The Max problem revisited: The importance of mutation in genetic programming

Kötzing, Timo; Sutton, Andrew M.; Neumann, Franz‐Josef; O’Reilly, Una-May

doi:10.1016/j.tcs.2013.06.014

Cited by 15 publications

(16 citation statements)

References 2 publications

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“…Furthermore, understanding how and when correct structures are evolved will be necessarily crucial in an analysis of more realistic GP scenarios. Recently, the same simple GP systems have been analysed on the MAX Problem [6] where, given a set of functions, a set of terminals and a bound D on the maximum depth of the solution, the goal is to evolve a tree that returns the maximum value given any combination of functions and terminals [3]. The analysis shows that the simple GP systems can efficiently evolve MAX with function set F = {+, * } and one constant as terminal set.…”

Section: Introductionmentioning

confidence: 99%

“…When the initial foundations for a systematic time complexity analysis of EAs were being set, very simple EAs were considered (eg., the (1+1) EA) for simple benchmark problems which are easy for EAs (eg., OneMax and LeadingOnes) and others which are hard (eg., Trap Functions and Needle-In-A-Haystack) [2]. In a similar fashion we will analyse the simple and minimalistic (1+1) GP considered in previous runtime analyses of GP [11,3] for simple Boolean functions with minimal function and terminal sets. Since under the evolvability notion in the PAC-learing framework it is well-understood that conjunctions (i.e., AND) are evolvable efficiently while parity problems (i.e., XOR) are not [19], we naturally choose these boolean functions as our starting points for the analysis.…”

Section: Introductionmentioning

confidence: 99%

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On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

Mambrini¹,

Oliveto²

2016

Lecture Notes in Computer Science

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Abstract. This work presents a first step towards a systematic time and space complexity analysis of genetic programming (GP) for evolving functions with desired input/output behaviour. Two simple GP algorithms, called (1+1) GP and (1+1) GP*, equipped with minimal function (F) and terminal (L) sets are considered for evolving two standard classes of Boolean functions. It is rigorously proved that both algorithms are efficient for the easy problem of evolving conjunctions of Boolean variables with the minimal sets. However, if an extra function (i.e. NOT) is added to F, then the algorithms require at least exponential time to evolve the conjunction of n variables. On the other hand, it is proved that both algorithms fail at evolving the difficult parity function in polynomial time with probability at least exponentially close to 1. Concerning generalisation, it is shown how the quality of the evolved conjunctions depends on the size of the training set s while the evolved exclusive disjunctions generalize equally badly independent of s.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

Mambrini¹,

Oliveto²

2016

Lecture Notes in Computer Science

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“…The theorem is proven using fitness-based partitions, exploiting the existence of at least one leaf in a tree of size n which could be selected by insertion to grow the tree. Experimental results suggesting that the true runtime of the (1 + 1) GP on MAX is also O(n log n) were also presented, and the authors of [19] noted that a more precise potential function based on the contents of the tree would be required to show this upper bound using drift analysis.…”

Section: The Max Problemmentioning

confidence: 99%

“…A more realistic problem where the program output, rather than structure, is used as the basis for determining solution quality is the MAX problem [19], originally introduced in [12]. The problem is to evolve a program which, given some mathematical operators and constants (the problem admits no variable inputs), outputs the maximum possible value subject to a constraint on program size.…”

Section: Introductionmentioning

confidence: 99%

Computational Complexity Analysis of Genetic Programming

Lissovoi

Oliveto

2019

Natural Computing Series

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Genetic programming (GP) is an evolutionary computation technique to solve problems in an automated, domain-independent way. Rather than identifying the optimum of a function as in more traditional evolutionary optimization, the aim of GP is to evolve computer programs with a given functionality. While many GP applications have produced human competitive results, the theoretical understanding of what problem characteristics and algorithm properties allow GP to be effective is comparatively limited. Compared with traditional evolutionary algorithms for function optimization, GP applications are further complicated by two additional factors: the variable-length representation of candidate programs, and the difficulty of evaluating their quality efficiently. Such difficulties considerably impact the runtime analysis of GP, where space complexity also comes into play. As a result, initial complexity analyses of GP have focused on restricted settings such as the evolution of trees with given structures or the estimation of solution quality using only a small polynomial number of input/output examples. However, the first computational complexity analyses of GP for evolving proper functions with defined input/output behavior have recently appeared. In this chapter, we present an overview of the state of the art.Algorithm 1: The (1+1) GP 1 Initialize a tree X;We will also consider the theoretical work on GSGP, where the variation operators used by the GP system are designed to modify program semantics rather than program syntax.This chapter presents an overview of the state of the art. It is structured as follows. In Section 2, we introduce the (1 + 1) GP, the GP system used for most of the available computational complexity analysis results. In Section 3, we present an overview of the analyses of GP systems for evolving tree structures with specific properties (the Order, Majority, and Sorting problems). In Section 4, we present results where GP systems evolve programs with limited functionality: the MAX problem is considered in Subsection 4.1, and the Identification problem in Subsection 4.2. Section 5 presents results for GP evolving proper Boolean functions of arity n. Section 6 presents a brief overview of the computational complexity results available for GSGP algorithms. Finally, Section 7 presents a summary of the presented results and discusses the open directions for future work.

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“…Both problems are simple enough to be analysed thoroughly, and they represent different aspects of problems solved through genetic programming, that is, including components in the correct order (ORDER), and including the correct set of components in a solution (MAJORITY). Additional recent computational complexity results are those of Kötzing et al (2012) on the MAX problem, and of Wagner and Neumann (2012) on the SORTING problem.…”

Section: Introductionmentioning

confidence: 99%

Single- and multi-objective genetic programming

Nguyen

Urli

Wagner

2013

Proceedings of the Twelfth Workshop on Foundations of Genetic Algorithms XII

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We consolidate the existing computational complexity analysis of genetic programming (GP) by bringing together sound theoretical proofs and empirical analysis. In particular, we address computational complexity issues arising when coupling algorithms using variable length representation, such as GP itself, with different bloat-control techniques. In order to accomplish this, we first introduce several novel upper bounds for two single-and multi-objective GP algorithms on the generalised Weighted ORDER and MAJORITY problems. To obtain these, we employ well-established computational complexity analysis techniques such as fitness-based partitions, and for the first time, additive and multiplicative drift.The bounds we identify depend on two measures, the maximum tree size and the maximum population size, that arise during the optimization run and that have a key relevance in determining the runtime of the studied GP algorithms. In order to understand the impact of these measures on a typical run, we study their magnitude experimentally, and we discuss the obtained findings.

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The Max problem revisited: The importance of mutation in genetic programming

Cited by 15 publications

References 2 publications

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

Computational Complexity Analysis of Genetic Programming

Single- and multi-objective genetic programming

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