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1 #+title: Discovering Effective Pok\eacute{}mon Types Using Linear Optimization
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2 #+author: Robert McIntyre & Dylan Holmes
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3 #+EMAIL: rlm@mit.edu
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4 #+SETUPFILE: ../../aurellem/org/setup.org
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5 #+INCLUDE: ../../aurellem/org/level-0.org
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6
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7
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8
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9 * Introduction
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10 In the [[./types.org][previous post]], we used the best-first search algorithm to
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11 locate the most effective Pok\eacute{}mon type
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12 combinations. Afterwards, we realized that we could transform this
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13 search problem into a /linear optimization problem/. This conversion
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14 offeres several advantages: first, search algorithms are comparatively
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15 slow, whereas linear optimization algorithms are extremely fast;
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16 second, it is difficult to determine whether a search problem has any
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17 solution, whereas it is straightforward to determine whether a linear
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18 optimization problem has any solution; finally, because systems of
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19 linear equations are so common, many programming languages have linear
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20 equation solvers written for them.
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21
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22 In this article, we will
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23 - Solve a simple linear optimization problem in C :: We demonstrate
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24 how to use the linear programming C library, =lp_solve=, to
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25 solve a simple linear
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26 optimization problem.
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27 - Incorporate a C library into Clojure :: We will show how we gave
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28 Clojure access to the linear programming C library, =lp_solve=.
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29 - Find effective Pokemon types using linear programming :: Building
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30 on our earlier code, (...)
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31 - Present our results ::
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32
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33 #which can be represented and solved as a system of linear equations.
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34
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35 * COMMENT
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36 This post continues the [[./types.org][previous one]] about pok\eacute{}mon types.
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37 Pok\eacute{}mon is a game in which adorable creatures battle each
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38 other using fantastic attacks. It was made into a several gameboy
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39 games that all share the same battle system. Every pok\eacute{}mon in the
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40 gameboy game has one or two /types/, such as Ground, Fire, Water,
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41 etc. Every pok\eacute{}mon attack has exactly one type. Certain defending
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42 types are weak or strong to other attacking types. For example, Water
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43 attacks are strong against Fire pok\eacute{}mon, while Electric attacks are
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44 weak against Ground Pok\eacute{}mon. In the games, attacks can be either
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45 twice as effective as normal (Water vs. Fire), neutrally effective
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46 (Normal vs. Normal), half as effective (Fire vs. Water), or not
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47 effective at all (Electric vs. Ground). Thus the range of defense
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48 values for a particular type is the set 0, 1/2, 1, 2. These are
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49 referred to in the game as being immune, resistant, neutral, and
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50 weak, respectively. I call them the /susceptance/ of one type to another.
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51
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52 If a pokemon has two types, then the strengths and weakness of each
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53 type are /multiplied/ together. Thus Electric (2x weak to Ground)
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54 combined with Flying (immune to Ground (0x)) is immune to
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55 Ground. Fire (2x weak to Water) combined with Water (1/2x resistant
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56 to Water) is neutral to Water. If both types are resistant to another
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57 type, then the combination is doubly-resistant (1/4x) to that type. If
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58 both types are weak to a certain type then the combination is
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59 double-weak (4x) to that type.
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60
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61 ** Immortal Types
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62
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63 In the game, pok\eacute{}mon can have either one type, or two types. If this
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64 restriction is lifted, is there any combination of types that is
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65 resistant to all types? I call such a combination an /Immortal Type/,
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66 since if that type's pattern was repeated over and over again towards
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67 infinity, the resulting type would be immune to all attack types.
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68
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69 * Linear Programming
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70
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71 ** Terminology
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72 Linear programming is the process of finding an optimal solution to a
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73 linear equation of several variables which are constrained by some linear
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74 inequalities.
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75
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76 In linear programming terminology, the function to be extremized is
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77 the /objective function/.
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78
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79 ** COMMENT
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80 First, we'll give a small example of a linear optimization problem,
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81 and show how it can be solved with Clojure and =lp_solve=. Then,
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82 we'll show how finding immortal pok\eacute{}mon types can be converted
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83 into a linear problem suitable for solving in the same way.
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84
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85 ** The Farmer's Problem
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86
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87 Let's solve the Farmer's Problem, an example linear programming problem
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88 borrowed from http://lpsolve.sourceforge.net/5.5/formulate.htm.
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89
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90
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91 #+BEGIN_QUOTE
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92 *The Farmer's Problem:* Suppose a farmer has 75 acres on which to
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93 plant two crops: wheat and barley. To produce these crops, it costs
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94 the farmer (for seed, fertilizer, etc.) $120 per acre for the wheat
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95 and $210 per acre for the barley. The farmer has $15000 available for
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96 expenses. But after the harvest, the farmer must store the crops while
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97 awaiting favorable market conditions. The farmer has storage space
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98 for 4000 bushels. Each acre yields an average of 110 bushels of wheat
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99 or 30 bushels of barley. If the net profit per bushel of wheat (after
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100 all expenses have been subtracted) is $1.30 and for barley is $2.00,
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101 how should the farmer plant the 75 acres to maximize profit?
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102 #+END_QUOTE
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103
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104 The Farmer's Problem is to maximize profit subject to constraints on
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105 available farmland, funds for expenses, and storage space.
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106
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107 | | Wheat | Barley | Maximum total |
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108 |----------+----------------------+---------------------+--------------|
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109 | / | < | > | <> |
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110 | Farmland | \(w\) acres | \(b\) acres | 75 acres |
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111 | Expense | $120 per acre | $210 per acre | $15000 |
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112 | Storage | 110 bushels per acre | 30 bushels per acre | 4000 bushels |
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113 |----------+----------------------+---------------------+--------------|
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114 | Profit | $1.30 per bushel | $2.00 per bushel | |
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115
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116 *** COMMENT
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117 can be represented as a linear optimization
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118 problem. In this form, it is a problem with two variables\mdash{}the number of
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119 acres of wheat, \(w\), and the number of acres of barley, \(b\). The
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120 aim is to maximize profit, which
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121
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122 subject to three constraints: the farmer can't spend more money
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123 than he has, the farmer can't use more acres than he owns, and the harvest has
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124 to fit in his storage space.
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125
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126 We can express these constraints succinctly using matrix
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127 notation. Denoting the number of acres of barley and wheat by \(b\) and \(w\),
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128 we want to maximize the expression \(143 w + 60 b\) subject to
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129
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130 \(
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131 \begin{cases}
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132 120 w + 210 b & \leq & 1500\\
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133 110 w + 30 b & \leq & 4000\\
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134 1 w + 1 w & \leq & 75
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135 \end{cases}
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136 \)
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137
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138 #\(\begin{bmatrix}120&210\\110&30\\1 &
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139 #1\end{bmatrix}\;\begin{bmatrix}w\\b\end{bmatrix}
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140 #\leq \begin{bmatrix}\$15000\\4000\text{ bushels}\\75\text{ acres}\end{bmatrix}\)
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141
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142 ** Solution using LP Solve
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143 #(LP solve is available at http://www.example.com.)
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144 In a new file, =farmer.lp=, we list the variables and constraints
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145 of our problem using LP Solve syntax.
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146
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147 #+begin_src lpsolve :tangle ../lp/farmer.lp
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148 /* Maximize Total Profit */
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149 max: +143 wheat +60 barley;
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150
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151
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152 /* -------- Constraints --------*/
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153
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154 /* the farmer can't spend more money than he has */
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155 +120 wheat +210 barley <= 15000;
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156
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157 /* the harvest has to fit in his storage space */
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158 +110 wheat +30 barley <= 4000;
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159
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160 /* he can't use more acres than he owns */
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161 +wheat +barley <= 75;
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162 #+end_src
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163
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164
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165 #This is a set of linear equations ideal for solving using a program like
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166 #=lp_solve=. In Linear Algebra terms we are maximizing the linear function
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167
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168 #\(\text{profit} = 143\text{ wheat} + 60\text{ barley}\), subject to the constraints
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169
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170 #Ax <= b,
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171
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172 #where A is [120 210 110 30 1 1], x is [wheat barley] and b is [15000
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173 #4000 75].
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174
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175 Running the =lp_solve= program on =farmer.lp= yields the following output.
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176
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177 #+begin_src sh :exports both :results scalar
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178 ~/roBin/lpsolve/lp_solve ~/aurellem/src/pokemon/farmer.lp
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179 #+end_src
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180
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181 #+results:
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182 :
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183 : Value of objective function: 6315.62500000
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184 :
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185 : Actual values of the variables:
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186 : wheat 21.875
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187 : barley 53.125
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188
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189 This shows that the farmer can maximize his profit by planting 21.875
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190 of the available acres with wheat and the remaining 53.125 acres with
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191 barley; by doing so, he will make $6315.62(5) in profit.
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192
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193
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194 #The farmer can make a profit of $6315.62 by planting 21.875 acres of
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195 #his land with wheat and the other 53.125 acres of his land with barley.
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196
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197 * Incorporating =lp_solve= into Clojure
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198
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199 There is a Java API available which enables Java programs to use Lp
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200 Solve. Although Clojure can use this Java API directly, the
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201 interaction between Java, C, and Clojure is clumsy:
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202
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203
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204 The Java API for LP Solve makes it possible to use Lp Solve algorithms
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205 within Java. Although Clojure can use this Java API directly,
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206
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207
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208 ** The Farmer's Problem in Clojure
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209 We are going to solve the same problem involving wheat and barley,
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210 that we did above, but this time using clojure and the lpsolve API.
