Hierarchical Grammatical Evolution Eric Medvet

DIA - Universit`a di Trieste Trieste, Italy [email protected]

ABSTRACT We present Hierarchical Grammatical Evolution (HGE) and its variant Weighted HGE (WHGE), two novel genotype-phenotype mapping procedures to be used in the Grammatical Evolution (GE) framework. HGE/WHGE are designed to exhibit better variational inheritance than standard GE without imposing any constraint on the structure of the genotype nor on the genetic operators. Our proposal considers the phenotype as a hierarchy of non-terminal expansions and is based on two key ideas: (i) the closer the nonterminal to be expanded to the root of the hierarchy, the larger the genotype substring determining its expansion, and (ii) upon expansion, a non-terminal divides its genotype substring among the resulting non-terminals. We experimentally evaluate our proposals on a set of benchmark problems and show that for the majority of them WHGE outperforms GE (and its variant π GE).

CCS CONCEPTS •Computing methodologies → Genetic programming;

KEYWORDS Genotype-phenotype mapping, Representation, Locality, Redundancy ACM Reference format: Eric Medvet. 2017. Hierarchical Grammatical Evolution. In Proceedings of The Genetic and Evolutionary Computation Conference, Berlin, Germany, July 15–19, 2017 (GECCO’17), 2 pages. DOI: http://dx.doi.org/10.1145/3067695.3075972

1

INTRODUCTION

The distinctive feature of Grammatical Evolution (GE) [7] is its ability to evolve programs in any language, using a user-provided context-free grammar (CFG). In GE, each individual is represented as a string of bits (the genotype) which is then mapped to a program in the language defined by the grammar (the phenotype) by means of a genotype-phenotype mapping procedure. This mechanism relieves from the user the burden of adapting the internals of the evolutionary algorithm to his specific problem, hence favoring GE usage in a wide range of applications (e.g., [1, 6, 10]). However, the mapping procedure of GE has been often deemed to weakly comply to the variational inheritance principle, stating Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. GECCO’17, Berlin, Germany © 2017 ACM. 978-1-4503-4939-0/17/07. . . $15.00 DOI: http://dx.doi.org/10.1145/3067695.3075972

that offspring should closely resemble, but not be identical to their parents [2]. In particular, many studies elaborate on GE locality and redundancy (e.g., [4, 8, 9]) and highlight its limitations in terms of these properties. In this paper, we describe a novel genotype-phenotype mapping procedure, called Hierarchical Grammatical Evolution (HGE), and its variant Weighted HGE (WHGE), which we designed precisely with the aim of addressing GE shortcomings in terms of variational inheritance. We experimentally compare HGE/WHGE with GE showing that WHGE outperforms GE in most of the benchmark problems. Since its original introduction, many variants of GE have been proposed (e.g., [3, 5]). The most relevant variant, w.r.t. our proposal, is Structured Grammatical Evolution (SGE) [3], which has been motivated by the same need of addressing GE poor locality and high redundancy. However, unlike our HGE/WHGE, SGE operates on an integer string genotype, whose length is a function of the grammar and the maximum depth of phenotypes (a parameter which has to be set by the user in advance), using specific genetic operators.

2

HGE/WHGE MAPPING

We consider the problem of mapping a genotype д (a bit string) to a phenotype p ∈ L(G), i.e., the language defined by the CFG G = (N ,T , s 0 , R), where N is the set of non-terminal symbols, T is the set of terminal symbols, s 0 ∈ N is the starting symbol, and R is the set of production rules. HGE mapping consists in iterating the following steps, starting with the association of the full genotype д with the only symbol s 0 = s 0 (i.e., the starting symbol) initially present in the phenotype p: (1) Let s 0 be a non-terminal s 0 in p, д 0 the genotype substring associated with s 0 , and r s 0 the rule for s 0 . If the size of д 0 is greater of equal than the number of options in r s 0 (i.e., if |д 0 | ≥ |r s 0 |), then: (a) split д 0 in |r s 0 | substrings of equal length or, if not possible, in a way such that the variance of the lengths is the lowest; (b) find the index i for which the relative cardinality (i.e., count of bits set to 1 divided by the number of all bits) of the i-th substring of д 0 is the largest or, in case of tie, the lowest index among ties; (c) use the i-th option of r s 0 to expand s 0 . Otherwise, i.e., if |д 0 | < |r s 0 |, use the option of r s 0 which leads to a sequence of terminals in the lowest number of expansions starting from s 0 . (2) After the expansion of s 0 , split д 0 in a number ns 0 of substrings of equal length (or with lengths determined as above), with ns 0 being the number of non-terminals resulting from the expansion of s 0 . Then, associate with each resulting j-th non-terminal the j-th substring of д 0 . The procedure stops when no more non-terminals are present in the phenotype. WHGE mapping differs from HGE in step 2, where the sizes of the substrings are proportional to the expressive powers of the

