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