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211
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212 #Because we ultimately intend to use Lp Solve to find optimal Pokemon type combinations.
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213 # we want to solve linear optimization problems within Clojure, the language
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214
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215 ** Setup
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216 =lp_solve= is a crufty =C= program which already comes with a JNI
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217 interface written by Juergen Ebert. It's API is not
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218 particularly friendly from a functional/immutable perspective, but
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219 with a little work, we can build something that works great with
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220 clojure.
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221
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222 #+srcname: intro
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223 #+begin_src clojure :results silent
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224 (ns pokemon.lpsolve
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225 (:use rlm.ns-rlm))
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226 (rlm.ns-rlm/ns-clone rlm.light-base)
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227 (use 'clojure.set)
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228 (import 'lpsolve.LpSolve)
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229 (use '[clojure [pprint :only [pprint]]])
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230 #+end_src
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231
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232 The LpSolve Java interface is available from the same site as
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233 =lp_solve= itself, http://lpsolve.sourceforge.net/
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234 Using it is the same as many other =C=
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235 programs. There are excellent instructions to get set
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236 up. The short version is that you must call Java with
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237 =-Djava.library.path=/path/to/lpsolve/libraries= and also add the
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238 libraries to your export =LD_LIBRARY_PATH= if you are using Linux. For
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239 example, in my =.bashrc= file, I have the line
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240 =LD_LIBRARY_PATH=$HOME/roBin/lpsolve:$LD_LIBRARY_PATH=.
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241 If everything is set-up correctly,
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242
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243 #+srcname: body
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244 #+begin_src clojure :results verbatim :exports both
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245 (import 'lpsolve.LpSolve)
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246 #+end_src
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247
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248 #+results: body
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249 : lpsolve.LpSolve
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250
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251 should run with no problems.
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252
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253 ** Making a DSL to talk with LpSolve
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254 *** Problems
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255 Since we are using a =C= wrapper, we have to deal with manual memory
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256 management for the =C= structures which are wrapped by the =LpSolve=
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257 object. Memory leaks in =LpSolve= instances can crash the JVM, so it's
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258 very important to get it right. Also, the Java wrapper follows the
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259 =C= tradition closely and defines many =static final int= constants
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260 for the different states of the =LpSolve= instance instead of using Java
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261 enums. The calling convention for adding rows and columns to
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262 the constraint matrix is rather complicated and must be done column by
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263 column or row by row, which can be error prone. Finally, I'd like to
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264 gather all the important output information from the =LpSolve= instance
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265 into a final, immutable structure.
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266
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267 In summary, the issues I'd like to address are:
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268
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269 - reliable memory management
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270 - functional interface to =LpSolve=
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271 - intelligible, immutable output
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272
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273 To deal with these issues I'll create four functions for interfacing
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274 with =LpSolve=
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275
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276 #+srcname: declares
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277 #+begin_src clojure :results silent
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278 (in-ns 'pokemon.lpsolve)
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279
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280 ;; deal with automatic memory management for LpSolve instance.
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281 (declare linear-program)
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282
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283 ;; functional interface to LpSolve
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284 (declare lp-solve)
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285
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286 ;; immutable output from lp-solve
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287 (declare solve get-results)
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288 #+end_src
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289
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290 *** Memory Management
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291
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292 Every instance of =LpSolve= must be manually garbage collected via a
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293 call to =deleteLP=. I use a non-hygienic macro similar to =with-open=
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294 to ensure that =deleteLP= is always called.
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295
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296 #+srcname: memory-management
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297 #+begin_src clojure :results silent
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298 (in-ns 'pokemon.lpsolve)
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299 (defmacro linear-program
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300 "solve a linear programming problem using LpSolve syntax.
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301 within the macro, the variable =lps= is bound to the LpSolve instance."
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302 [& statements]
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303 (list 'let '[lps (LpSolve/makeLp 0 0)]
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304 (concat '(try)
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305 statements
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306 ;; always free the =C= data structures.
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307 '((finally (.deleteLp lps))))))
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308 #+end_src
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309
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310
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311 The macro captures the variable =lps= within its body, providing for a
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312 convenient way to access the object using any of the methods of the
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313 =LpSolve= API without having to worry about when to call
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314 =deleteLP=.
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315
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316 *** Sensible Results
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317 The =linear-program= macro deletes the actual =lps= object once it is
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318 done working, so it's important to collect the important results and
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319 add return them in an immutable structure at the end.
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320
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321 #+srcname: get-results
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322 #+begin_src clojure :results silent
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323 (in-ns 'pokemon.lpsolve)
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324
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325 (defrecord LpSolution
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326 [objective-value
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327 optimal-values
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328 variable-names
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329 solution
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330 status
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331 model])
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332
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333 (defn model
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334 "Returns a textual representation of the problem suitable for
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335 direct input to the =lp_solve= program (lps format)"
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336 [#^LpSolve lps]
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337 (let [target (java.io.File/createTempFile "lps" ".lp")]
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338 (.writeLp lps (.getPath target))
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339 (slurp target)))
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340
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341 (defn results
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342 "given an LpSolve object, solves the object and returns a map of the
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343 essential values which compose the solution."
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344 [#^LpSolve lps]
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345 (locking lps
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346 (let [status (solve lps)
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347 number-of-variables (.getNcolumns lps)
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348 optimal-values (double-array number-of-variables)
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349 optimal-values (do (.getVariables lps optimal-values)
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350 (seq optimal-values))
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351 variable-names
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352 (doall ;; the doall is necessary since the lps object might
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353 ;; soon be deleted
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354 (map
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355 #(.getColName lps (inc %))
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356 (range number-of-variables)))
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357 model (model lps)]
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358 (LpSolution.
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359 (.getObjective lps)
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360 optimal-values
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361 variable-names
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362 (zipmap variable-names optimal-values)
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363 status
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364 model))))
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365
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366 #+end_src
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367
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368 Here I've created an object called =LpSolution= which stores the
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369 important results from a session with =lp_solve=. Of note is the
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370 =model= function which returns the problem in a form that can be
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371 solved by other linear programming packages.
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372
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373 *** Solution Status of an LpSolve Object
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374
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375 #+srcname: solve
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376 #+begin_src clojure :results silent
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377 (in-ns 'pokemon.lpsolve)
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378
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379 (defn static-integer?
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380 "does the field represent a static integer constant?"
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381 [#^java.lang.reflect.Field field]
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382 (and (java.lang.reflect.Modifier/isStatic (.getModifiers field))
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383 (integer? (.get field nil))))
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384
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385 (defn integer-constants [class]
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386 (filter static-integer? (.getFields class)))
|
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387
|
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388 (defn-memo constant-map
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389 "Takes a class and creates a map of the static constant integer
|
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390 fields with their names. This helps with C wrappers where they have
|
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391 just defined a bunch of integer constants instead of enums"
|
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392 [class]
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393 (let [integer-fields (integer-constants class)]
|
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394 (into (sorted-map)
|
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395 (zipmap (map #(.get % nil) integer-fields)
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396 (map #(.getName %) integer-fields)))))
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397
|
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398 (defn solve
|
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399 "Solve an instance of LpSolve and return a string representing the
|
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400 status of the computation. Will only solve a particular LpSolve
|
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401 instance once."
|
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402 [#^LpSolve lps]
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403 ((constant-map LpSolve)
|
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404 (.solve lps)))
|
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405
|
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|
406 #+end_src
|
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407
|
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|
408 The =.solve= method of an =LpSolve= object only returns an integer code
|
|
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|
409 to specify the status of the computation. The =solve= method here
|
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410 uses reflection to look up the actual name of the status code and
|
|
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411 returns a more helpful status message that is also resistant to
|
|
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|
412 changes in the meanings of the code numbers.
|
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413
|
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|
414 *** The Farmer Example in Clojure, Pass 1
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415
|
|
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|
416 Now we can implement a nicer version of the examples from the
|
|
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|
417 [[http://lpsolve.sourceforge.net/][=lp\_solve= website]]. The following is a more or less
|
|
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|
418 line-by-line translation of the Java code from that example.
