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