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Towards Practical Autoconstructive Evolution: Self-Evolution of Problem-Solving Genetic Programming Systems

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Abstract

Most genetic programming systems use hard-coded genetic operators that are applied according to user-specified parameters. Because it is unlikely that the provided operators or the default parameters will be ideal for all problems or all program representations, practitioners often devote considerable energy to experimentation with alternatives. Attempts to bring choices about operators and parameters under evolutionary control, through self-adaptative algorithms or meta-genetic programming, have been explored in the literature and have produced interesting results. However, no systems based on such principles have yet been demonstrated to have greater practical problem-solving power than the more-standard alternatives. This chapter explores the prospects for extending the practical power of genetic programming through the refinement of an approach called autoconstructive evolution, in which the algorithms used for the reproduction and variation of evolving programs are encoded in the programs themselves, and are thereby subject to variation and evolution in tandem with their problem-solving components. We present the motivation for the autoconstructive evolution approach, show how it can be instantiated using the Push programming language, summarize previous results with the Pushpop system, outline the more recent AutoPush system, and chart a course for future work focused on the production of practical systems that can solve hard problems. Keywordsgenetic programming-meta-genetic programming-autoconstructive evolution-Push-PushGP-Pushpop-AutoPush
TOWARDS PRACTICAL AUTOCONSTRUCTIVE
EVOLUTION: SELF-EVOLUTION OF PROBLEM-
SOLVING GENETIC PROGRAMMING SYSTEMS
Lee Spector
Cognitive Science, Hampshire College, Amherst, MA, 01002-3359 USA.
Abstract
Most genetic programming systems use hard-coded genetic operators
that are applied according to user-specified parameters. Because it is un-
likely that the provided operators or the default parameters will be ideal
for all problems or all program representations, practitioners often devote
considerable energy to experimentation with alternatives. Attempts to
bring choices about operators and parameters under evolutionary control,
through self-adaptative algorithms or meta-genetic programming, have
been explored in the literature and have produced interesting results.
However, no systems based on such principles have yet been demonstrated
to have greater practical problem-solving power than the more-standard
alternatives. This chapter explores the prospects for extending the practi-
cal power of genetic programming through the refinement of an approach
called autoconstructive evolution, in which the algorithms used for the
reproduction and variation of evolving programs are encoded in the pro-
grams themselves, and are thereby subject to variation and evolution in
tandem with their problem-solving components. We present the moti-
vation for the autoconstructive evolution approach, show how it can be
instantiated using the Push programming language, summarize previous
results with the Pushpop system, outline the more recent AutoPush sys-
tem, and chart a course for future work focused on the production of
practical systems that can solve hard problems.
Keywords: genetic programming, meta-genetic programming, autoconstructive evo-
lution, Push, PushGP, Pushpop, AutoPush
To appear in Riolo, Rick L., McConaghy, Trent, and Vladislavleva,
Ekaterina, editors, Genetic Programming Theory and Practice VIII.
Springer. 2010.
2GENETIC PROGRAMMING THEORY AND PRACTICE III
1. Introduction
The work described in this chapter is motivated both by features of
biological evolution and by the requirements for the high-performance
problem-solving systems of the future.
Under common conceptions of biological evolution the variation of
genotypes from parents to children, and hence the diversification of phe-
notypes from progenitors to their descendants, is essentially random
prior to selection. Offspring vary randomly, it is said, and selection acts
on the resulting diversity by allowing the better-adapted random vari-
ants to survive and reproduce. Such conceptions are held not only by
the lay public but also by theorists such as Jerry Fodor and Massimo
Piattelli-Palmarini who, in their book What Darwin Got Wrong, criti-
cize Darwinian theory in part on the grounds that the random “generate
and test” algorithm at its core is insufficiently powerful to account for
the facts of natural history (Fodor and Piattelli-Palmarini, 2010).