|
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|
419
|
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|
420 #+srcname: farmer-example
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421 #+begin_src clojure :results silent
|
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422 (in-ns 'pokemon.lpsolve)
|
|
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|
423 (defn farmer-example []
|
|
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|
424 (linear-program
|
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|
425 (results
|
|
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|
426 (doto lps
|
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|
427 ;; name the columns
|
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|
428 (.setColName 1 "wheat")
|
|
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|
429 (.setColName 2 "barley")
|
|
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|
430 (.setAddRowmode true)
|
|
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|
431 ;; row 1 : 120x + 210y <= 15000
|
|
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|
432 (.addConstraintex 2
|
|
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|
433 (double-array [120 210])
|
|
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434 (int-array [1 2])
|
|
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435 LpSolve/LE
|
|
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|
436 15e3)
|
|
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|
437 ;; row 2 : 110x + 30y <= 4000
|
|
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|
438 (.addConstraintex 2
|
|
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|
439 (double-array [110 30])
|
|
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|
440 (int-array [1 2])
|
|
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|
441 LpSolve/LE
|
|
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|
442 4e3)
|
|
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|
443 ;; ;; row 3 : x + y <= 75
|
|
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|
444 (.addConstraintex 2
|
|
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|
445 (double-array [1 1])
|
|
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|
446 (int-array [1 2])
|
|
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|
447 LpSolve/LE
|
|
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|
448 75)
|
|
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|
449 (.setAddRowmode false)
|
|
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|
450
|
|
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|
451 ;; add constraints
|
|
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|
452 (.setObjFnex 2
|
|
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|
453 (double-array [143 60])
|
|
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|
454 (int-array [1 2]))
|
|
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|
455
|
|
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|
456 ;; set this as a maximization problem
|
|
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|
457 (.setMaxim)))))
|
|
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|
458
|
|
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|
459 #+end_src
|
|
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|
460
|
|
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|
461 #+begin_src clojure :results output :exports both
|
|
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|
462 (clojure.pprint/pprint
|
|
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|
463 (:solution (pokemon.lpsolve/farmer-example)))
|
|
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|
464 #+end_src
|
|
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|
465
|
|
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|
466 #+results:
|
|
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|
467 : {"barley" 53.12499999999999, "wheat" 21.875}
|
|
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|
468
|
|
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|
469 And it works as expected!
|
|
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|
470
|
|
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|
471 *** The Farmer Example in Clojure, Pass 2
|
|
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|
472 We don't have to worry about memory management anymore, and the farmer
|
|
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|
473 example is about half as long as the example from the =LpSolve=
|
|
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|
474 website, but we can still do better. Solving linear problems is all
|
|
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|
475 about the constraint matrix $A$ , the objective function $c$, and the
|
|
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|
476 right-hand-side $b$, plus whatever other options one cares to set for
|
|
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|
477 the particular instance of =lp_solve=. Why not make a version of
|
|
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|
478 =linear-program= that takes care of initialization?
|
|
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|
479
|
|
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|
480
|
|
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|
481
|
|
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|
482 #+srcname: lp-solve
|
|
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|
483 #+begin_src clojure :results silent
|
|
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|
484 (in-ns 'pokemon.lpsolve)
|
|
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|
485 (defn initialize-lpsolve-row-oriented
|
|
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|
486 "fill in an lpsolve instance using a constraint matrix =A=, the
|
|
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|
487 objective function =c=, and the right-hand-side =b="
|
|
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|
488 [#^LpSolve lps A b c]
|
|
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|
489 ;; set the name of the last column to _something_
|
|
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|
490 ;; this appears to be necessary to ensure proper initialization.
|
|
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|
491 (.setColName lps (count c) (str "C" (count c)))
|
|
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|
492
|
|
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|
493 ;; This is the recommended way to "fill-in" an lps instance from the
|
|
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|
494 ;; documentation. First, set row mode, then set the objective
|
|
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|
495 ;; function, then set each row of the problem, and then turn off row
|
|
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|
496 ;; mode.
|
|
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|
497 (.setAddRowmode lps true)
|
|
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|
498 (.setObjFnex lps (count c)
|
|
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|
499 (double-array c)
|
|
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|
500 (int-array (range 1 (inc (count c)))))
|
|
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|
501 (dorun
|
|
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|
502 (for [n (range (count A))]
|
|
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|
503 (let [row (nth A n)
|
|
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|
504 row-length (int (count row))]
|
|
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|
505 (.addConstraintex lps
|
|
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|
506 row-length
|
|
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|
507 (double-array row)
|
|
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|
508 (int-array (range 1 (inc row-length)))
|
|
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|
509 LpSolve/LE
|
|
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|
510 (double (nth b n))))))
|
|
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|
511 (.setAddRowmode lps false)
|
|
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|
512 lps)
|
|
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|
513
|
|
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|
514
|
|
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|
515 (defmacro lp-solve
|
|
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|
516 "by default:,
|
|
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|
517 minimize (* c x), subject to (<= (* A x) b),
|
|
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|
518 using continuous variables. You may set any number of
|
|
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|
519 other options as in the LpSolve API."
|
|
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|
520 [A b c & lp-solve-forms]
|
|
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|
521 ;; assume that A is a vector of vectors
|
|
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|
522 (concat
|
|
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|
523 (list 'linear-program
|
|
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|
524 (list 'initialize-lpsolve-row-oriented 'lps A b c))
|
|
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|
525 `~lp-solve-forms))
|
|
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|
526 #+end_src
|
|
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|
527
|
|
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|
528 Now, we can use a much more functional approach to solving the
|
|
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|
529 farmer's problem:
|
|
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|
530
|
|
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|
531 #+srcname: better-farmer
|
|
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|
532 #+begin_src clojure :results silent
|
|
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|
533 (in-ns 'pokemon.lpsolve)
|
|
rlm@0
|
534 (defn better-farmer-example []
|
|
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|
535 (lp-solve [[120 210]
|
|
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|
536 [110 30]
|
|
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|
537 [1 1]]
|
|
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|
538 [15000
|
|
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|
539 4000
|
|
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|
540 75]
|
|
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|
541 [143 60]
|
|
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|
542 (.setColName lps 1 "wheat")
|
|
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|
543 (.setColName lps 2 "barley")
|
|
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|
544 (.setMaxim lps)
|
|
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|
545 (results lps)))
|
|
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|
546 #+end_src
|
|
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|
547
|
|
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|
548 #+begin_src clojure :exports both :results verbatim
|
|
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|
549 (vec (:solution (pokemon.lpsolve/better-farmer-example)))
|
|
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|
550 #+end_src
|
|
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|
551
|
|
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|
552 #+results:
|
|
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|
553 : [["barley" 53.12499999999999] ["wheat" 21.875]]
|
|
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|
554
|
|
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|
555 Notice that both the inputs to =better-farmer-example= and the results
|
|
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|
556 are immutable.
|
|
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|
557
|
|
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|
558 * Using LpSolve to find Immortal Types
|
|
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|
559 ** Converting the Pokemon problem into a linear form
|
|
rlm@0
|
560 How can the original question about pok\eacute{}mon types be converted
|
|
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|
561 into a linear problem?
|
|
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|
562
|
|
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|
563 Pokemon types can be considered to be vectors of numbers representing
|
|
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|
564 their susceptances to various attacking types, so Water might look
|
|
rlm@0
|
565 something like this.
|
|
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|
566
|
|
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|
567 #+begin_src clojure :results scalar :exports both
|
|
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|
568 (:water (pokemon.types/defense-strengths))
|
|
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|
569 #+end_src
|
|
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|
570
|
|
rlm@0
|
571 #+results:
|
|
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|
572 : [1 0.5 0.5 2 2 0.5 1 1 1 1 1 1 1 1 1 1 0.5]
|
|
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|
573
|
|
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|
574 Where the numbers represent the susceptibility of Water to the
|
|
rlm@0
|
575 attacking types in the following order:
|
|
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|
576
|
|
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|
577 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
578 (clojure.pprint/pprint
|
|
rlm@0
|
579 (pokemon.types/type-names))
|
|
rlm@0
|
580 #+end_src
|
|
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|
581
|
|
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|
582 #+results:
|
|
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|
583 #+begin_example
|
|
rlm@0
|
584 [:normal
|
|
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|
585 :fire
|
|
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|
586 :water
|
|
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|
587 :electric
|
|
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|
588 :grass
|
|
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|
589 :ice
|
|
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|
590 :fighting
|
|
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|
591 :poison
|
|
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|
592 :ground
|
|
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|
593 :flying
|
|
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|
594 :psychic
|
|
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|
595 :bug
|
|
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|
596 :rock
|
|
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|
597 :ghost
|
|
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|
598 :dragon
|
|
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|
599 :dark
|
|
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|
600 :steel]
|
|
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|
601 #+end_example
|
|
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|
602
|
|
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|
603
|
|
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|
604 So, for example, Water is /resistant/ (x0.5) against Fire, which is
|
|
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|
605 the second element in the list.
|
|
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|
606
|
|
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|
607 To combine types, these sorts of vectors are multiplied together
|
|
rlm@0
|
608 pair-wise to yield the resulting combination.
|
|
rlm@0
|
609
|
|
rlm@0
|
610 Unfortunately, we need some way to add two type vectors together
|
|
rlm@0
|
611 instead of multiplying them if we want to solve the problem with
|
|
rlm@0
|
612 =lp_solve=. Taking the log of the vector does just the trick.
|
|
rlm@0
|
613
|
|
rlm@0
|
614 If we make a matrix with each column being the log (base 2) of the
|
|
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|
615 susceptance of each type, then finding an immortal type corresponds to
|
|
rlm@0
|
616 setting each constraint (the $b$ vector) to -1 (since log_2(1/2) = -1)
|
|
rlm@0
|
617 and setting the constraint vector $c$ to all ones, which means that we
|
|
rlm@0
|
618 want to find the immortal type which uses the least amount of types.
|
|
rlm@0
|
619
|
|
rlm@0
|
620 #+srcname: pokemon-lp
|
|
rlm@0
|
621 #+begin_src clojure :results silent
|
|
rlm@0
|
622 (in-ns 'pokemon.lpsolve)
|
|
rlm@0
|
623
|
|
rlm@0
|
624 (require 'pokemon.types)
|
|
rlm@0
|
625 (require 'incanter.core)
|
|
rlm@0
|
626
|
|
rlm@0
|
627 (defn log-clamp-matrix [matrix]
|
|
rlm@0
|
628 ;; we have to clamp the Infinities to a more reasonable negative
|
|
rlm@0
|
629 ;; value because lp_solve does not play well with infinities in its
|
|
rlm@0
|
630 ;; constraint matrix.
