=Paper=
{{Paper
|id=Vol-2472/p2
|storemode=property
|title=Takagi–Sugeno fuzzy systems as a method of acting opponents in games
|pdfUrl=https://ceur-ws.org/Vol-2472/p2.pdf
|volume=Vol-2472
|authors=Karolina Kesik
}}
==Takagi–Sugeno fuzzy systems as a method of acting opponents in games==
<pdf width="1500px">https://ceur-ws.org/Vol-2472/p2.pdf</pdf>
<pre>
Takagi–Sugeno fuzzy systems as a method of acting
              opponents in games
                                                                Karolina K˛esik
                                                          Institute of Mathematics
                                                     Silesian University of Technology
                                                   Kaszubska 23, 44-100 Gliwice, Poland
                                                       Email: karola.ksk@gmail.com


   Abstract—Controlling a hero in 2D or 3D games can occur on                   networks and tree searching were used to obtain impressive
different rules. To increase the level of gameplay, there are objects           results. A year later, the authors presented a learning algorithm
in the entire game with which the player can interact. Such                     without the need for knowledge given by programmers [6].
an action is quite often implemented using artificial intelligence,
which has been growing in recent years. In this paper, we propose               Again in [7], the authors presented idea to train deep neural
using a model of a fuzzy system to move objects in the games.                   networks to play othello ie a game on a board with 64 fields
Movement takes place on the principle of moving between two                     and black and white verticals. An interesting idea is to extract
points, where the first one is the player’s position and the second             knowledge from how others play the game [8], or using the
is destination point. The paper presents the mathematical model                 idea of games in other areas, such as medicine. An example
of the Takagi–Sugeno system and tests for the correctness of the
movement.                                                                       is creating decision systems using game theory and rough sets
                                                                                [9].
                         I. I NTRODUCTION                                          The idea of multi-agents and collective using many objects
   The field of artificial intelligence is developing rapidly,                  is a frequent use in games. The results of research on this
which is visible in various applications around us. Especially                  topic are presented in [10], [11] using fuzzy logic. This field
in devices of the Internet of Things, whose times are coming                    of science can also be used in solving various problems [12],
thanks to the 5G network. Enabling communication between a                      [13].
huge number of devices results in acquiring and generating a
                                                                                                      II. F UZZY LOGIC
large amount of digital information. This information must be
processed so that it can be forwarded or take some decisions                      Fuzzy logic was proposed by L. A. Zadeh in 1965, where
based on it. More often, such problems are solved using                         unlike classical sets, partial membership is allowed [14]. So
artificial intelligence (AI) techniques.                                        the sets can be described using the adaptation function
   In practice, almost everyone uses a smartphone, on which
various applications are installed. Especially often of these                                          µA : X → [0, 1].                      (1)
applications are games that use AI methods. Their practical
                                                                                Fuzzy set A on space X can also be described by a set
use is based on the movement of opponents or the genera-
                                                                                of ordered pairs (x, µA (x)), where µA (x) is the degree of
tion of boards/environment. The movement can be simulated
                                                                                ownership of the x object to the fuzzy set A
using various tools such as artificial neural networks, fuzzy
controllers and heuristic algorithms. In this paper, we show                                A = {(x, µA (x)); x ∈ X, µA ∈ [0, 1]}.           (2)
the simple use of the Takagi-Sugeno system to simulate the
movement of objects.                                                            Zadeh suggested another notation for fuzzy sets. For the
                                                                                discrete space X, we have
A. Related works                                                                                          X
   Heuristic methods are used only to simulate player action                                        A=      µA (x)/x                 (3)
[1], but also to create a game environment. An example is the                                               x∈X

creation of labyrinths [2]. The same algorithms can be used                     Again, for continuous space X
for play-testing in various games, especially in card ones [3].                                           Z
Quite often, there is a need for the opponent’s intelligence to                                      A=      µA (x)/x                        (4)
be trained. This is a popular situation with games that have                                                 X
large amounts of different rules, such as chess [4] or go [5].
                                                                                  These markings should be treated as the sum of elements x
The second of these games is much more difficult, which made
                                                                                having a degree of belonging µA (x). Mark / should be treated
training AI quite problematic. In [6] deep learning of neural
                                                                                as a separator.
  c 2019 for this paper by its authors. Use permitted under Creative Com-         The most commonly used affiliation functions that describe
mons License Attribution 4.0 International (CC BY 4.0).                         the sets are as follows