But diversification in nature, while certainly random in some respects,
is also clearly non-random in several others. If one were to modify DNA
molecules in truly random ways, considering all chemical bonds to be
equally good candidates for breakage and re-connection, then one would
not end up with DNA molecules at all but instead with some other sort
of organic soup. Cellular machinery copies DNA, and repairs copying
errors, in ways that allow for certain kinds of “errors” but only within
tightly constrained bounds. At higher levels of organization variation
is constrained by genetic regulatory processes, the mechanics of sexual
recombination, cell division and development, and, at a much higher
level of organization, by social structures that guide non-random mate
selection. All of these constraints emerge from reproductive processes
that have themselves evolved over the course of natural history. There
is a large literature on such constraints, including a recent theory of
“facilitated variation” (Gerhart and Kirschner, 2007), and summaries of
the evolution of variation from pre-biotic Earth to the present (Maynard
Smith and Szathm´ary, 1999).
Whether or not the evolved-non-randomness of biological variation
constitutes a significant critique of neo-Darwinism or of the historical
Darwin, as claimed by Fodor and Piattelli-Palmarini, is beyond the scope
of the present discussion. For our purposes, however, two related points
should be made. First, while truly random variation, filtered by selec-
tion, may be too weak of a mechanism to have produced the sequence
of phenotypes observed over time in the historical record, it is possi-
ble for random variation, when acting on the reproductive mechanisms
themselves, to produce variation mechanisms that are not purely ran-
Towards practical autoconstructive evolution 3
dom. This is presumably what happened in natural history. Second,
this bootstrapping process, of the evolution of adaptive, not-entirely-
random variation by means of the initially random variation of the vari-
ation mechanisms, might also be applied to evolutionary problem-solving
technologies.
Why would we want to do this? One reason is that the problem-
solving power of current evolutionary computing technologies is limited
by the nature of the variation mechanisms that we build into these sys-
tems by hand. Consider, for example, the standard mutation operators
used in genetic programming. Subtree replacement, applied uniformly
to the nodes in a program tree (or uniformly to interior vs. leaf nodes
with a specified probability), involving the replacement of subtrees with
newly-generated random subtrees, provides a form of variation that leads
to solutions in some but not all problem environments. This has led to
the development of a wide range of alternative mutation operators; see,
for example, the “Mutation Cookbook” section of (Poli et al., 2008, pp.
42–44). But which of these will be most helpful in which circumstances,
and which others, perhaps not yet invented, may be needed to solve
challenging new problems?
The field currently has no satisfying answer to this question, which
will become all the more pressing as genetic programming systems incor-
porate more expressive and heterogeneous program representations. In
the context of such representations it may well make sense for different
program elements or program locations to have different variation rates
or procedures, and it will not be obvious, in advance, how to make these
choices. The question will also become all the more pressing as genetic
programming systems are applied to ever more complex problems, about
which the system designers will have less knowledge and intuition. And
the question will be raised with even greater urgency with respect to re-
combination operators such as crossover, for which there even more open
questions (e.g. about how to choose crossover partners) that currently
require the user to make choices that may not be optimal.
Two approaches to these general issues that have previously been
explored in the literature are “self-adaptation” and “meta-genetic pro-
gramming.” Many forms of self-adaptation have been investigated, both
within genetic programming and in other areas of evolutionary computa-
tion (with many examples including (Angeline, 1995; Spears, 1995; An-
geline, 1996; Eiben et al., 1999; MacCallum, 2003; Fry et al., 2005; Beyer
and Meyer-Nieberg, 2006; Vafaee et al., 2008; Silva and Dignum, 2009)).
In all of these systems the parameters of the evolutionary algorithm are
varied and subjected to some form of selection, whether the variation
and selection is accomplished by means of the overarching evolution-
4GENETIC PROGRAMMING THEORY AND PRACTICE III
ary algorithm, by a secondary evolutionary algorithm, or by some other
machine learning technique. In some cases the parameters are adapted
on an individual basis, while in others the self-adaptive system modifies
global parameters that apply to an entire population. In general, how-
ever, these systems vary only pre-selected parameters of the variation
operators in pre-specified ways, and they do not allow for the evolution
of arbitrary methods of variation.