|
|
rlm@0
|
631 (map (fn [row] (map #(if (= Double/NEGATIVE_INFINITY %) -1e3 %) row))
|
|
rlm@0
|
632 (incanter.core/log2
|
|
rlm@0
|
633 (incanter.core/trans
|
|
rlm@0
|
634 matrix))))
|
|
rlm@0
|
635
|
|
rlm@0
|
636 ;; constraint matrices
|
|
rlm@0
|
637 (defn log-defense-matrix []
|
|
rlm@0
|
638 (log-clamp-matrix
|
|
rlm@0
|
639 (doall (map (pokemon.types/defense-strengths)
|
|
rlm@0
|
640 (pokemon.types/type-names)))))
|
|
rlm@0
|
641
|
|
rlm@0
|
642 (defn log-attack-matrix []
|
|
rlm@0
|
643 (incanter.core/trans (log-defense-matrix)))
|
|
rlm@0
|
644
|
|
rlm@0
|
645 ;; target vectors
|
|
rlm@0
|
646 (defn all-resistant []
|
|
rlm@0
|
647 (doall (map (constantly -1) (pokemon.types/type-names))))
|
|
rlm@0
|
648
|
|
rlm@0
|
649 (defn all-weak []
|
|
rlm@0
|
650 (doall (map (constantly 1) (pokemon.types/type-names))))
|
|
rlm@0
|
651
|
|
rlm@0
|
652 (defn all-neutral []
|
|
rlm@0
|
653 (doall (map (constantly 0) (pokemon.types/type-names))))
|
|
rlm@0
|
654
|
|
rlm@0
|
655
|
|
rlm@0
|
656 ;; objective functions
|
|
rlm@0
|
657 (defn number-of-types []
|
|
rlm@0
|
658 (doall (map (constantly 1) (pokemon.types/type-names))))
|
|
rlm@0
|
659
|
|
rlm@0
|
660 (defn set-constraints
|
|
rlm@0
|
661 "sets all the constraints for an lpsolve instance to the given
|
|
rlm@0
|
662 constraint. =constraint= here is one of the LpSolve constants such
|
|
rlm@0
|
663 as LpSolve/EQ."
|
|
rlm@0
|
664 [#^LpSolve lps constraint]
|
|
rlm@0
|
665 (dorun (map (fn [index] (.setConstrType lps index constraint))
|
|
rlm@0
|
666 ;; ONE based indexing!!!
|
|
rlm@0
|
667 (range 1 (inc (.getNrows lps))))))
|
|
rlm@0
|
668
|
|
rlm@0
|
669
|
|
rlm@0
|
670 (defn set-discrete
|
|
rlm@0
|
671 "sets every variable in an lps problem to be a discrete rather than
|
|
rlm@0
|
672 continuous variable"
|
|
rlm@0
|
673 [#^LpSolve lps]
|
|
rlm@0
|
674 (dorun (map (fn [index] (.setInt lps index true))
|
|
rlm@0
|
675 ;; ONE based indexing!!!
|
|
rlm@0
|
676 (range 1 (inc (.getNcolumns lps))))))
|
|
rlm@0
|
677
|
|
rlm@0
|
678 (defn set-variable-names
|
|
rlm@0
|
679 "sets the variable names of the problem given a vector of names"
|
|
rlm@0
|
680 [#^LpSolve lps names]
|
|
rlm@0
|
681 (dorun
|
|
rlm@0
|
682 (map (fn [[index name]]
|
|
rlm@0
|
683 (.setColName lps (inc index) (str name)))
|
|
rlm@0
|
684 ;; ONE based indexing!!!
|
|
rlm@0
|
685 (indexed names))))
|
|
rlm@0
|
686
|
|
rlm@0
|
687 (defn poke-solve
|
|
rlm@0
|
688 ([poke-matrix target objective-function constraint min-num-types]
|
|
rlm@0
|
689 ;; must have at least one type
|
|
rlm@0
|
690 (let [poke-matrix
|
|
rlm@0
|
691 (concat poke-matrix
|
|
rlm@0
|
692 [(map (constantly 1)
|
|
rlm@0
|
693 (range (count (first poke-matrix))))])
|
|
rlm@0
|
694 target (concat target [min-num-types])]
|
|
rlm@0
|
695 (lp-solve poke-matrix target objective-function
|
|
rlm@0
|
696 (set-constraints lps constraint)
|
|
rlm@0
|
697 ;; must have more than min-num-types
|
|
rlm@0
|
698 (.setConstrType lps (count target) LpSolve/GE)
|
|
rlm@0
|
699 (set-discrete lps)
|
|
rlm@0
|
700 (set-variable-names lps (pokemon.types/type-names))
|
|
rlm@0
|
701 (results lps))))
|
|
rlm@0
|
702 ([poke-matrix target objective-function constraint]
|
|
rlm@0
|
703 ;; at least one type
|
|
rlm@0
|
704 (poke-solve poke-matrix target objective-function constraint 1)))
|
|
rlm@0
|
705
|
|
rlm@0
|
706 (defn solution
|
|
rlm@0
|
707 "If the results of an lpsolve operation are feasible, returns the
|
|
rlm@0
|
708 results. Otherwise, returns the error."
|
|
rlm@0
|
709 [results]
|
|
rlm@0
|
710 (if (not (= (:status results) "OPTIMAL"))
|
|
rlm@0
|
711 (:status results)
|
|
rlm@0
|
712 (:solution results)))
|
|
rlm@0
|
713
|
|
rlm@0
|
714 #+end_src
|
|
rlm@0
|
715
|
|
rlm@0
|
716 With this, we are finally able to get some results.
|
|
rlm@0
|
717
|
|
rlm@0
|
718 ** Results
|
|
rlm@0
|
719 #+srcname: results
|
|
rlm@0
|
720 #+begin_src clojure :results silent
|
|
rlm@0
|
721 (in-ns 'pokemon.lpsolve)
|
|
rlm@0
|
722
|
|
rlm@0
|
723 (defn best-defense-type
|
|
rlm@0
|
724 "finds a type combination which is resistant to all attacks."
|
|
rlm@0
|
725 []
|
|
rlm@0
|
726 (poke-solve
|
|
rlm@0
|
727 (log-defense-matrix) (all-resistant) (number-of-types) LpSolve/LE))
|
|
rlm@0
|
728
|
|
rlm@0
|
729 (defn worst-attack-type
|
|
rlm@0
|
730 "finds the attack type which is not-very-effective against all pure
|
|
rlm@0
|
731 defending types (each single defending type is resistant to this
|
|
rlm@0
|
732 attack combination"
|
|
rlm@0
|
733 []
|
|
rlm@0
|
734 (poke-solve
|
|
rlm@0
|
735 (log-attack-matrix) (all-resistant) (number-of-types) LpSolve/LE))
|
|
rlm@0
|
736
|
|
rlm@0
|
737 (defn worst-defense-type
|
|
rlm@0
|
738 "finds a defending type that is weak to all single attacking types."
|
|
rlm@0
|
739 []
|
|
rlm@0
|
740 (poke-solve
|
|
rlm@0
|
741 (log-defense-matrix) (all-weak) (number-of-types) LpSolve/GE))
|
|
rlm@0
|
742
|
|
rlm@0
|
743 (defn best-attack-type
|
|
rlm@0
|
744 "finds an attack type which is super effective against all single
|
|
rlm@0
|
745 defending types"
|
|
rlm@0
|
746 []
|
|
rlm@0
|
747 (poke-solve
|
|
rlm@0
|
748 (log-attack-matrix) (all-weak) (number-of-types) LpSolve/GE))
|
|
rlm@0
|
749
|
|
rlm@0
|
750 (defn solid-defense-type
|
|
rlm@0
|
751 "finds a defense type which is either neutral or resistant to all
|
|
rlm@0
|
752 single attacking types"
|
|
rlm@0
|
753 []
|
|
rlm@0
|
754 (poke-solve
|
|
rlm@0
|
755 (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/LE))
|
|
rlm@0
|
756
|
|
rlm@0
|
757 (defn solid-attack-type
|
|
rlm@0
|
758 "finds an attack type which is either neutral or super-effective to
|
|
rlm@0
|
759 all single attacking types."
|
|
rlm@0
|
760 []
|
|
rlm@0
|
761 (poke-solve
|
|
rlm@0
|
762 (log-attack-matrix) (all-neutral) (number-of-types) LpSolve/GE))
|
|
rlm@0
|
763
|
|
rlm@0
|
764 (defn weak-defense-type
|
|
rlm@0
|
765 "finds a defense type which is either neutral or weak to all single
|
|
rlm@0
|
766 attacking types"
|
|
rlm@0
|
767 []
|
|
rlm@0
|
768 (poke-solve
|
|
rlm@0
|
769 (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/GE))
|
|
rlm@0
|
770
|
|
rlm@0
|
771 (defn neutral-defense-type
|
|
rlm@0
|
772 "finds a defense type which is perfectly neutral to all attacking
|
|
rlm@0
|
773 types."
