#+title: Discovering Effective Pok\eacute{}mon Types Using Linear Optimization
#+author: Robert McIntyre & Dylan Holmes
#+EMAIL:     rlm@mit.edu
#+description: Using Lpsolve to find effective pokemon types in clojure.
#+keywords: Pokemon, clojure, linear optimization, lp_solve, LpSolve
#+SETUPFILE: ../../aurellem/org/setup.org
#+INCLUDE: ../../aurellem/org/level-0.org

* Introduction
This post continues the [[./types.org][previous one]] about pok\eacute{}mon types.
Pok\eacute{}mon is a game in which adorable creatures battle each
other using fantastic attacks.  It was made into a several gameboy
games that all share the same battle system.  Every pok\eacute{}mon in
the gameboy game has one or two /types/, such as Ground, Fire, Water,
etc. Every pok\eacute{}mon attack has exactly one type. Certain
defending types are weak or strong to other attacking types.  For
example, Water attacks are strong against Fire pok\eacute{}mon, while
Electric attacks are weak against Ground Pok\eacute{}mon.  In the
games, attacks can be either twice as effective as normal (Water
vs. Fire), neutrally effective (Normal vs. Normal), half as effective
(Fire vs. Water), or not effective at all (Electric vs. Ground). We
represent these strengths and weaknesses as the numbers 2, 1,
$\frac{1}{2}$, and 0, and call them the /susceptance/ of one type to
another.

If a pokemon has two types, then the strengths and weakness of each
type are /multiplied/ together.  Thus Electric (2x weak to Ground)
combined with Flying (immune to Ground (0x)) is immune to Ground.
Fire (2x weak to Water) combined with Water (1/2x resistant to Water)
is neutral to Water. If both types are resistant to another type, then
the combination is doubly-resistant (1/4x) to that type. If both types
are weak to a certain type then the combination is double-weak (4x) to
that type.

In the [[./types.org][previous post]], we used the best-first search algorithm to find
the most effective Pok\eacute{}mon type combinations. Afterwards, we
realized that we could transform this search problem into a /linear
optimization problem/.  This conversion offers several advantages:
first, search algorithms are comparatively slow, whereas linear
optimization algorithms are extremely fast; second, it is difficult to
determine whether a search problem has any solution, whereas it is
straightforward to determine whether a linear optimization problem has
any solution; finally, because systems of linear equations are so
common, many programming languages have linear equation solvers
written for them.

In this article, we will:
 - Solve a simple linear optimization problem in C :: We demonstrate
      how to use the linear programming C library, =lp_solve=, to
      solve a simple linear optimization problem.
 - Incorporate a C library into Clojure :: We will show how we gave
      Clojure access to the linear programming C library, =lp_solve=.
 - Find effective Pokemon types using linear programming :: Building
      on our earlier code, we answer some questions that were
      impossible to answer using best-first search.
- Present our results :: We found some cool examples and learned a lot
     about the pok\eacute{}mon type system as a whole.
 

** Immortal Types
   
In the game, pok\eacute{}mon can have either one type or two types. If
this restriction is lifted, is there any combination of types that is
resistant to all types?  I call such a combination an /Immortal Type/,
since if that type's pattern was repeated over and over again towards
infinity, the resulting type would be immune to all attack types.

* Linear Programming

Linear programming is the process of finding an optimal solution to a
linear equation of several variables which are constrained by some linear
inequalities.

** The Farmer's Problem

Let's solve the Farmer's Problem, an example linear programming problem
borrowed from http://lpsolve.sourceforge.net/5.5/formulate.htm.


#+BEGIN_QUOTE 
*The Farmer's Problem:* Suppose a farmer has 75 acres on which to
plant two crops: wheat and barley. To produce these crops, it costs
the farmer (for seed, fertilizer, etc.)  $120 per acre for the wheat
and $210 per acre for the barley. The farmer has $15000 available for
expenses. But after the harvest, the farmer must store the crops while
awaiting favorable market conditions. The farmer has storage space
for 4000 bushels. Each acre yields an average of 110 bushels of wheat
or 30 bushels of barley.  If the net profit per bushel of wheat (after
all expenses have been subtracted) is $1.30 and for barley is $2.00,
how should the farmer plant the 75 acres to maximize profit?
#+END_QUOTE

The Farmer's Problem is to maximize profit subject to constraints on
available farmland, funds for expenses, and storage space.

|          | Wheat                | Barley              | Maximum total |
|----------+----------------------+---------------------+--------------|
| /        | <                    | >                   | <>           |
| Farmland | \(w\) acres          | \(b\) acres         | 75 acres     |
| Expense  | $120 per acre        | $210 per acre       | $15000       |
| Storage  | 110 bushels per acre | 30 bushels per acre | 4000 bushels |
|----------+----------------------+---------------------+--------------|
| Profit   | $1.30 per bushel     | $2.00 per bushel    |              |

** Solution using LP Solve
In a new file,  =farmer.lp=, we list the variables and constraints 
of our problem using LP Solve syntax.