By contrast, the “meta-genetic programming” approach leverages the
program-space search capabilities of genetic programming to search for
variation operators—which are, after all, themselves programs—during
the search for problem-solving programs (Schmidhuber, 1987; Kantschik
et al., 1999; Edmonds, 2001; Tavares et al., 2004; Diosan and Oltean,
2009). These systems would appear to have more potential to evolve
adaptive variation algorithms, but they have generally been subject to
one or both of the following two significant limitations:
The evolving genetic operators are not associated with specific
evolving problem-solving programs; they are expected to apply to
all evolving problem-solving programs equally well.
The evolving genetic operators are restricted to being compositions
of a small number of pre-designed components; many conceivable
genetic operators will not be representable using these components.
The first of these limitations contrasts with some of the self-adaptive
evolutionary algorithms mentioned previously, in which the values of
parameters for genetic operators are encoded in individuals. That this
“global” conception of the applicability of genetic operators might be a
limitation should be evident from a cursory examination of the diver-
sity of reproductive strategies in nature. For example, the reproductive
strategies of the dandelion are quite different from those of the tiger, the
oyster mushroom, and Escherichia coli; nobody would expect the strate-
gies of any of these organisms to work particularly well for any of the
others. Of course the diversity present in the Earth’s biosphere dwarfs
that of any current genetic programming system, but it would nonethe-
less be quite surprising if the same genetic operators worked equally well
across a genetic programming population with any significant diversity.
One could well imagine, for example, that a subset of the population
might share one particular subtree in which a high degree of mutation is
adaptive and a second subtree in which mutation is always deleterious.
Other individuals in the population might lack either or both of these
subtrees, or they might contain additional code that changes the effects
of mutations within these particular subtrees.
Towards practical autoconstructive evolution 5
The second of these limitations is probably mostly a reflection of the
fact that most genetic programming representations limit the expres-
siveness of the programs that they can evolve more generally. Although
several Turing complete representations have been described (for ex-
ample, (Teller, 1994; Nordin and Banzhaf, 1995; Spector and Robin-
son, 2002a; Woodward, 2003; Yabuki and Iba, 2004; Langdon and Poli,
2006)), such representations are relatively rare and representations that
can easily perform arbitrary transformations on variable-sized programs
are rarer still. Nature appears to be quite flexible and inventive in the
variation mechanisms that it employs (e.g., mechanisms involving gene
duplication), and we can easily imagine cases in which genetic program-
ming systems would benefit from the use of genetic operators that are
not simple compositions of hand-designed operator components.
Another line of research that bears on the approach presented here
generally appears in the artificial life literature. Systems such as Tierra
(Ray, 1991), Avida (Ofria and Wilke, 2004), and SeMar (Suzuki, 2004)
all involve the evolution of programs that are partially responsible for
their own reproduction, and in which the reproductive mechanisms (in-
cluding genetic operators) are therefore subject to variation and se-
lection. However, in these systems diversification is generally driven
by hand-designed “ancestor” replicators and/or by the effects of hand-
designed mutation algorithms that are applied automatically to the re-
sults of all code manipulation operations. Furthermore, while some of
these systems have been used to solve computational problems their
problem-solving power has been quite limited; they have been used to
evolve simple logic gates and arithmetic functions, but they have not
been applied to the kinds of difficult problems that genetic program-
ming practitioners are interested in solving. This is not surprising, as
these systems have generally been developed primarily to study biologi-
cal evolution, not to solve difficult computational problems.
Additional related work has been conducted in the context of evolved
self-reproduction (Taylor, 1999; Sipper and Reggia, 2001) although most
of this work has been focused on the evolution of exact replication rather
than the evolution of adaptive variation. An exception, and the closest
work to that described below, is Koza’s work on the “Spontaneous Emer-
gence of Self-Replicating and Evolutionarily Self-Improving Computer
Programs” (Koza, 1994). In that work Koza evolved programs that
simultaneously solved problems (albeit simple Boolean problems) and
produced variant offspring using a template-based code self-modification
in a “sea” or “Turing gas” of programs (Fontana, 1992).