|
|
rlm@0
|
774 []
|
|
rlm@0
|
775 (poke-solve
|
|
rlm@0
|
776 (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/EQ))
|
|
rlm@0
|
777
|
|
rlm@0
|
778 #+end_src
|
|
rlm@0
|
779
|
|
rlm@0
|
780 *** Strongest Attack/Defense Combinations
|
|
rlm@0
|
781
|
|
rlm@0
|
782 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
783 (clojure.pprint/pprint
|
|
rlm@0
|
784 (pokemon.lpsolve/solution (pokemon.lpsolve/best-defense-type)))
|
|
rlm@0
|
785 #+end_src
|
|
rlm@0
|
786
|
|
rlm@0
|
787 #+results:
|
|
rlm@0
|
788 #+begin_example
|
|
rlm@0
|
789 {":normal" 0.0,
|
|
rlm@0
|
790 ":ground" 1.0,
|
|
rlm@0
|
791 ":poison" 2.0,
|
|
rlm@0
|
792 ":flying" 1.0,
|
|
rlm@0
|
793 ":fighting" 0.0,
|
|
rlm@0
|
794 ":dragon" 0.0,
|
|
rlm@0
|
795 ":fire" 0.0,
|
|
rlm@0
|
796 ":dark" 1.0,
|
|
rlm@0
|
797 ":ice" 0.0,
|
|
rlm@0
|
798 ":steel" 1.0,
|
|
rlm@0
|
799 ":ghost" 0.0,
|
|
rlm@0
|
800 ":electric" 0.0,
|
|
rlm@0
|
801 ":bug" 0.0,
|
|
rlm@0
|
802 ":psychic" 0.0,
|
|
rlm@0
|
803 ":grass" 0.0,
|
|
rlm@0
|
804 ":water" 2.0,
|
|
rlm@0
|
805 ":rock" 0.0}
|
|
rlm@0
|
806 #+end_example
|
|
rlm@0
|
807
|
|
rlm@0
|
808 # #+results-old:
|
|
rlm@0
|
809 # : [[":normal" 0.0] [":ground" 1.0] [":poison" 0.0] [":flying" 1.0] [":fighting" 0.0] [":dragon" 1.0] [":fire" 0.0] [":dark" 0.0] [":ice" 0.0] [":steel" 2.0] [":ghost" 1.0] [":electric" 0.0] [":bug" 0.0] [":psychic" 0.0] [":grass" 0.0] [":water" 2.0] [":rock" 0.0]]
|
|
rlm@0
|
810
|
|
rlm@0
|
811
|
|
rlm@0
|
812 This is the immortal type combination we've been looking for. By
|
|
rlm@0
|
813 combining Steel, Water, Poison, and three types which each have complete
|
|
rlm@0
|
814 immunities to various other types, we've created a type that is resistant to
|
|
rlm@0
|
815 all attacking types.
|
|
rlm@0
|
816
|
|
rlm@0
|
817 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
818 (clojure.pprint/pprint
|
|
rlm@0
|
819 (pokemon.types/susceptibility
|
|
rlm@0
|
820 [:poison :poison :water :water :steel :ground :flying :dark]))
|
|
rlm@0
|
821 #+end_src
|
|
rlm@0
|
822
|
|
rlm@0
|
823 #+results:
|
|
rlm@0
|
824 #+begin_example
|
|
rlm@0
|
825 {:water 1/2,
|
|
rlm@0
|
826 :psychic 0,
|
|
rlm@0
|
827 :dragon 1/2,
|
|
rlm@0
|
828 :fire 1/2,
|
|
rlm@0
|
829 :ice 1/2,
|
|
rlm@0
|
830 :grass 1/2,
|
|
rlm@0
|
831 :ghost 1/4,
|
|
rlm@0
|
832 :poison 0,
|
|
rlm@0
|
833 :flying 1/2,
|
|
rlm@0
|
834 :normal 1/2,
|
|
rlm@0
|
835 :rock 1/2,
|
|
rlm@0
|
836 :electric 0,
|
|
rlm@0
|
837 :ground 0,
|
|
rlm@0
|
838 :fighting 1/2,
|
|
rlm@0
|
839 :dark 1/4,
|
|
rlm@0
|
840 :steel 1/8,
|
|
rlm@0
|
841 :bug 1/8}
|
|
rlm@0
|
842 #+end_example
|
|
rlm@0
|
843
|
|
rlm@0
|
844 # #+results-old:
|
|
rlm@0
|
845 # : {:water 1/4, :psychic 1/4, :dragon 1/2, :fire 1/2, :ice 1/2, :grass 1/2, :ghost 1/2, :poison 0, :flying 1/4, :normal 0, :rock 1/4, :electric 0, :ground 0, :fighting 0, :dark 1/2, :steel 1/16, :bug 1/16}
|
|
rlm@0
|
846
|
|
rlm@0
|
847
|
|
rlm@0
|
848 Cool!
|
|
rlm@0
|
849
|
|
rlm@0
|
850 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
851 (clojure.pprint/pprint
|
|
rlm@0
|
852 (pokemon.lpsolve/solution (pokemon.lpsolve/solid-defense-type)))
|
|
rlm@0
|
853 #+end_src
|
|
rlm@0
|
854
|
|
rlm@0
|
855 #+results:
|
|
rlm@0
|
856 #+begin_example
|
|
rlm@0
|
857 {":normal" 0.0,
|
|
rlm@0
|
858 ":ground" 0.0,
|
|
rlm@0
|
859 ":poison" 0.0,
|
|
rlm@0
|
860 ":flying" 0.0,
|
|
rlm@0
|
861 ":fighting" 0.0,
|
|
rlm@0
|
862 ":dragon" 0.0,
|
|
rlm@0
|
863 ":fire" 0.0,
|
|
rlm@0
|
864 ":dark" 1.0,
|
|
rlm@0
|
865 ":ice" 0.0,
|
|
rlm@0
|
866 ":steel" 0.0,
|
|
rlm@0
|
867 ":ghost" 1.0,
|
|
rlm@0
|
868 ":electric" 0.0,
|
|
rlm@0
|
869 ":bug" 0.0,
|
|
rlm@0
|
870 ":psychic" 0.0,
|
|
rlm@0
|
871 ":grass" 0.0,
|
|
rlm@0
|
872 ":water" 0.0,
|
|
rlm@0
|
873 ":rock" 0.0}
|
|
rlm@0
|
874 #+end_example
|
|
rlm@0
|
875
|
|
rlm@0
|
876 Dark and Ghost are the best dual-type combo, and are resistant or
|
|
rlm@0
|
877 neutral to all types.
|
|
rlm@0
|
878
|
|
rlm@0
|
879 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
880 (clojure.pprint/pprint
|
|
rlm@0
|
881 (pokemon.types/old-school
|
|
rlm@0
|
882 (pokemon.lpsolve/solution (pokemon.lpsolve/solid-defense-type))))
|
|
rlm@0
|
883 #+end_src
|
|
rlm@0
|
884
|
|
rlm@0
|
885 #+results:
|
|
rlm@0
|
886 #+begin_example
|
|
rlm@0
|
887 {":normal" 0.0,
|
|
rlm@0
|
888 ":ground" 0.0,
|
|
rlm@0
|
889 ":poison" 0.0,
|
|
rlm@0
|
890 ":flying" 0.0,
|
|
rlm@0
|
891 ":fighting" 0.0,
|
|
rlm@0
|
892 ":dragon" 0.0,
|
|
rlm@0
|
893 ":fire" 0.0,
|
|
rlm@0
|
894 ":ice" 0.0,
|
|
rlm@0
|
895 ":ghost" 1.0,
|
|
rlm@0
|
896 ":electric" 0.0,
|
|
rlm@0
|
897 ":bug" 0.0,
|
|
rlm@0
|
898 ":psychic" 1.0,
|
|
rlm@0
|
899 ":grass" 0.0,
|
|
rlm@0
|
900 ":water" 0.0,
|
|
rlm@0
|
901 ":rock" 0.0}
|
|
rlm@0
|
902 #+end_example
|
|
rlm@0
|
903
|
|
rlm@0
|
904 Ghost and Psychic are a powerful dual type combo in the original games,
|
|
rlm@0
|
905 due to a glitch which made Psychic immune to Ghost type attacks, even
|
|
rlm@0
|
906 though the game claims that Ghost is strong to Psychic.