                                                                            5
  •   triangular                                                      variables of the system into the values of its output variables.
                                 0,      x ≤ a,
                               
                                 x−a
                               
                               
                               
                               
                                      , a < x ≤ b,
             µA (x; a, b, c) =   b−a                ,       (5)
                                 c − x
                               
                                      , b < x ≤ c,
                                c−b
                               
                               
                                 0,      x>c
      where a, b, c are parameters (a ≤ b ≤ c).
  •   Gaussian
                                       (x−m)2
                      µA (x; m, σ) = e− 2σ2 ,               (6)
    where m, σ are parameters. For x = m, this function
    assumes the value 1 and the parameter σ > 0 determines
    the width of the fuzzy set.
  • inverse trapezoid
                               
                                b − x a < x ≤ b,
                               
                                b−a
                               
          µA (x; a, b, c, d) =   0     b < x ≤ c, , (7)
                               
                                x−c
                               
                                      c<x<d
                                 d−c
     where a, b, c and d are parameters (a ≤ b ≤ c ≤ d).
   In addition, we distinguish two types of fuzzy sets. Type-
1 fuzzy sets on the one hand describe uncertain terms, and
on the other hand, the values of the membership function
are precise, because they are specific numbers. The idea was
born that the values of the membership function were fuzzy
sets and were called fuzzy sets of type-2. Fuzzy sets that
represent the values of membership functions are described
precisely. Hence, the further generalization that they can
be represented again by fuzzy sets. This reasoning can be
continued indefinitely, creating the concept of fuzzy sets of
type-m.
   Apart from the harvest itself, we define the concept of a
fuzzy number Ã, which is a convex and normal fuzzy set
with a limited medium, defined on the axis of real numbers
R.
A. Fuzzy system
   A fuzzy system is a set of interrelated elements separated
from the environment. Characteristic for the system is the fact
that internal connections are dominant in relation to connec-
tions with elements from outside the system. If connections           Figure 1: Sample screenshots using Takagi–Sugeno to drive a
with elements outside the system occur, then we understand            car on a short distance.
them as inputs and outputs from the system. We say then that
the system processes the input values to the output ones. At
the entrance of such a system we define                                          III. TAKAGI –S UGENO FUZZY SYSTEMS
   • linguistic values (in the form of fuzzy sets),
                                                                         The knowledge base of this system [16] is a set of if-then
   • numerical values - where is the need to apply some fuzzy
                                                                      rules
     operations.                                                               n    ol
Each fuzzy system contains a knowledge base written in                    R = R(i)
                                                                                     i=1
the form of fuzzy conditional rules if–then and so-called an                   n                                       ol        (8)
approximate motor of approximation based on fuzzy set theory                = if ∧n=1 x0n is A(i)
                                                                                    N
                                                                                                 n , then y = fi (x0 )     ,
                                                                                                                            i=1
and fuzzy reasoning [15]. The aforementioned motor combines
parts of knowledge contained in the rules if-then and with            where l is the amount of rules in knowledge base, x0n is input
                                                                                                                     (i)
approximate inference methods transforms the values of input          singleton, x0 = [x01 , x02 , . . . , x0N ]T , An is the linguistic