This chapter describes an approach to self-adaptive genetic program-
ming, called autoconstructive evolution, that combines several features
6GENETIC PROGRAMMING THEORY AND PRACTICE III
of the approaches described above, with the long-term goal of producing
a new generation of powerful problem solving systems. The potential ad-
vantage of the autoconstructive evolution approach is that it will allow
variation mechanisms to co-evolve with the programs to which they are
applied, thereby allowing the evolutionary system itself to adapt to its
problem environments in significant ways. The autoconstructive evolu-
tion approach was first described in 2001 and 2002 (Spector, 2001; Spec-
tor, 2002; Spector and Robinson, 2002a; Spector and Robinson, 2002b),
using the Pushpop system that leveraged features of the Push program-
ming language for evolved programs. In the next section this earlier
work is briefly described. The subsequent section describes more recent
work on the approach, using better technology and a more explicit fo-
cus on the goal of high performance problem solving, implemented in a
newer system called AutoPush. The final section of the chapter offers
some brief conclusions.
2. Push and Pushpop
An autoconstructive evolution system was defined in (Spector and
Robinson, 2002a) as “any evolutionary computation system that adap-
tively constructs its own mechanisms of reproduction and diversification
as it runs.” In the context of the present discussion, however, that def-
inition is too general, and a more specific definition that captures both
the past and present usage would be “any genetic programming system
in which the methods for reproduction and diversification are encoded
in the individual programs themselves, and are thereby subject to vari-
ation and evolution.” The goal in the previous work, as in the work
described here, is for the ways in which children are produced to be
evolved along with the programs to which they will be applied. This is
done by encoding the mechanisms for reproduction and diversification
within the programs themselves, which must be capable of producing
children and, in principle, of solving the problem to which the genetic
programming system is being applied. The space of possible repro-
duction and diversification methods is vast and an ideal system would
allow evolving programs to reach new and uncharted reaches of this
space. Human-designed diversification mechanisms, including human-
designed genetic operators, human-specified automatic mutation during
code-manipulation, and human-written ancestor programs, should all be
avoided.
Of course it will generally be necessary for some features of any evo-
lutionary system to be pre-specified; for example, all of the systems de-
scribed here borrow several pre-specified elements of traditional genetic
Towards practical autoconstructive evolution 7
programming systems, including a generation-based evolutionary loop,
a fixed-size population, and tournament selection with a pre-specified
tournament size. The focus here is on the evolution of the means by
which children are produced from parents, and it is this task for which
we currently seek autoconstructive methods.
A prerequisite for this approach is a program representation in which
problem-solving functions and child-production functions can both be
easily expressed. The Push programming language was originally de-
signed specifically for this purpose (Spector, 2001). Push is a stack-
based language roughly in the tradition of Forth, but for which each
data type has its own stack. Instructions generally take their arguments
from the appropriate stacks and push their results onto the appropriate
stacks.1If an instruction requires arguments that are not present on
the appropriate stacks when it is called then it does nothing (it acts as
a “no-op”). These specifications mean that even though multiple data
types may be present in a program no instruction will ever be called on
arguments of the wrong type, regardless of its syntactic position in the
program. Among other benefits, this means that there are essentially
no syntax constraints on Push programs aside from a requirement that
parentheses be balanced. This is particularly useful for systems in which
child programs will be produced by evolving programs.
One of Push’s most important features for autoconstructive evolution,
and for genetic programming more generally, is the fact that “code” is a
first-class data type. When a Push program is being executed the code
that is queued for execution is stored on a special stack called the “exec”
stack, and exec instructions in the program can manipulate the queued
instructions in order to implement a wide variety of evolved control struc-
tures (Spector et al., 2005). Additional code stacks (including one called
simply “code,” and in some implementations others with names such as
“child”) can be used to store and manipulate code for a variety of other
purposes. This feature has significant benefits for genetic programming
even in a non-autoconstructive context (that is, even when standard,
hard-coded genetic operators are used, as in the PushGP system), but
here we focus on the use of Push for autoconstructive evolution. Space
limits prevent full exposition of the Push language here; see (Spector
et al., 2005) and the references therein for further details.2
1Exceptions are instructions that draw their inputs from external data structures, for example
instructions that access inputs, and instructions that act on external data structures, for exam-
ple “developmental” instructions that add components to externally-developing representations of
circuits or other structured objects.