|
|
rlm@0
|
907
|
|
rlm@0
|
908 #+begin_src clojure :results verbatim :exports both
|
|
rlm@0
|
909 (pokemon.lpsolve/solution (pokemon.lpsolve/best-attack-type))
|
|
rlm@0
|
910 #+end_src
|
|
rlm@0
|
911
|
|
rlm@0
|
912 #+results:
|
|
rlm@0
|
913 : INFEASIBLE
|
|
rlm@0
|
914
|
|
rlm@0
|
915 #+begin_src clojure :results verbatim :exports both
|
|
rlm@0
|
916 (pokemon.lpsolve/solution (pokemon.lpsolve/solid-attack-type))
|
|
rlm@0
|
917 #+end_src
|
|
rlm@0
|
918
|
|
rlm@0
|
919 #+results:
|
|
rlm@0
|
920 : INFEASIBLE
|
|
rlm@0
|
921
|
|
rlm@0
|
922
|
|
rlm@0
|
923 #+begin_src clojure :results verbatim :exports both
|
|
rlm@0
|
924 (pokemon.types/old-school
|
|
rlm@0
|
925 (pokemon.lpsolve/solution (pokemon.lpsolve/best-attack-type)))
|
|
rlm@0
|
926 #+end_src
|
|
rlm@0
|
927
|
|
rlm@0
|
928 #+results:
|
|
rlm@0
|
929 : INFEASIBLE
|
|
rlm@0
|
930
|
|
rlm@0
|
931
|
|
rlm@0
|
932 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
933 (clojure.pprint/pprint
|
|
rlm@0
|
934 (pokemon.types/old-school
|
|
rlm@0
|
935 (pokemon.lpsolve/solution (pokemon.lpsolve/solid-attack-type))))
|
|
rlm@0
|
936 #+end_src
|
|
rlm@0
|
937
|
|
rlm@0
|
938 #+results:
|
|
rlm@0
|
939 #+begin_example
|
|
rlm@0
|
940 {":normal" 0.0,
|
|
rlm@0
|
941 ":ground" 0.0,
|
|
rlm@0
|
942 ":poison" 0.0,
|
|
rlm@0
|
943 ":flying" 0.0,
|
|
rlm@0
|
944 ":fighting" 0.0,
|
|
rlm@0
|
945 ":dragon" 1.0,
|
|
rlm@0
|
946 ":fire" 0.0,
|
|
rlm@0
|
947 ":ice" 0.0,
|
|
rlm@0
|
948 ":ghost" 0.0,
|
|
rlm@0
|
949 ":electric" 0.0,
|
|
rlm@0
|
950 ":bug" 0.0,
|
|
rlm@0
|
951 ":psychic" 0.0,
|
|
rlm@0
|
952 ":grass" 0.0,
|
|
rlm@0
|
953 ":water" 0.0,
|
|
rlm@0
|
954 ":rock" 0.0}
|
|
rlm@0
|
955 #+end_example
|
|
rlm@0
|
956
|
|
rlm@0
|
957 The best attacking type combination is strangely Dragon from the
|
|
rlm@0
|
958 original games. It is neutral against all the original types except
|
|
rlm@0
|
959 for Dragon, which it is strong against. There is no way to make an
|
|
rlm@0
|
960 attacking type that is strong against every type, or even one that is
|
|
rlm@0
|
961 strong or neutral against every type, in the new games.
|
|
rlm@0
|
962
|
|
rlm@0
|
963
|
|
rlm@0
|
964 *** Weakest Attack/Defense Combinations
|
|
rlm@0
|
965
|
|
rlm@0
|
966 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
967 (clojure.pprint/pprint
|
|
rlm@0
|
968 (pokemon.types/old-school
|
|
rlm@0
|
969 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-attack-type))))
|
|
rlm@0
|
970 #+end_src
|
|
rlm@0
|
971
|
|
rlm@0
|
972 #+results:
|
|
rlm@0
|
973 #+begin_example
|
|
rlm@0
|
974 {":normal" 5.0,
|
|
rlm@0
|
975 ":ground" 0.0,
|
|
rlm@0
|
976 ":poison" 0.0,
|
|
rlm@0
|
977 ":flying" 0.0,
|
|
rlm@0
|
978 ":fighting" 0.0,
|
|
rlm@0
|
979 ":dragon" 0.0,
|
|
rlm@0
|
980 ":fire" 1.0,
|
|
rlm@0
|
981 ":ice" 2.0,
|
|
rlm@0
|
982 ":ghost" 1.0,
|
|
rlm@0
|
983 ":electric" 1.0,
|
|
rlm@0
|
984 ":bug" 1.0,
|
|
rlm@0
|
985 ":psychic" 0.0,
|
|
rlm@0
|
986 ":grass" 3.0,
|
|
rlm@0
|
987 ":water" 2.0,
|
|
rlm@0
|
988 ":rock" 0.0}
|
|
rlm@0
|
989 #+end_example
|
|
rlm@0
|
990
|
|
rlm@0
|
991 # #+results-old:
|
|
rlm@0
|
992 # : [[":normal" 5.0] [":ground" 1.0] [":poison" 0.0] [":flying" 0.0] [":fighting" 2.0] [":dragon" 0.0] [":fire" 0.0] [":ice" 4.0] [":ghost" 1.0] [":electric" 4.0] [":bug" 0.0] [":psychic" 0.0] [":grass" 0.0] [":water" 1.0] [":rock" 1.0]]
|
|
rlm@0
|
993
|
|
rlm@0
|
994 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
995 (clojure.pprint/pprint
|
|
rlm@0
|
996 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-attack-type)))
|
|
rlm@0
|
997 #+end_src
|
|
rlm@0
|
998
|
|
rlm@0
|
999 #+results:
|
|
rlm@0
|
1000 #+begin_example
|
|
rlm@0
|
1001 {":normal" 4.0,
|
|
rlm@0
|
1002 ":ground" 1.0,
|
|
rlm@0
|
1003 ":poison" 1.0,
|
|
rlm@0
|
1004 ":flying" 0.0,
|
|
rlm@0
|
1005 ":fighting" 1.0,
|
|
rlm@0
|
1006 ":dragon" 0.0,
|
|
rlm@0
|
1007 ":fire" 0.0,
|
|
rlm@0
|
1008 ":dark" 0.0,
|
|
rlm@0
|
1009 ":ice" 4.0,
|
|
rlm@0
|
1010 ":steel" 0.0,
|
|
rlm@0
|
1011 ":ghost" 1.0,
|
|
rlm@0
|
1012 ":electric" 3.0,
|
|
rlm@0
|
1013 ":bug" 0.0,
|
|
rlm@0
|
1014 ":psychic" 1.0,
|
|
rlm@0
|
1015 ":grass" 1.0,
|
|
rlm@0
|
1016 ":water" 1.0,
|
|
rlm@0
|
1017 ":rock" 2.0}
|
|
rlm@0
|
1018 #+end_example
|
|
rlm@0
|
1019
|
|
rlm@0
|
1020 # #+results-old:
|
|
rlm@0
|
1021 # : [[":normal" 4.0] [":ground" 1.0] [":poison" 1.0] [":flying" 0.0] [":fighting" 2.0] [":dragon" 0.0] [":fire" 0.0] [":dark" 0.0] [":ice" 5.0] [":steel" 0.0] [":ghost" 1.0] [":electric" 5.0] [":bug" 0.0] [":psychic" 1.0] [":grass" 0.0] [":water" 1.0] [":rock" 2.0]]
|
|
rlm@0
|
1022
|
|
rlm@0
|
1023
|
|
rlm@0
|
1024 This is an extremely interesting type combination, in that it uses
|
|
rlm@0
|
1025 quite a few types.
|
|
rlm@0
|
1026
|
|
rlm@0
|
1027 #+begin_src clojure :results verbatim :exports both
|
|
rlm@0
|
1028 (reduce + (vals (:solution (pokemon.lpsolve/worst-attack-type))))
|
|
rlm@0
|
1029 #+end_src
|
|
rlm@0
|
1030
|
|
rlm@0
|
1031 #+results:
|
|
rlm@0
|
1032 : 20.0
|
|
rlm@0
|
1033
|
|
rlm@0
|
1034 20 types is the /minimum/ number of types before the attacking
|
|
rlm@0
|
1035 combination is not-very-effective or worse against all defending
|
|
rlm@0
|
1036 types. This would probably have been impossible to discover using
|
|
rlm@0
|
1037 best-first search, since it involves such an intricate type
|
|
rlm@0
|
1038 combination.
|
|
rlm@0
|
1039
|
|
rlm@0
|
1040 It's so interesting that it takes 20 types to make an attack type that
|
|
rlm@0
|
1041 is weak to all types that the combination merits further investigation.
|
|
rlm@0
|
1042
|
|
rlm@0
|
1043 Unfortunately, all of the tools that we've written so far are focused
|
|
rlm@0
|
1044 on defense type combinations. However, it is possible to make every
|
|
rlm@0
|
1045 tool attack-oriented via a simple macro.
|
|
rlm@0
|
1046
|
|
rlm@0
|
1047 #+srcname: attack-oriented
|
|
rlm@0
|
1048 #+begin_src clojure :results silent
|
|
rlm@0
|
1049 (in-ns 'pokemon.lpsolve)
|
|
rlm@0
|
1050
|
|
rlm@0
|
1051 (defmacro attack-mode [& forms]
|
|
rlm@0
|
1052 `(let [attack-strengths# pokemon.types/attack-strengths
|
|
rlm@0
|
1053 defense-strengths# pokemon.types/defense-strengths]
|
|
rlm@0
|
1054 (binding [pokemon.types/attack-strengths
|
|
rlm@0
|
1055 defense-strengths#
|
|
rlm@0
|
1056 pokemon.types/defense-strengths
|
|
rlm@0
|
1057 attack-strengths#]
|
|
rlm@0
|
1058 ~@forms)))
|
|
rlm@0
|
1059 #+end_src
|
|
rlm@0
|
1060
|
|
rlm@0
|
1061 Now all the tools from =pokemon.types= will work for attack
|
|
rlm@0
|
1062 combinations.