                                                                  6
value and y = f (x0 ) is a function in the conclusion i-th if-
then rule.
  Each of the conditional rules leads to the numeric type
output value, and output is determined as a weighted average
calculate using the following formula
                        I
                        X
                            F (i) (x0 ),        y (i) (x0 )
                 y0 = i=1     I
                                                                 (9)
                              X
                                        (j)
                                    F         (x0 )
                              j=1

where y (i) (x0 ) = fi (x0 ) determines the position of the
singleton in conclusion of i-th rule or or numerical result of
inference using i-th rule, F (i) (x0 ) is activation degree of i-th
rule defined as
 F (i) (x0 ) = µA(i) (x01 ) ?T µA(i) (x02 ) ?T . . . ?T µA(i) (x0N ),       Figure 2: The movement of the car depending on the desti-
                                    2                      N
                                                                 (10)       nation located in a very far distance ahead of the car on the
where ?T means t-norm.                                                      black surface.
  A function y = fi (x0 ) in conclusion of i-th rule can be
understand as fuzzy singleton with the following membership
function                                                                    and the second one is a movement angle. The variable has one
                                    
                                      1, y = y (i) (x0 ),                   of two membership functions – triangular or inverse trapezoid,
       µB (i) (y) = δy,y(i) (x0 ) =                              (11)       where the points of the bend will be the angles given in the
                                      0, y 6= y (i) (x0 ).
                                                                            numerical angles from the range h0, 360i. Rules were created
                                                                            based on these two variables and given functions. Each rule
  Start;                                                                    takes input and output values and returns value of the weight.
  Define the object;                                                        The created rule model works on the principle of adjusting the
  Define the goal;                                                          values of variables.
  Create rules;                                                                Using the Takagi–Sugeno system described in the previous
  Create a system based on previously created rules;                        section, it was implemented and described in Algorithm 1.
  Create a set composed of angles and the current angle                     Examples of screenshots from the car’s movement on the
   of the object, distances and distance between the                        surface are shown on Fig. 1.
   target and the current position;
  while the target did not reach its destination do                                               V. E XPERIMENTS
      Calculate the angle value using the system and the
                                                                               Proposed system was tested with car placed on the road
        current object parameters;
                                                                            with two lanes. The car was supposed to fit in the lanes at the
      Correct the angle of the object;
                                                                            turn. During the testing, different target distances were used
      Move the object forward;
                                                                            to see which option would be most advantageous. The distant
  end
                                                                            goal turned out to be ineffective, what can be seen in Fig.
  Stop;
 Algorithm 1: The code of moving the object in the game                     2. The car left the road choosing the shortest path. Then the
 using the Takagi–Sugeno system.                                            target distances were reduced and set at a distance of 1 − 4
                                                                            unit in UNITY. The effect for one field was the best, what is
                                                                            visible on Fig. 3.
     IV. TAKAGI –S UGENO FUZZY SYSTEM FOR GAMES
                                                                                                 VI. C ONCLUSIONS
   Takagi–Sugeno systems can be responsible for the move-
ment of certain objects in 2D or even 3D games. The main                       In this paper, the idea of using the Takagi-Sugeno system to
idea is that the movement takes place between two points                    make movement of objects in games. The simplicity of imple-
– the location of the object, and the planned destination                   mentation and creation of rules means that such systems are
position. To make this possible, a base of rules must be                    accurate under additional conditions, as noted in the previous
created. As previously stated, the rules are based on if-then.              section. Setting close goals allows to move objects on arcs,
The simplest way is to create two variables – distance and                  which is the correct movement on the road. Unfortunately,
angle. The first one is responsible for the moving distance                 this is not an ideal solution which is visible by finding new
and contains three linguistic variables from the following set              goals and setting them. However, it is a solution that does not
{lef t, straight, right} and a triangular membership function,              require training as opposed to neural networks[17].



                                                                        7
                                     (a)                                                                              (b)




                                     (c)                                                                              (d)
      Figure 3: The movement of the car depending on the destination located very close to the car on a black surface.


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