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Evolution of -Expressions Through Genetic Programming

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Evolution of -Expressions Through Genetic Programming Evolution of expressions through Genetic Programming z x Dorothea HeissCzedik and Walter Fontana Theoretical Chemistry University of Vienna Wahringerstrae A Vienna Austria z Institute for Information Systems E Vienna University of Technology A Vienna Austria x International Institute for Applied Systems Analysis I IASA A Laxenburg Austria fdorothea waltergtbiunivieacat Abstract We illustrate a minimal version of Genetic Programming op erating with calculus byevolving the predecessor function The expression ob tained works dierently than the original version of Kleene In those runs that were successful hundreds of dierent expressions realizing the pre decessor function were found indicating a large degree of neutrality We suggest that the study of the calculus landscap e holds promise for a more rigorous and systematic understanding of the p ower and limita tions of Genetic Programming as they derive from the language that maps syntactical constructs into functional b ehaviors keywords genetic programming calculus predecessor function landscap es Intro duction The Darwinian principle of adaptation through replication heritable variation and selection is not limited to a p opulation of biological entities It is applicable to any ob ject that can b e copied and varied and for which at least some of the variants are distinguishable by an arbitrary criterion called tness Genetic Programming GP applies the logical structure of the Darwinian scheme to computer programs searching for those that compute a desired function In the realm of computation the phenotyp e corresp onds to the behavior of a program ie the graph of a function in the settheoretic sense and the genotyp e corresp onds to that which behaves ie a syntactical construct expressing a function Genetic Programming Koza has indeed b ecome a p owerful machine learning metho d Its success has been demonstrated in a variety of applica tions such as solving symbolic regression problems Koza discovering gameplaying strategies Koza Angeline inducing decision trees Koza generating controllers eg for rob ots Koza Reynolds Sp encer Handley cracking and evolving randomizers Koza Jannink A question however remains When is the Darwinian pro cess eective The search for an answer is still a frontier in biology The question is hard b ecause replication and variation apply to the carrier of b ehavior genotyp e while selection applies to the b ehavior phenotyp e and at least in biology we dont suciently understand the mapping b etween the two GP applications typically start with a cleverly crafted representation of a problem domain from which suitable highlevel primitives are inferred It may be argued that in such cases GP plays a minor role in nding the solution compared to the contribution of the user Abb ott Taylor The p oint relevant here is that the problemsp ecic comp onents of applications make a rigorous and general theoretical exploration of GP nearly imp ossible OReilly and Oppacher An analysis is needed of the intrinsic constraints and opp ortunities of GP deriving from the specic language that maps syntactical constructs genotyp es into functional b ehaviors phenotyp es Such an analysis could b egin byturn ing to an abstract universe of functions that is i transparent ii suciently formal to encourage mathematical analysis and iii canonical ie it should capture a programming language paradigm One such universe is calculus invented by Alonzo Church Church Church to study the prop erties of functions As is well known calculus is the syntactically unsug ared core of functional programming languages The notion of denabilityis equivalenttoTurings notion of computability and the HerbrandGodel notion of general recursiveness An exploration of the characteristic features of the calculus landscape that is the programtofunction mapping in calculus would greatly help in framing what is p ossible and what is not with GP within the paradigm of functional programming This then is the motivation for our use of calculus The presentpaper bynomeanscharacterizes the features of the calculus landscap e That is a longterm challenge Our contribution merely consists in illustrating GP within this semantically elegant and minimalist framework raising a few questions and providing supp ort for the claim that the study of the calculus landscap e holds promise for a more rigorous and systematic understanding of GP As an example we attempt to evolve the predecessor function Churchhad just ab out convinced himself that there is no denition of the predecessor function when Kleene found a representation for it Kleene Revesz It was an imp ortant result b ecause otherwise a computable function would exist that is not denable Although the predecessor function seems dicult to nd there is no particular practical b enet in nding it since it is already known Section briey intro duces the calculus Section intro duces genetic op erators that naturally t the syntactical machinery of calculus In Section the predecessor function is evolved as an example In Sections and the inuence of dierent parameter settings is discussed Section concludes and raises a few issues calculus In calculus functions are represented as expressions The simplest expression is a variable without typ e or sort of which there is an innite supply To build more complex expressions there are only two constructions application and abstraction The syntax of a expression following Revesz is h expr essioni hv ar iabl eijhabstr actionijhappl icationi habstr actioni hv ar iabl eihexpr essioni happl icationi hexpr essionih expr essioni and turns a Abstraction intro duces a formal parameter hv ar iabl ei eg x given hexpr essioni into a unary function For example x is a expression by virtue of and abstracting x via yields xx which is the identity function I dened in usual notation as I xx The x after the dot in xx corresp onds to the body of the function ie the right hand side in the dening equation I x x In xx the preceding the x declares it as a formal parameter corresp onding to the left hand side in the equation I xx Of course we could also abstract some other variable say y to get y x In an expression of the form hv ar iabl eihexpr essioni all o ccurrences of the hv ar iabl ei in hexpr essioni are called b ound Avariable that is not b ound is termed free Names of b ound variables dont matter and we identify ex pressions that dier only in the names of b ound variables happl icationi is intended to express the application of an op erator here enclosed in parentheses to an op erand There is no syntactical distinction between op erator and op erand b oth are arbitrary expressions calculus is meant to be a theory of the evaluation of functions This is achieved by dening a reduction relation b etween expressions that captures the notion of substitution xP Q QxP where P and Q are expressions and x is a variable QxP means the substi tution of Q for all o ccurrences of x in P We assume unique names for b ound variables distinct from names of free variables The situation in equation corresp onds to the substitution of the actual parameter Q for the formal pa rameter x in the b o dy P of a function xP The reduced form corresp onds to def the result of the evaluation For example I y xxy yxx y An expression that contains no subexpression to whichscheme applies is called a normal form and the pro cess of rewriting an expression into normal form is a normalization Not every expression do es have a normal form One mayencounter an innite sequence of reductions corresp onding to a non terminating computation Everything computable can b e dened with just the syntax and the rule Recall however that there is no syntactical distinction b etween functions and arguments Natural numb ers for example are functions to o and they can b e represented in a varietyofways WewillusetheChurchnumerals here and henceforth refer to them simply as numerals def f xx def f xf x def f xf f x def f xf f f x In general the numeral representing the number n iterates its rst argumentto its second argument n times In the following we use the shorthand notation def def n x f xf x n f x f f z n times Note that in untyp ed calculus every expression can act via as a map sending any expression into some expression that may or may not p ossess a normal form In recursion theory however the notion of function is a map from nonnegativeintegers to nonnegativeintegers Thus only the b ehavior of expressions restricted to a representation of numbers ie numerals is considered In other words arithmetic functions are expressions whose re duced form is a numeral when they are applied to one or several numerals expressions that are arithmetic functions dont contain free variables Such expressions are termed closed expressions also known as combinators As an example take the successor function which increments its argument def by one succ ng y ng g y Normalizing the application of succ to the numeral yields def succ ng y ng g y f xf f x z z succ g y f xf f xg g y g y xg g xg y g y g g g y A more detailed intro duction of calculus can be found for example in Revesz Hankin Barendregt Genetic programming in calculus In GP the task is to nd a program with a presp ecied behavior in a given language A tness function is dened to grade the actual b ehavior of a program with resp ect to the desired target b ehavior In the present case GP is not op erating on conventional programs but on expressions Since wewanttoevolve arithmetic
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