2See also http://hampshire.edu/lspector/push.html.
8GENETIC PROGRAMMING THEORY AND PRACTICE III
The first autoconstructive evolution system built using Push, called
Pushpop, can best be understood as an extension of a more-standard
genetic programming system such as PushGP. In PushGP, when a pro-
gram is being tested for fitness on a particular fitness case it is run and
then the problem-solving outputs are collected from the relevant data
stacks (typically integer or float) and tested for errors; Pushpop does
this as well, but it also simultaneously collects a potential child from the
child stack. If the problem to which the system is being applied involves
nfitness cases then the testing of each program in the population will
produce npotential children. In the reproductive phase tournaments
are conducted among parents and children are selected randomly from
the set of potential children of the winning parents. If there are insuffi-
cient children to fill the child population then newly generated random
individuals are used.
In Pushpop, as in any autoconstructive evolution system, care must be
taken to prevent the takeover of the population by perfect replicators or
other pathological replicants. Because there is no automatic mutation in
Pushpop a perfect replicator can rapidly fill the population with copies
of itself, after which no evolution (and indeed no change at all) will
occur. The production of perfect replicators in Push is generally trivial,
because programs are pushed onto the CODE stack prior to execution.
For this reason Pushpop includes a “no cloning” rule that specifies that
exact clones will not be allowed into the child population. Settings are
also available that prohibit children that are identical to any of their
ancestors or any other individuals in the population. The “no cloning”
rule forces programs to diversify in some way, but it does not dictate the
mode or extent of diversification. The pathology of perfect replicators in
nature was presumably overcome with the aid of vast stretches of time
and over vast expanses of the Earth, within which perfect replicators
may have arisen but later been eliminated when changes occurred to
which they could not adapt. Our resources are much more constrained,
however, and so we must proactively cull the individuals that we know
cannot possibly evolve.
Programs in a Pushpop population can reproduce using evolved forms
of multi-parent recombination, accessing other individuals in the popu-
lation through the use of a variety of instructions provided for this pur-
pose and using them in any computable way to produce their children
(Spector and Robinson, 2002a). In fact, evolving Pushpop programs
can access and then execute code from other individuals in the popula-
tion, which means that evolved programs may not work correctly when
executed outside of the populations within which they evolved. This
is unfortunate from the perspective of a practitioner who is primarily
Towards practical autoconstructive evolution 9
interested in producing a program that will solve a particular problem,
since the “solution” may require the entire population to work and it
may be exceptionally difficult to understand. The mechanisms for pop-
ulation access in Pushpop are also somewhat complex, and the presence
of these mechanisms makes it particularly difficult to analyze the per-
formance of the system. For these reasons the new work described here
does not allow executing programs to access the other programs in the
population; see below for further discussion.
Pushpop is capable of solving simple symbolic regression problems,
and it has served as the basis for studies of the evolution of diversifi-
cation. For example, one study showed that evolving populations that
produce adaptive Pushpop programs—that is, programs that actually
solve the problems presented to the system—are reliably more diverse
than is required by the “no cloning” rule alone (Spector, 2002). But
Pushpop’s utility as a problem-solving system is limited, and the focus
of the Push project in subsequent years has been on more traditional ge-
netic programming systems such as PushGP. PushGP uses traditional
genetic operators but the code-manipulation features of Push nonethe-
less provide benefits, for example by simplifying the evolution of novel
control structures and modular architectures.
More recently, however, the use of Push for autoconstructive evolution
has been revisited in light of improvements to the Push language (Spec-
tor et al., 2005), the availability of substantially faster hardware, and a
clarified focus on the long-term potential of autoconstructive evolution
to solve problems that cannot be solved with hand-coded diversification
mechanisms.