|
|
rlm@0
|
1063
|
|
rlm@0
|
1064 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1065 (clojure.pprint/pprint
|
|
rlm@0
|
1066 (pokemon.types/susceptibility [:water]))
|
|
rlm@0
|
1067 #+end_src
|
|
rlm@0
|
1068
|
|
rlm@0
|
1069 #+results:
|
|
rlm@0
|
1070 #+begin_example
|
|
rlm@0
|
1071 {:water 1/2,
|
|
rlm@0
|
1072 :psychic 1,
|
|
rlm@0
|
1073 :dragon 1,
|
|
rlm@0
|
1074 :fire 1/2,
|
|
rlm@0
|
1075 :ice 1/2,
|
|
rlm@0
|
1076 :grass 2,
|
|
rlm@0
|
1077 :ghost 1,
|
|
rlm@0
|
1078 :poison 1,
|
|
rlm@0
|
1079 :flying 1,
|
|
rlm@0
|
1080 :normal 1,
|
|
rlm@0
|
1081 :rock 1,
|
|
rlm@0
|
1082 :electric 2,
|
|
rlm@0
|
1083 :ground 1,
|
|
rlm@0
|
1084 :fighting 1,
|
|
rlm@0
|
1085 :dark 1,
|
|
rlm@0
|
1086 :steel 1/2,
|
|
rlm@0
|
1087 :bug 1}
|
|
rlm@0
|
1088 #+end_example
|
|
rlm@0
|
1089
|
|
rlm@0
|
1090
|
|
rlm@0
|
1091 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1092 (clojure.pprint/pprint
|
|
rlm@0
|
1093 (pokemon.lpsolve/attack-mode
|
|
rlm@0
|
1094 (pokemon.types/susceptibility [:water])))
|
|
rlm@0
|
1095 #+end_src
|
|
rlm@0
|
1096
|
|
rlm@0
|
1097 #+results:
|
|
rlm@0
|
1098 #+begin_example
|
|
rlm@0
|
1099 {:water 1/2,
|
|
rlm@0
|
1100 :psychic 1,
|
|
rlm@0
|
1101 :dragon 1/2,
|
|
rlm@0
|
1102 :fire 2,
|
|
rlm@0
|
1103 :ice 1,
|
|
rlm@0
|
1104 :grass 1/2,
|
|
rlm@0
|
1105 :ghost 1,
|
|
rlm@0
|
1106 :poison 1,
|
|
rlm@0
|
1107 :flying 1,
|
|
rlm@0
|
1108 :normal 1,
|
|
rlm@0
|
1109 :rock 2,
|
|
rlm@0
|
1110 :electric 1,
|
|
rlm@0
|
1111 :ground 2,
|
|
rlm@0
|
1112 :fighting 1,
|
|
rlm@0
|
1113 :dark 1,
|
|
rlm@0
|
1114 :steel 1,
|
|
rlm@0
|
1115 :bug 1}
|
|
rlm@0
|
1116 #+end_example
|
|
rlm@0
|
1117
|
|
rlm@0
|
1118 Now =pokemon.types/susceptibility= reports the /attack-type/
|
|
rlm@0
|
1119 combination's effectiveness against other types.
|
|
rlm@0
|
1120
|
|
rlm@0
|
1121 The 20 type combo achieves its goal in a very clever way.
|
|
rlm@0
|
1122
|
|
rlm@0
|
1123 First, it weakens its effectiveness to other types at the expense of
|
|
rlm@0
|
1124 making it very strong against flying.
|
|
rlm@0
|
1125
|
|
rlm@0
|
1126 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1127 (clojure.pprint/pprint
|
|
rlm@0
|
1128 (pokemon.lpsolve/attack-mode
|
|
rlm@0
|
1129 (pokemon.types/susceptibility
|
|
rlm@0
|
1130 [:normal :normal :normal :normal
|
|
rlm@0
|
1131 :ice :ice :ice :ice
|
|
rlm@0
|
1132 :electric :electric :electric
|
|
rlm@0
|
1133 :rock :rock])))
|
|
rlm@0
|
1134 #+end_src
|
|
rlm@0
|
1135
|
|
rlm@0
|
1136 #+results:
|
|
rlm@0
|
1137 #+begin_example
|
|
rlm@0
|
1138 {:water 1/2,
|
|
rlm@0
|
1139 :psychic 1,
|
|
rlm@0
|
1140 :dragon 2,
|
|
rlm@0
|
1141 :fire 1/4,
|
|
rlm@0
|
1142 :ice 1/4,
|
|
rlm@0
|
1143 :grass 2,
|
|
rlm@0
|
1144 :ghost 0,
|
|
rlm@0
|
1145 :poison 1,
|
|
rlm@0
|
1146 :flying 512,
|
|
rlm@0
|
1147 :normal 1,
|
|
rlm@0
|
1148 :rock 1/16,
|
|
rlm@0
|
1149 :electric 1/8,
|
|
rlm@0
|
1150 :ground 0,
|
|
rlm@0
|
1151 :fighting 1/4,
|
|
rlm@0
|
1152 :dark 1,
|
|
rlm@0
|
1153 :steel 1/1024,
|
|
rlm@0
|
1154 :bug 4}
|
|
rlm@0
|
1155 #+end_example
|
|
rlm@0
|
1156
|
|
rlm@0
|
1157 Then, it removes it's strengths against Flying, Normal, and Fighting
|
|
rlm@0
|
1158 by adding Ghost and Ground.
|
|
rlm@0
|
1159
|
|
rlm@0
|
1160 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1161 (clojure.pprint/pprint
|
|
rlm@0
|
1162 (pokemon.lpsolve/attack-mode
|
|
rlm@0
|
1163 (pokemon.types/susceptibility
|
|
rlm@0
|
1164 [:normal :normal :normal :normal
|
|
rlm@0
|
1165 :ice :ice :ice :ice
|
|
rlm@0
|
1166 :electric :electric :electric
|
|
rlm@0
|
1167 :rock :rock
|
|
rlm@0
|
1168 ;; Spot resistances
|
|
rlm@0
|
1169 :ghost :ground])))
|
|
rlm@0
|
1170 #+end_src
|
|
rlm@0
|
1171
|
|
rlm@0
|
1172 #+results:
|
|
rlm@0
|
1173 #+begin_example
|
|
rlm@0
|
1174 {:water 1/2,
|
|
rlm@0
|
1175 :psychic 2,
|
|
rlm@0
|
1176 :dragon 2,
|
|
rlm@0
|
1177 :fire 1/2,
|
|
rlm@0
|
1178 :ice 1/4,
|
|
rlm@0
|
1179 :grass 1,
|
|
rlm@0
|
1180 :ghost 0,
|
|
rlm@0
|
1181 :poison 2,
|
|
rlm@0
|
1182 :flying 0,
|
|
rlm@0
|
1183 :normal 0,
|
|
rlm@0
|
1184 :rock 1/8,
|
|
rlm@0
|
1185 :electric 1/4,
|
|
rlm@0
|
1186 :ground 0,
|
|
rlm@0
|
1187 :fighting 1/4,
|
|
rlm@0
|
1188 :dark 1/2,
|
|
rlm@0
|
1189 :steel 1/1024,
|
|
rlm@0
|
1190 :bug 2}
|
|
rlm@0
|
1191 #+end_example
|
|
rlm@0
|
1192
|
|
rlm@0
|
1193 Adding the pair Psychic and Fighting takes care of its strength
|
|
rlm@0
|
1194 against Psychic and makes it ineffective against Dark, which is immune
|
|
rlm@0
|
1195 to Psychic.
|
|
rlm@0
|
1196
|
|
rlm@0
|
1197 Adding the pair Grass and Poison makes takes care of its strength
|
|
rlm@0
|
1198 against poison and makes it ineffective against Steel, which is immune
|
|
rlm@0
|
1199 to poison.
|
|
rlm@0
|
1200
|
|
rlm@0
|
1201 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1202 (clojure.pprint/pprint
|
|
rlm@0
|
1203 (pokemon.lpsolve/attack-mode
|
|
rlm@0
|
1204 (pokemon.types/susceptibility
|
|
rlm@0
|
1205 [;; setup
|
|
rlm@0
|
1206 :normal :normal :normal :normal
|
|
rlm@0
|
1207 :ice :ice :ice :ice
|
|
rlm@0
|
1208 :electric :electric :electric
|
|
rlm@0
|
1209 :rock :rock
|
|
rlm@0
|
1210 ;; Spot resistances
|
|
rlm@0
|
1211 :ghost :ground
|
|
rlm@0
|
1212 ;; Pair resistances
|
|
rlm@0
|
1213 :psychic :fighting
|
|
rlm@0
|
1214 :grass :poison])))
|
|
rlm@0
|
1215 #+end_src
|
|
rlm@0
|
1216
|
|
rlm@0
|
1217 #+results:
|
|
rlm@0
|
1218 #+begin_example
|
|
rlm@0
|
1219 {:water 1,
|
|
rlm@0
|
1220 :psychic 1/2,
|
|
rlm@0
|
1221 :dragon 1,
|
|
rlm@0
|
1222 :fire 1/4,
|
|
rlm@0
|
1223 :ice 1/2,
|
|
rlm@0
|
1224 :grass 1,
|
|
rlm@0
|
1225 :ghost 0,
|
|
rlm@0
|
1226 :poison 1/2,
|
|
rlm@0
|
1227 :flying 0,
|
|
rlm@0
|
1228 :normal 0,
|
|
rlm@0
|
1229 :rock 1/4,
|
|
rlm@0
|
1230 :electric 1/4,
|
|
rlm@0
|
1231 :ground 0,
|
|
rlm@0
|
1232 :fighting 1/2,
|
|
rlm@0
|
1233 :dark 0,
|
|
rlm@0
|
1234 :steel 0,
|
|
rlm@0
|
1235 :bug 1/2}
|
|
rlm@0
|
1236 #+end_example
|
|
rlm@0
|
1237
|
|
rlm@0
|
1238 Can you see the final step?
|
|
rlm@0
|
1239
|
|
rlm@0
|
1240 It's adding the Water type, which is weak against Water and Dragon and
|
|
rlm@0
|
1241 strong against Rock and Fire.