GECCO’17, July 15–19, 2017, Berlin, Germany corresponding non-terminals. We omit the details due to space constraints. An implementation of HGE/WHGE is publicly available at https://github.com/ericmedvet/evolved-ge. Concerning locality and redundancy, some differences arise between HGE/WHGE and GE. W.r.t. locality, it can be noted that in HGE/WHGE the expansion option for a node close to the root is chosen by processing a large genotype substring, and hence it is unlikely that a single bit leads to a different choice; on the other hand, for a node far from the root, the single bit may impact on the choice. In GE a modification of a single bit could instead result in a very different phenotype. W.r.t. redundancy, it can be noted that in HGE/WHGE each bit of the genotype concurs in determining the mapping, whereas in GE, if the mapping procedure ends before the first wrapping, all the bits of the codons which were not used do not concur in the mapping.

3

EXPERIMENTAL EVALUATION

We experimentally assessed HGE/WHGE and compared them against GE and π GE [5] on 4 benchmark problems (Harmonic, Polynomial, Santa-Fe, and Text, fully described in [4]) with 3 different genotype sizes |д|. For each GE variant, problem, and genotype size, we performed 30 independent executions of the evolutionary search by varying the random seed and with the following evolution parameters: population of 500 individuals randomly initialized, 50 generations, one-point independent cut point crossover for GE/π GE or two-points same cut points crossover for HGE/WHGE with 0.8 rate, bit flip mutation with pmut = 0.01 and 0.2 rate, tournament selection with size 5, m + 1 replacement strategy. Moreover, we set, for GE/π GE, the codon size to n = 8 and the maximum number of wrappings to nw = 5. Table 1 presents the results of the experimentation in terms of the fitness of the best individuals at the end of the evolution, averaged across the 30 repetitions, for different problems and mappers. For π GE, HGE, and WHGE, the table also shows the statistical significance of the Mann-Withney test performed on the best fitness values of the variant and those of GE. It can be seen that WHGE outperforms the other variants in 11 on 16 problem/genotype size combinations. In particular, the performance gap is very large for the Harmonic problem, for which both HGE and WGHE greatly improve the best fitness at the end of the evolution with all the values for |д|, the differences being also statistical significant. WGHE is the best variant also in the Polynomial problem with all |д| values, but the improvements are smaller. Concerning the two other problems, our experiments do not allow to declare a clear winner among the variants: however, it can be noted that in both cases the gap between HGE and WHGE is large. Our explanation for this finding is that Santa-Fe and Text grammars exhibit a larger difference in expressive power among non-terminals than the symbolic regression grammars. This aspect likely highlights the HGE limitation concerning the splitting of the genotype among children, which is addressed in WHGE, hence justifying the performance gap.

4

CONCLUDING REMARKS

We presented HGE/WHGE, two variants of a novel genotype-phenotype mapping procedure for Grammatical Evolution. HGE/WHGE

Eric Medvet Table 1: Best fitness at the end of the evolution. In each row, the best figure is highlighted in bold. For each variant other than GE, the statistical significance of the Mann-Whitney test w.r.t. GE is shown: ∗ means p < 0.1, ∗∗ means p < 0.05, ∗∗∗ means p < 0.01, no subscript means p ≥ 0.1. Problem Harmonic

Polynomial

Santa-Fe

Text

|д|

GE

π GE

HGE

WHGE

256 512 1024

10.78 13.53 11.74

11.20 11.05 12.09

7.67 6.89∗∗∗ 6.70

6.24∗∗∗ 5.87∗∗∗ 5.83∗∗∗

256 512 1024

2.50 2.55 2.54

2.70 2.64 2.58

2.31 2.62 2.94

2.19 2.36 1.80∗∗∗

256 512 1024

44.13 42.30 40.63

41.03∗∗ 41.40 43.07

68.00∗∗∗ 62.30∗∗∗ 53.57∗∗∗

43.60 41.43 37.90∗∗

256 512 1024

4.70 4.87 5.17

5.27∗∗ 5.02 5.20

6.40∗∗∗ 5.87∗∗∗ 5.40

5.27∗∗ 4.93 4.87

are designed to exhibit a better variational inheritance than standard GE and, in particular, to improve its locality and redundancy. We assessed experimentally our proposal and showed that WHGE is more effective than GE in the majority of the considered benchmark problems.

