,29]. Such nodes modify the phenotype. ...
... This allows SMCGP to be applied to classes of problems that non-developmental encodings can not solve. Indeed, it has been shown that SMCGP could find provably general solutions to certain classes of problems: parity, binary addition [26], computation of and e [27] Fig. 2 In ICCGP nodes are called enzymes. An enzyme has two binding vectors and a shape vector. ...
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Cartesian genetic programming, a well-established method of genetic programming, is approximately 20 years old. It represents solutions to computational problems as graphs. Its genetic encoding includes explicitly redundant genes which are well-known to assist in effective evolutionary search. In this article, we review and compare many of the important aspects of the method and findings discussed since its inception. In the process, we make many suggestions for further work which could improve the efficiency of the CGP for solving computational problems.
... SMCGP make use of these functions to handle inputs during execution. The outputs are taken from the OUTPUT function (Harding et al. 2010a). Harding et al. (2007) provided solutions in the cases where the number of outputs and values of OUTPUT function do not match. ...
... Self-Modification procedure. [Adapted fromHarding et al. 2010a.] ...
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Cartesian Genetic Programming (CGP) is a variant of Genetic Programming with several advantages. During the last one and a half decades, CGP has been further extended to several other forms with lots of promising advantages and applications. This article formally discusses the classical form of CGP and its six different variants proposed so far, which include Embedded CGP, Self-Modifying CGP, Recurrent CGP, Mixed-Type CGP, Balanced CGP, and Differential CGP. Also, this article makes a comparison among these variants in terms of population representations, various constraints in representation, operators and functions applied, and algorithms used. Further, future work directions and open problems in the area have been discussed.
... 10) yet do not improve subsequently. It is worth contrasting the 561 model discussed in this chapter with previous work on Self-Modifying CGP [14]. 562 In SMCGP phenotypes can be iterated to produce a sequence of programs or 563 phenotypes. ...
... If Incr opt is one or two, the updated values of the 715 soma are changed from the parent neuron's values in an incremental way. This is 716 either a linear or nonlinear increment or decrement depending on whether the soma 717 program's outputs are greater than or less than or equal to zero (lines [8][9][10][11][12][13][14][15][16]. The 718 magnitudes of the increments is defined by the user-defined constants: δ sh , δ sp , δ sb 719 and sigmoid slope parameter, α (see Table 8.1). ...
Chapter
A developmental model of an artificial neuron is presented. In this model, a pair of neural developmental programs develop an entire artificial neural network of arbitrary size. The pair of neural chromosomes are evolved using Cartesian Genetic Programming. During development, neurons and their connections can move, change, die or be created. We show that this two-chromosome genotype can be evolved to develop into a single neural network from which multiple conventional artificial neural networks can be extracted. The extracted conventional ANNs share some neurons across tasks. We have evaluated the performance of this method on three standard classification problems: cancer, diabetes and the glass datasets. The evolved pair of neuron programs can generate artificial neural networks that perform reasonably well on all three benchmark problems simultaneously. It appears to be the first attempt to solve multiple standard classification problems using a developmental approach.
... With the intention to use a representation in which the specific location of genes within the chromosome has no direct or indirect influence on the phenotype, Smith et al. [139] introduced an implicit context representation for CGP. Self-Modifying-CGP was developed by Harding et al. [36]. It uses functions that cause the evolved programs to change themselves as a function of time. ...
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This dissertation aims on analyzing fundamental concepts and dogmas of a graph-based genetic programming approach called Cartesian Genetic Programming (CGP) and introduces advanced genetic operators for CGP. The results of the experiments presented in this thesis lead to more knowledge about the algorithmic use of CGP and its underlying working mechanisms. CGP has been mostly used with a parametrization pattern, which has been prematurely generalized as the most efficient pattern for standard CGP and its variants. Several parametrization patterns are evaluated and analyzed with more detailed and comprehensive experiments by using meta-optimization. This thesis also presents a first runtime analysis of CGP. The time complexity of a simple (1+1)-CGP algorithm is analyzed with a simple mathematical problem and a simple Boolean function problem. In the subfield of genetic operators for CGP, new recombination and mutation techniques that work on a phenotypic level are presented. The effectiveness of these operators is demonstrated on a widespread set of popular benchmark problems. Especially the role of recombination can be seen as a big open question in the field of CGP, since the lack of an effective recombination operator limits CGP to mutation-only use. Phenotypic exploration analysis is used to analyze the effects caused by the presented operators. This type of analysis also leads to new insights into the search behavior of CGP in continuous and discrete fitness spaces. Overall, the outcome of this thesis leads to a reconsideration of how CGP is effectively used and extends its adaption from Darwin's and Lamarck's theories of evolution.
... Related to hyper-heuristics are evolutionary algorithms (EAs) [34], which can discover the optimal EA to optimize problem solutions. Self-modification Cartesian genetic programming [35] encodes low-level heuristics with self-modifying operators, and the exponentially expanding hyper-heuristic [36] employs a meta-hyper-heuristic algorithm to seek the heuristic space for better performance. The leader-following consensus problem can be characterized as a combination of heuristics, where the leader is the heuristic with the greatest fitness and is pursued by other heuristics. ...