|
|
rlm@0
|
1242
|
|
rlm@0
|
1243 #+begin_src clojure :results output :exports both
|
|
rlm@0
|
1244 (clojure.pprint/pprint
|
|
rlm@0
|
1245 (pokemon.lpsolve/attack-mode
|
|
rlm@0
|
1246 (pokemon.types/susceptibility
|
|
rlm@0
|
1247 [;; setup
|
|
rlm@0
|
1248 :normal :normal :normal :normal
|
|
rlm@0
|
1249 :ice :ice :ice :ice
|
|
rlm@0
|
1250 :electric :electric :electric
|
|
rlm@0
|
1251 :rock :rock
|
|
rlm@0
|
1252 ;; Spot resistances
|
|
rlm@0
|
1253 :ghost :ground
|
|
rlm@0
|
1254 ;; Pair resistances
|
|
rlm@0
|
1255 :psychic :fighting
|
|
rlm@0
|
1256 :grass :poison
|
|
rlm@0
|
1257 ;; completion
|
|
rlm@0
|
1258 :water])))
|
|
rlm@0
|
1259 #+end_src
|
|
rlm@0
|
1260
|
|
rlm@0
|
1261 #+results:
|
|
rlm@0
|
1262 #+begin_example
|
|
rlm@0
|
1263 {:water 1/2,
|
|
rlm@0
|
1264 :psychic 1/2,
|
|
rlm@0
|
1265 :dragon 1/2,
|
|
rlm@0
|
1266 :fire 1/2,
|
|
rlm@0
|
1267 :ice 1/2,
|
|
rlm@0
|
1268 :grass 1/2,
|
|
rlm@0
|
1269 :ghost 0,
|
|
rlm@0
|
1270 :poison 1/2,
|
|
rlm@0
|
1271 :flying 0,
|
|
rlm@0
|
1272 :normal 0,
|
|
rlm@0
|
1273 :rock 1/2,
|
|
rlm@0
|
1274 :electric 1/4,
|
|
rlm@0
|
1275 :ground 0,
|
|
rlm@0
|
1276 :fighting 1/2,
|
|
rlm@0
|
1277 :dark 0,
|
|
rlm@0
|
1278 :steel 0,
|
|
rlm@0
|
1279 :bug 1/2}
|
|
rlm@0
|
1280 #+end_example
|
|
rlm@0
|
1281
|
|
rlm@0
|
1282 Which makes a particularly beautiful combination which is ineffective
|
|
rlm@0
|
1283 against all defending types.
|
|
rlm@0
|
1284
|
|
rlm@0
|
1285
|
|
rlm@0
|
1286 # #+begin_src clojure :results scalar :exports both
|
|
rlm@0
|
1287 # (with-out-str (clojure.contrib.pprint/pprint (seq (attack-mode (pokemon.types/susceptibility [:normal :normal :normal :normal :ice :ice :ice :ice :electric :electric :electric :rock :rock :ground :ghost :psychic :fighting :grass :poison])))))
|
|
rlm@0
|
1288 # #+end_src
|
|
rlm@0
|
1289
|
|
rlm@0
|
1290 # #+results:
|
|
rlm@0
|
1291 # | [:water 1] | [:psychic 1/2] | [:dragon 1] | [:fire 1/4] | [:ice 1/2] | [:grass 1] | [:ghost 0] | [:poison 1/2] | [:flying 0] | [:normal 0] | [:rock 1/4] | [:electric 1/4] | [:ground 0] | [:fighting 1/2] | [:dark 0] | [:steel 0] | [:bug 1/2] |
|
|
rlm@0
|
1292
|
|
rlm@0
|
1293
|
|
rlm@0
|
1294 Is there anything else that's interesting?
|
|
rlm@0
|
1295
|
|
rlm@0
|
1296 #+begin_src clojure :exports both
|
|
rlm@0
|
1297 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-defense-type))
|
|
rlm@0
|
1298 #+end_src
|
|
rlm@0
|
1299
|
|
rlm@0
|
1300 #+results:
|
|
rlm@0
|
1301 : INFEASIBLE
|
|
rlm@0
|
1302
|
|
rlm@0
|
1303 #+begin_src clojure :exports both
|
|
rlm@0
|
1304 (pokemon.types/old-school
|
|
rlm@0
|
1305 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-defense-type)))
|
|
rlm@0
|
1306 #+end_src
|
|
rlm@0
|
1307
|
|
rlm@0
|
1308 #+results:
|
|
rlm@0
|
1309 : INFEASIBLE
|
|
rlm@0
|
1310
|
|
rlm@0
|
1311 #+begin_src clojure :exports both
|
|
rlm@0
|
1312 (pokemon.lpsolve/solution (pokemon.lpsolve/weak-defense-type))
|
|
rlm@0
|
1313 #+end_src
|
|
rlm@0
|
1314
|
|
rlm@0
|
1315 #+results:
|
|
rlm@0
|
1316 : INFEASIBLE
|
|
rlm@0
|
1317
|
|
rlm@0
|
1318 #+begin_src clojure :exports both
|
|
rlm@0
|
1319 (pokemon.types/old-school
|
|
rlm@0
|
1320 (pokemon.lpsolve/solution (pokemon.lpsolve/weak-defense-type)))
|
|
rlm@0
|
1321 #+end_src
|
|
rlm@0
|
1322
|
|
rlm@0
|
1323 #+results:
|
|
rlm@0
|
1324 : INFEASIBLE
|
|
rlm@0
|
1325
|
|
rlm@0
|
1326 #+begin_src clojure :exports both
|
|
rlm@0
|
1327 (pokemon.lpsolve/solution (pokemon.lpsolve/neutral-defense-type))
|
|
rlm@0
|
1328 #+end_src
|
|
rlm@0
|
1329
|
|
rlm@0
|
1330 #+results:
|
|
rlm@0
|
1331 : INFEASIBLE
|
|
rlm@0
|
1332
|
|
rlm@0
|
1333 #+begin_src clojure :exports both
|
|
rlm@0
|
1334 (pokemon.types/old-school
|
|
rlm@0
|
1335 (pokemon.lpsolve/solution (pokemon.lpsolve/neutral-defense-type)))
|
|
rlm@0
|
1336 #+end_src
|
|
rlm@0
|
1337
|
|
rlm@0
|
1338 #+results:
|
|
rlm@0
|
1339 : INFEASIBLE
|
|
rlm@0
|
1340
|
|
rlm@0
|
1341 There is no way to produce a defense-type that is weak to all types.
|
|
rlm@0
|
1342 This is probably because there are many types that are completely
|
|
rlm@0
|
1343 immune to some types, such as Flying, which is immune to Ground. A
|
|
rlm@0
|
1344 perfectly weak type could not use any of these types.
|
|
rlm@0
|
1345
|
|
rlm@0
|
1346 * Summary
|
|
rlm@0
|
1347
|
|
rlm@0
|
1348 Overall, the pok\eacute{}mon type system is slanted more towards defense
|
|
rlm@0
|
1349 rather than offense. While it is possible to create superior
|
|
rlm@0
|
1350 defensive types and exceptionally weak attack types, it is not possible to
|
|
rlm@0
|
1351 create exceptionally weak defensive types or very powerful attack
|
|
rlm@0
|
1352 types.
|
|
rlm@0
|
1353
|
|
rlm@0
|
1354 Using the =lp_solve= library was more complicated than the best-first
|
|
rlm@0
|
1355 search, but yielded results quickly and efficiently. Expressing the
|
|
rlm@0
|
1356 problem in a linear form does have its drawbacks, however --- it's
|
|
rlm@0
|
1357 hard to ask questions such as "what is the best 3-type defensive combo
|
|
rlm@0
|
1358 in terms of susceptibility?", since susceptibility is not a linear
|
|
rlm@0
|
1359 function of a combo's types. It is also hard to get all the solutions
|
|
rlm@0
|
1360 to a particular problem, such as all the pokemon type combinations of
|
|
rlm@0
|
1361 length 8 which are immortal defense types.
|
|
rlm@0
|
1362
|
|
rlm@0
|
1363
|
|
rlm@0
|
1364 * COMMENT main-program
|
|
rlm@0
|
1365 #+begin_src clojure :tangle ../src/pokemon/lpsolve.clj :noweb yes :exports none
|
|
rlm@0
|
1366 <<intro>>
|
|
rlm@0
|
1367 <<body>>
|
|
rlm@0
|
1368 <<declares>>
|
|
rlm@0
|
1369 <<memory-management>>
|
|
rlm@0
|
1370 <<get-results>>
|
|
rlm@0
|
1371 <<solve>>
|
|
rlm@0
|
1372 <<farmer-example>>
|
|
rlm@0
|
1373 <<lp-solve>>
|
|
rlm@0
|
1374 <<better-farmer>>
|
|
rlm@0
|
1375 <<pokemon-lp>>
|
|
rlm@0
|
1376 <<results>>
|
|
rlm@0
|
1377 <<attack-oriented>>
|
|
rlm@0
|
1378 #+end_src
|
|
rlm@0
|
1379
|
|
rlm@0
|
1380
|
|
rlm@0
|
1381 * COMMENT Stuff to do.
|
|
rlm@0
|
1382 ** TODO fix namespaces to not use rlm.light-base
|
|
rlm@0
|
1383
|