REFERENCES [1] Alberto Bartoli, Andrea De Lorenzo, Eric Medvet, and Fabiano Tarlao. 2016. Syntactical Similarity Learning by Means of Grammatical Evolution. In Parallel Problem Solving from Nature – PPSN XIV: 14th International Conference, Edinburgh, UK, September 17-21, 2016, Proceedings. Springer International Publishing, Cham, 260–269. DOI:http://dx.doi.org/10.1007/978-3-319-45823-6 24 [2] Kenneth A De Jong. 2006. Evolutionary computation: a unified approach. MIT press. [3] Nuno Lourenc¸o, Francisco B Pereira, and Ernesto Costa. 2015. SGE: a structured representation for grammatical evolution. In International Conference on Artificial Evolution (Evolution Artificielle). Springer, 136–148. [4] Eric Medvet. 2017. A Comparative Analysis of Dynamic Locality and Redundancy in Grammatical Evolution. In Genetic Programming: 20th European Conference, EuroGP 2017, Amsterdam, Netherlands, April 19-21, 2017, Proceedings. Springer International Publishing, Cham, to appear. [5] Michael O’Neill, Anthony Brabazon, Miguel Nicolau, Sean Mc Garraghy, and Peter Keenan. 2004. π Grammatical Evolution. Springer Berlin Heidelberg, Berlin, Heidelberg, 617–629. DOI:http://dx.doi.org/10.1007/978-3-540-24855-2 70 [6] Michael O’Neill, James McDermott, John Mark Swafford, Jonathan Byrne, Erik Hemberg, Anthony Brabazon, Elizabeth Shotton, Ciaran McNally, and Martin Hemberg. 2010. Evolutionary design using grammatical evolution and shape grammars: Designing a shelter. International Journal of Design Engineering 3, 1 (2010), 4–24. [7] Conor Ryan, JJ Collins, and Michael O. Neill. 1998. Grammatical evolution: Evolving programs for an arbitrary language. Springer Berlin Heidelberg, Berlin, Heidelberg, 83–96. DOI:http://dx.doi.org/10.1007/BFb0055930 [8] Ann Thorhauer. 2016. On the Non-uniform Redundancy in Grammatical Evolution. In International Conference on Parallel Problem Solving from Nature. Springer, 292–302. [9] Ann Thorhauer and Franz Rothlauf. 2014. On the locality of standard search operators in grammatical evolution. In International Conference on Parallel Problem Solving from Nature. Springer, 465–475. [10] Marina Zapater, Jos´e L. Risco-Mart´ın, Patricia Arroba, Jos´e L. Ayala, Jos´e M. Moya, and Rom´an Hermida. 2016. Runtime data center temperature prediction using Grammatical Evolution techniques. Applied Soft Computing 49 (2016), 94 – 107. DOI:http://dx.doi.org/10.1016/j.asoc.2016.07.042

Hierarchical Grammatical Evolution

in the language de ned by the grammar (the phenotype) by means ... to weakly comply to the variational inheritance principle, stating. Permission to make digital ...

516KB Sizes 3 Downloads 298 Views

Recommend Documents

Hierarchical Grammatical Evolution
Jul 19, 2017 - ant Weighted HGE (WHGE), two novel genotype-phenotype map- ... ability to evolve programs in any language, using a user-provided.

Evolvability in Grammatical Evolution
and context-free languages; •Computing methodologies → Heuris- ... classroom use is granted without fee provided that copies are not made or distributed ... Figure 1: From Fitness Cloud to Fitness-Probability Cloud. ..... Figure 3: AEP vs. genoty

Grammatical Evolution and Corporate Failure ... Accounts
Kingston Business School, London. Conor Ryan .... business, to legal bankruptcy followed by liquidation of the firm's .... representing the programs as parse trees, as in traditional .... table that each model only employed a small subset of these.

Evolvability in Grammatical Evolution
each candidate solution. ... Figure 1: From Fitness Cloud to Fitness-Probability Cloud. ... the crossover operator, a pair of parents is needed but only the best.