3. Practical Autoconstructive Evolution
AutoPush is a new autoconstructive genetic programming system,
a successor to Pushpop built on the more expressive version 3 of the
Push programming language and designed with a more explicit focus
on problem-solving power. To that end, several sources of inessential
complexity in Pushpop have been removed to aid in the analysis of
AutoPush runs and their results.
AutoPush, like Pushpop, uses the basic generational loop of a stan-
dard genetic programming system and tournament selection with a pre-
specified tournament size. Also like Pushpop it uses no pre-specified ge-
netic operators, no ancestor replicators, and no pre-specified, automatic
mutation. And like Pushpop it represents its programs in a Turing com-
plete language so that children may be produced from parents by means
of any computable function, modulo limits on execution steps or time.
10 GENETIC PROGRAMMING THEORY AND PRACTICE III
The current version of AutoPush is asexual—that is, parents must
construct their children without having access to other programs in the
population—because this eliminates the complexity that may not be
necessary and it also simplifies analysis. Asexual programs may be run
in isolation, both to solve the target problem and to study the range of
children that they produce, and it is easy to store all of their ancestors (of
which there will be only as many as there have been generations, while
each individual in a sexually-reproducing population may have exponen-
tially many ancestors). Future versions of AutoPush may reintroduce
the possibility of recombination by reintroducing instructions that pro-
vide access to other individuals in the population; it is our intention to
explore this option once the dynamics of the asexual version are better
understood. It is also worth noting that the role of sex in biological
diversification is a subject of considerable debate, and that asexual or-
ganisms diversify in complex and significant ways (Barraclough et al.,
2003).
The processes by which programs are tested for problem-solving per-
formance and used to produce children also differ between Pushpop and
AutoPush. In Pushpop a potential child is produced for each fitness
case, during the calculation of the problem-solving answer for that fit-
ness case. This means that the number of children may depend on the
number of fitness cases, which complicates analysis and also changes the
way that the algorithm will perform on problems with different numbers
of fitness cases. By contrast, in AutoPush no children are produced dur-
ing fitness testing; any code left on the code stack after a fitness-testing
run is ignored.3Instead, when an individual is selected for autoconstruc-
tive reproduction in a tournament it is run again, with an input of 0, to
produce a child program for the next generation.4
The most significant innovation in AutoPush is a new approach to
constraints on birth and selection. Pushpop incorporates a “no cloning”
rule but AutoPush goes further, adding more constraints on birth and
selection to facilitate the evolution of adaptive diversification. Follow-
ing the lead of meta-genetic programming developers who judged the
fitness of evolving operators by “some measure of success in increasing
the fitness of the population they operate on” (Edmonds, 2001), Auto-
3In Pushpop a special child stack is used for the production of children because the code stack
is needed for the expression of evolved control structures in Push1, in which Pushpop was imple-
mented. AutoPush is implemented in Push3, in which the new exec stack can be used for evolved
control structures, freeing up the code stack for child production.
4The input value of 0is arbitrary, and an input value is provided only for the minor convenience of
avoiding re-definition of the input-pushing instruction. None of this should be significant as long
as we are consistent in the ways that we conduct the autoconstructive reproduction runs.
Towards practical autoconstructive evolution 11
Push incorporates factors based on the history of improvement within
the ancestry of an individual.
There are many ways in which one might measure “history of im-
provement” and many ways in which such measurements might be used
in an evolutionary algorithm. For example, Smits et al. define “ac-
tivity” or “potential to improve” as “the sum of the number of moves
[in the program search space] that either improved the fitness or neu-
tral moves that resulted in either no change in fitness or a change that
was less than a given (dynamic) tolerance limit” (Smits et al., 2010).
They use this measure to select candidates for further testing, crossover,
and replacement. Additional comments on varieties and measures of
self-improvement can be found in (Schmidhuber, 2006).
In AutoPush the history of improvement is a scalar that summarizes
the direction of problem-solving performance changes over the individ-
ual’s ancestry, with greater weight given to more recent changes (see
formula below). It would be tempting to use this measure of improve-
ment only in selection, possibly as a second objective—in addition to
problem-solving performance—in the context of a multi-objective selec-
tion scheme. But this, by itself, would not work well because selec-
tion cannot salvage a population that has become overrun by evolution-
ary “dead-enders” that can never produce improved descendants. Such
dead-enders include not only cloners but also programs of several other
categories. For example, consider a population full of programs that pro-
duce children that vary only in a subexpression that is never executed.
