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Weighted Hierarchical Grammatical Evolution: A Genotype-Phenotype Mapping with Better Properties

Context-free grammars of the benchmark problems in the Backus Naur Form

MOPM-3

<e> ::= .or <e> <e> | .and <e> <e> | .not <e> | <v>

Parity-5

<e> ::= .or <e> <e> | .xor <e> <e> | .and <e> <e> | .and1not <e> <e> | <v>

KLandscapes-3

<N> ::= <n> <N> <N> | <t>
<n> ::= n0 | n1
<t> ::= t0 | t1 | t2 | t3

KLandscapes-7

<N> ::= <n> <N> <N> | <t>
<n> ::= n0 | n1
<t> ::= t0 | t1 | t2 | t3

Text

<text> ::= <sentence> <text> | <sentence>
<sentence> ::= <Word> _ <sentence> | <word> _ <sentence> | <word> <punct> 
<word> ::= <letter> <word> | <letter>
<Word> ::= <Letter> <word>
<letter> ::= <vowel> | <consonant>
<vowel> ::= a | o | u | e | i
<consonant> ::= q | w | r | t | y | p | s | d | f | g | h | j | k | l | z | x | c | v | b | n | m
<Letter> ::= <Vowel> | <Consonant>
<Vowel> ::= A | O | U | E | I
<Consonant> ::= Q | W | R | T | Y | P | S | D | F | G | H | J | K | L | Z | X | C | V | B | N | M
<punct> ::= ! | ? | .

Keijzer-6

<expr> ::= <op> <expr> <expr> | <pre-op> <expr> | <var>
<op> ::= + | *
<pre-op> ::= _ | 1/ | √
<var> ::= x

Nguyen-7

<expr> ::= <op> <expr> <expr> | <pre-op> <expr> | <var>
<op> ::= + | - | * | p/
<pre-op> ::= sin | cos | exp | plog
<var> ::= x | 1.0

Pagie-1

<expr> ::= <op> <expr> <expr> | <pre-op> <expr> | <var>
<op> ::= + | - | p/ | *
<pre-op> ::= sin | cos | exp | plog
<var> ::= x | y | 1.0

Vladislavleva-4

<expr> ::= <op> <expr> <expr> | <pre-op> <expr> | <var> | <const>
<op> ::= + | - | p/ | *
<pre-op> ::= ²
<var> ::= x1 | x2 | x3 | x4 | x5
<const> ::= 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 | 9.0 | 10.0

Impact of the depth parameter

Impact of d_max (for SGE), n_d (for WHGE), maximum tree depth (for CFGGP) on the median best fitness at the last generation.