Evolvability in Grammatical Evolution
share the same phenotype? 4Often ..... Lu, Li, and Yao, “Fitness-probability cloud and a measure of problem hardness for evolutionary algorithms”. Medvet ...

Christiansen Grammar Evolution: grammatical ...
Computer Science. Chomsky ... they have been little used in computer science to formally ...... the requirements for the degree of Master of Science in Computer.

Grammatical evolution - Evolutionary Computation, IEEE ... - IEEE Xplore
definition are used in a genotype-to-phenotype mapping process to a program. ... evolutionary process on the actual programs, but rather on vari- able-length ...

Syntactical Similarity Learning by means of Grammatical Evolution
on input data which consist of triplets of data points (a, b, c) labelled with the .... when X and Y are perfectly separated, o(X, Y ; p)=0, Vp. The value of p is used.

Syntactical Similarity Learning by means of Grammatical Evolution
by a similarity learning algorithm have proven very powerful in many different application ... a virtual machine that we designed and implemented. The virtual ..... pattern (e.g., telephone numbers, or email addresses) from strings which do not.

Grammatical Errors
http://grammarist.com/articles/grammarly-review/. British​ ​English​ ​vs.​ ​American​ ​English​ ​Test. For this test, we'll create a series of sentences that contain distinctly British spelling and. grammatical structures. § The

About Grammatical Framework
Jan 8, 2003 - is a generic system used to write and uses mathematical theories with a logical calculus. .... command line interpreter : functions to read grammar files and use grammars in ... delete : replaces subtree with a metavariable.

Hierarchical Decomposition.pdf
Page 2 of 24. 2. Background. • Functional decomposition of models (1993). • Hierarchical modules (2005). – Many systems naturally hierarchical. – Easier to ...

Hierarchical networks
quantitative information we introduce a threshold T to convert the correlation matrix into ... similarity between this picture and the one obtained by plotting the ... 1 We refer the interested reader to http://www.ffn.ub.es/albert/synchro.html, wher

Hierarchical networks
systems, and in particular the paradigmatic analysis of large populations of coupled oscillators. [6–8]. The connection between ... a large variety of synchronization patterns and sufficiently flexible to be adapted to many different contexts [10].

Hierarchical networks
May 21, 2008 - parameter' to characterize the level of entrainment between oscillators. However, this definition, although suitable for mean-field models, is not efficient to identify local dynamic effects. In particular, it does not give information

Spandanam_Worksheet of various grammatical terms for SSLC ...
Vanka was writing a letter to his grandfather. 2. He posted the letter without writing the address. 3. ... Conditional Sentences. Conditionals If-clause Main clause. First conditional Simple present ... Spandanam_Worksheet of various grammatical term

Inferring universals from grammatical variation
plane) is the crucial element, since for any roll-call vote we are interested in who voted 'Yea' .... In two dimensions, there are only 24 errors across 1250 data points. ..... the Quotative near the center of the vertical area. ... by a Present or I

Peters McLuhan's Grammatical Theology.pdf
Technology is behind the Medieval Centre [at the University of Toronto]?” asked. McLuhan in 1973 (Gordon 1997, 323). In his Cambridge dissertation we have a ...

Hierarchical networks
May 21, 2008 - Online at stacks.iop.org/JPhysA/41/224007. Abstract .... Some of them are homogeneous in degree, whereas other networks have special ...

hierarchical mtmm - UdG
Application to the Measurement of Egocentered Social Networks. 1. Lluís Coromina .... Figure 1: Path diagram for the MTMM model for trait (Ti) and method (Mj).

Hierarchical Learning
The CSL Algorithm. CSL(ε,δ,s) ε'← ε/sK δ'← δ/2s m ← ln(K/δ')/ε' for i ← 1 to s get EX i. # advance to next lesson for j ← 1 to 2m # create filtered sample. (x,y) ← Push-Button(EX i. ) # draw exmple if F. 1. ,…,F i-1 agree on x a

hierarchical mtmm - UdG
Application to the Measurement of Egocentered Social Networks. 1 ..... 10. Therefore Muthén (1989,1990) proposes to utilise another estimator known as the ...

Bayesian Hierarchical Curve Registration
The analysis often proceeds by synchronization of the data through curve registration. In this article we propose a Bayesian hierarchical model for curve ...