This population is just as un-adaptive as a population of cloners, and it
will do no good to select among its individuals on any basis whatsoever.
Many other, more subtle categories of dead-enders exist, presenting chal-
lenges to any evolutionary system that relies only on selection to drive
adaptation. The alternative approach taken in AutoPush is to prevent
such dead-enders, when they can be detected, from reproducing at all,
and to make room in the population for the children of improvers or at
least for new random individuals.
As a result, we place a variety of constraints on birth and selection
which act collectively to promote the evolution of adaptive diversification
without specifying the form(s) that the actual diversification algorithms
will take. More specifically, we conduct selection using tournaments,
with comparisons within the tournament set computed as follows:5
5These constraints, and those mentioned for birth below, are stated using the numerical parameter
values that were chosen, more or less arbitrarily, for the runs described here. Other values may
perform better, and further study may provide guidance on setting these values or eliminating the
parameters altogether.
12 GENETIC PROGRAMMING THEORY AND PRACTICE III
Prefer reproductively competent parents: Individuals that were
generated by other individuals beat randomly-generated individu-
als, and individuals that are “grandchildren” beat all others that
are not. If both individuals being compared are grandchildren then
the lengths of their lineages are not otherwise decisive.
Prefer parents with non-stagnant lineages: A lineage is considered
stagnant if it has persisted for at least some preset number of
generations (6 in the experiments described here) and if problem-
solving performance has not changed in the most recent half of the
lineage.
Prefer parents with good problem-solving performance: If neither
reproductive competence nor lineage stagnation are decisive then
select the parent that does a better job on the target problem.
The constraints on birth make use of two auxiliary definitions, for
“improvement” and “code discrepancy.” Improvement is a measure of
how much the problem-solving performance of a lineage has improved,
with greater weight being given to the most recent steps in the lineage.
We first compute a normalized vector of changes in problem-solving per-
formance, with improvements represented as 1, declines represented as
1, and repeats of the same value represented as 0. The overall improve-
ment value is then calculated as the weighted average of the elements of
this vector, with the weights produced by following function (with decay
factor δ= 0.1 for the runs described here):
wg=currentgen = 1
wg1=wg(1 δ)
Code discrepancy is a measure of the difference between two programs,
calculated as the sum, over all unique expressions and sub-expressions
in either of the programs, of the difference between the numbers of oc-
currences of the expression in the two programs. In the context of these
definitions we can state the constraints on birth as follows:
Prevent birth from lineages with at least a preset threshold number
of ancestors (4 here) and an improvement of less than some preset
minimum (0.1 here).
Prevent birth from lineages with at least a preset threshold number
of ancestors (3 here) and constant discrepancy between parent and
child in all generations.
Prevent birth from parents that received disqualifying fitness penal-
ties, e.g. for nontermination or non-production of result values.
Towards practical autoconstructive evolution 13
Prevent birth of children with sizes outside of the specified legal
range (here 10–100 points).
Prevent birth of children that are identical to any of their ances-
tors.
Prevent birth of children that are identical to potential siblings;
for this test the parent program is run a second time to produce
an additional child that is used only for this comparison.
4. Preliminary results
While the approach described here has not yet been shown to solve
problems that are out of reach of more conventional genetic programming
systems—indeed, it is currently weaker than the more-standard PushGP
system—it has solved simple problems and produced illuminating data
that may help to deepen our understanding.