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This paper describes a unique meta-heuristic technique for hybridizing bio-inspired heuristic algorithms. The technique is based on altering the state of agents using a logistic probability function that is dependent on an agent’s fitness rank. An evaluation using two bio-inspired algorithms (bat algorithm (BA) and krill herd (KH)) and 12 optimization problems (cross-in-tray, rotated hyper-ellipsoid (RHE), sphere, sum of squares, sum of different powers, McCormick, Zakharov, Rosenbrock, De Jong No. 5, Easom, Branin, and Styblinski–Tang) is presented. Furthermore, an experimental evaluation of the proposed scheme using the industrial three-bar truss design problem is presented. The experimental results demonstrate that the hybrid scheme outperformed the baseline algorithms (mean rank for the hybrid BA-KH algorithm is 1.279 vs. 1.958 for KH and 2.763 for BA).
... Recurrent CGP [241,242] allows the existence of recurrent connections which can target any node in a graph, facilitating the induction of recursive solutions to problems such as generating the Fibonacci sequence or time-series forecasting. Self-modifying CGP [90] facilitates the inclusion of nodes that can create and delete other nodes, thereby allowing a graph to develop to solve a class of problems, such as computing π or e to arbitrary precision [91]. ...
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... Those for physical technologies such as nanobots or protocells need to be tailored for the physical capabilities of the embodying systems. Example of virtual technologies include Morphogenetic Engineering (ME) [5,6,27], Developmental Cellular Automata (DCA) [20,23], Generative Systems (GS) [11-13, 24, 25, 28], Self-Modifying Cartesian Genetic Programming (SMCGP) [8,21], and Spatial Computing (SC) [4,31,32]. ...
Book
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... Self-adaptation has been used previously in GP [8], however, in these implementations the self-adaptation was triggered at discrete preset intervals, usually at every generations. One of the drawbacks of this approach is the fact that the self-adaptation is triggered by external factors, rather than by internal factors, and therefore the performance or progress of the system is not considered. ...
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Chapter
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Genetic programming (GP) extends traditional genetic algorithms to automatically induce computer programs. GP has been applied in a wide range of applications such as software re-engineering, electrical circuits synthesis, knowledge engineering, and data mining. One of the most important and challenging research areas in GP is the investigation of ways to successfully evolve recursive programs. A recursive program is one that calls itself either directly or indirectly through other programs. Because recursions lead to compact and general programs and provide a mechanism for reusing program code, they facilitate GP to solve larger and more complicated problems. Nevertheless, it is commonly agreed that the recursive program learning problem is very difficult for GP. In this paper, we propose a technique to tackle the difficulties in learning recursive programs. The technique is incorporated into an adaptive Grammar Based Genetic Programming system (adaptive GBGP). A number of experiments have been performed to demonstrate that the system can evolve recursive programs efficiently and effectively.
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Book
On Growth and Form, the classic by the great D'Arcy Wentworth Thompson provides the general inspiration for this book. D'Arcy was not one to run with the herd; he was an original thinker and brilHant classicist, mathematician and physicist, who provided a source of fresh air to developmental biology by examining growth and form in the light of physics and mathematics, courageously ignoring chemistry and genetics. Despite this omission of what are now regarded as the main sciences in understanding growth and form, D'Arcy's message is not in the least bit impaired. Instead, in today's biochemistry dominant world D'Arcy's work highlights, as it did in his own time, the role physics plays in growth and form. This book takes its name from D'Arcy's magnum opus and hopes to do it justice.
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Biological chromosomes are replete with repetitive sequences, micro satellites, SSR tracts, ALU, etc. in their DNA base sequences. We started looking for similar phenomena in evolutionary computation. First studies find copious repeated sequences, which can be hierarchically decomposed into shorter sequences, in programs evolved using both homologous and two point crossover but not with headless chicken crossover or other mutations. In bloated programs the small number of effective or expressed instructions appear in both repeated and nonrepeated code. Hinting that building-blocks or code reuse may evolve in unplanned ways. Mackey-Glass chaotic time series prediction and eukaryotic protein localisation (both previously used as artificial intelligence machine learning benchmarks) demonstrate evolution of Shannon information (entropy) and lead to models capable of lossy Kolmogorov compression. Our findings with diverse benchmarks and GP systems suggest this emergent phenomenon may be widespread in genetic systems.
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We have developed an artificial ecology in the computer core, where one is able to evolve assembler-automaton code without any predefined evolutionary path. The system, in the present version has one dimension, is updated in parallel, the instructions are only able to communicate locally, and the system is continuously subjected to noise. The system also has a notion of computational resources. Depending on the specified parameters and the level of complexity of distance from a randomized core, this electronic garden is started at, we see different evolutionary paths. For several initial conditions the system is able to develop extremely viable cooperative programs (organisms ) which totally dominates the core. This demonstrates the emergence of complex functional properties in a computational environment. 21 refs., 7 figs., 2 tabs.