Discrete Set Handling - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 168

7.3.3 DSH Applications on Class of Genetic Programming Techniques

The term symbolic regression represents a process during which measured data sets are
fitted such thereby a corresponding mathematical formula is obtained in an analytical
way. An output of the symbolic expression could be, for example, x 2 + y 3 / K ,andthe
like. For a long time, symbolic regression was a domain of human calculations but in
the last few decades it involves computers for symbolic computation as well.
The initial idea of symbolic regression by means of a computer program was pro-
posed in Genetic Programming (GP) [8, 9]. The other approaches are Grammati-
cal Evolution (GE) developed in [19, 13] and Analytic Programming (AP) in [27].
Oher interesting investigations using symbolic regression were carried out in [6] on
Artificial Immune Systems and Probabilistic Incremental Program Evolution (PIPE),
which generates functional programs from an adaptive probability distribution over
all possible programs. As an extension of GE to the another algorithms is also [14],
where DE was used with the GE. Symbolic regression, generally speaking, is a pro-
cess which combines, evaluates and creates more complex structures based on some
elementary and noncomplex objects, in an evolutionary way. Such elementary ob-
jects are usually simple mathematical operators (+ , ,... ), simple functions (sin,

cos, And, Not,.), user-defined functions (simple commands for robots

− , ∗

− MoveLeft,

TurnRight,.), etc.
An output of symbolic regression is a more complex object (formula, function, com-
mand,.), solving a given problem like data fitting of the so-called Sextic and Quintic
problem described by Equation 7.7) [10, 26], randomly synthesized function by Equa-
tion 7.8 [26], Boolean problems of parity and symmetry solution (basically logical cir-
cuits synthesis) by Equation 7.9) [11, 27], synthesis of Chaos by utilizing DSH and
Evolutionary Algorithms [28] given in Table 7.1 and in Figs 7.7 7.10.

Synthesis of quite complex robot control command by Equation 7.10

−

[10, 15] is

also accomplished with DSH. Equation 7.7 7.10 mentioned are just a few samples

from numerous repeated experiments done by AP, which are used to demonstrate how
complex structures can be produced by symbolic regression in general for different
problems.

−

x K 1 +

x 2 K 3
K 4 ( K 5 + K 6 )

• ( − 1 + K 2 + 2 x ( − x − K 7 )) (7.7)

√ t

sec − 1 (1 . 28)

1
log ( t )

log sec − 1 (1 . 28) (sinh (sec (cos(1))))

(7.8)

Nor Nand Nand B B, B && A ,B && C && A && B,
Nor C && B && A A && C && B C&& B&& A &&
C && B && A A && C && B C&& B&& A
A&& C && B && A A && C && B C&& B&& A ,
C C && B && A A && C && B C&& B&& A && A (7.9)