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