For example, in one run on a symbolic regression problem with the
target function y=x32x2xAutoPush found a solution that de-
scended from the following randomly generated program:6
((code_if (code_noop) boolean_fromfloat (2) integer_fromfloat)
(code_rand integer_rot) exec_swap code_append integer_mult)
While it is difficult to tell from inspection how this program works,
even for those experienced in reading Push code, the specific code in-
structions that are included provide clues about how it constructs chil-
dren. For example, the code rand instruction generates new random code,
and the code append instruction combines two pieces of code on the code
stack. It is even more revealing to look at the code outputs from several
runs of this program. In this case they are all of the form:
(RANDOM-INSTRUCTION (code_if (code_noop) boolean_fromfloat
(2) integer_fromfloat) (code_rand integer_rot) exec_swap code_append
integer_mult)
where “RANDOM-INSTRUCTION” is some particular randomly chosen instruc-
tion. So this program’s reproductive strategy is merely to add a new,
random instruction to the beginning of itself.
This strategy continues for several generations, with several improve-
ments in problem-solving performance, until something new and inter-
esting happens. In the sixth generation a child is produced with a new
6Space limitations prevent full description of the run parameters or the instruction set; see (Spector
et al., 2005) and the source code at http://hampshire.edu/lspector/gptp10 for more information.
14 GENETIC PROGRAMMING THEORY AND PRACTICE III
list added, rather than just a new instruction, and it also has a new
reproductive strategy: it adds something new to the beginning of both
of its top-level lists. In other words, the sixth-generation individual is
of this form:
(SUB-EXPRESSION-1 SUB-EXPRESSION-2)
where each “SUB-EXPRESSION-n” is a different sub-expression, and the
seventh-generation children of this program are all of the form:
((RANDOM-INSTRUCTION-1 (SUB-EXPRESSION-1))
(RANDOM-INSTRUCTION-2 (SUB-EXPRESSION-2)))
where each “RANDOM-INSTRUCTION-n” is some particular randomly chosen
instruction.
One generation later the problem was solved, by the following pro-
gram:
((integer_stackdepth (boolean_and code_map)) (integer_sub
(integer_stackdepth (integer_sub (in (code_wrap (code_if (code_noop)
boolean_fromfloat (2) integer_fromfloat) (code_rand integer_rot)
exec_swap code_append integer_mult))))))
This program inherits the altered reproductive strategy of its parent,
augmenting both of its primary sub-expressions with new initial instruc-
tions in its children.
In the run described above the only available code-manipulation in-
structions were those in the standard Push specification, which are mod-
eled loosely on Lisp list-manipulation primitives. In some runs, however,
we have added a “perturb” instruction that changes symbols and con-
stants in a program to other random symbols or constants with a prob-
ability derived from an integer popped from the integer stack. Perturb,
which was also used in some Pushpop runs, is itself a powerful mutation
operator, but its availability does not dictate if or how or where it will
be used; for example, it would be possible for an evolved reproductive
strategy to use perturb on only one part of its code, or to use it with dif-
ferent probabilities on different parts of its code, or to use it conditionally
or in conjunction with other code-manipulation instructions. With the
perturb instruction included we have been able to solve somewhat more
difficult problems such as the symbolic regression of y=x62x4+x22,
and we are actively exploring application to more difficult problems and
analysis of the resulting programs and lineages, with the hypothesis that
more complex and adaptive reproductive strategies will emerge in the
context of more challenging problem environments.
Towards practical autoconstructive evolution 15
5. Conclusions
The specific results reported here are preliminary, and the hypothesis
that autoconstructive evolution will extend the problem-solving power
of genetic programming is still speculative. However, the hypothesis has
been refined, the means for testing it have been simplified, the prin-
ciples that underlie it have been better articulated, and the prospects
for analysis of incremental results have been improved. We have shown
(again) that mechanisms of adaptive variation can evolve as components
of evolving problem-solving systems, and we have described reasons to
believe that the best problem-solving systems of the future will make
use of some such techniques. Only further experimentation will deter-
mine whether and when autoconstructive evolution will become the most
appropriate technique for solving difficult problems of practical signifi-
cance.
Acknowledgments
Kyle Harrington, Paul Sawaya, Thomas Helmuth, Brian Martin, Scott
Niekum and Rebecca Neimark contributed to conversations in which
some of the ideas used in this work were refined. Thanks also to the
GPTP reviewers, to William Josiah Erikson for superb technical sup-
port, and to Hampshire College for support for the Hampshire College
Institute for Computational Intelligence.
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