#+begin_src lpsolve :tangle ../lp/farmer.lp
/* Maximize Total Profit */
max: +143 wheat +60 barley;


/* --------  Constraints  --------*/

/* the farmer can't spend more money than he has */
+120 wheat +210 barley <= 15000;

/* the harvest has to fit in his storage space */
+110 wheat +30 barley <= 4000;

/* he can't use more acres than he owns */
+wheat +barley <= 75;
#+end_src

Running the =lp_solve= program on =farmer.lp= yields the following output.

#+begin_src sh :exports both :results scalar
lp_solve ~/proj/pokemon-types/lp/farmer.lp
#+end_src

#+results:
: 
: Value of objective function: 6315.62500000
: 
: Actual values of the variables:
: wheat                      21.875
: barley                     53.125


This shows that the farmer can maximize his profit by planting 21.875
of the available acres with wheat and the remaining 53.125 acres with
barley; by doing so, he will make $6315.62(5) in profit.

* Incorporating  =lp_solve= into Clojure 

There is a [[http://lpsolve.sourceforge.net/5.5/Java/README.html][Java API]] written by Juergen Ebert which enables Java
programs to use =lp_solve=. Although Clojure can use this Java API
directly, the interaction between Java, C, and Clojure is clumsy:

** The Farmer's Problem in Clojure

We are going to solve the same problem involving wheat and barley,
that we did above, but this time using clojure and the =lp_solve= API.

#+name: intro
#+begin_src clojure :results silent
(ns pokemon.lpsolve
  (:import lpsolve.LpSolve)
  (:require pokemon.types)
  ;;(:require incanter.core)
  (:require rlm.map-utils))
#+end_src

The =lp_solve= Java interface is available from the same site as
=lp_solve= itself, http://lpsolve.sourceforge.net/ Using it is the
same as many other =C= programs. There are excellent instructions to
get set up.  The short version is that you must call Java with
=-Djava.library.path=/path/to/lpsolve/libraries= and also add the
libraries to your export =LD_LIBRARY_PATH= if you are using Linux. For
example, in my =.bashrc= file, I have the line
=LD_LIBRARY_PATH=$HOME/roBin/lpsolve:$LD_LIBRARY_PATH=.  If everything
is set-up correctly,

#+name: body 
#+begin_src clojure :results verbatim :exports both
(import 'lpsolve.LpSolve)
#+end_src

#+results:
: lpsolve.LpSolve

should run with no problems.

** Making a DSL to talk with LpSolve
*** Problems
Since we are using a =C= wrapper, we have to deal with manual memory
management for the =C= structures which are wrapped by the =LpSolve=
object. Memory leaks in =LpSolve= instances can crash the JVM, so it's
very important to get it right.  Also, the Java wrapper follows the
=C= tradition closely and defines many =static final int= constants
for the different states of the =LpSolve= instance instead of using Java
enums.  The calling convention for adding rows and columns to
the constraint matrix is rather complicated and must be done column by
column or row by row, which can be error prone.  Finally, I'd like to
gather all the important output information from the =LpSolve= instance
into a final, immutable structure.

In summary, the issues I'd like to address are:

- reliable memory management
- functional interface to =LpSolve=
- intelligible, immutable output

To deal with these issues I'll create four functions for interfacing
with =LpSolve=

#+name: declares
#+begin_src clojure :results silent 
(in-ns 'pokemon.lpsolve)

;; deal with automatic memory management for LpSolve instance.
(declare linear-program)

;; functional interface to LpSolve
(declare lp-solve)

;; immutable output from lp-solve
(declare solve get-results)
#+end_src


*** Memory Management

Every instance of =LpSolve= must be manually garbage collected via a
call to =deleteLP=. I use a non-hygienic macro similar to =with-open=
to ensure that =deleteLP= is always called.

#+name: memory-management
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)
(defmacro linear-program
  "solve a linear programming problem using LpSolve syntax.
   within the macro, the variable =lps= is bound to the LpSolve instance."
  [& statements]
  (list 'let '[lps (LpSolve/makeLp 0 0)]
	(concat '(try)
		statements
		;; always free the =C= data structures.
		'((finally (.deleteLp lps))))))
#+end_src


The macro captures the variable =lps= within its body, providing for a
convenient way to access the object using any of the methods of the
=LpSolve= API without having to worry about when to call
=deleteLP=. 

*** Sensible Results
The =linear-program= macro deletes the actual =lps= object once it is
done working, so it's important to collect the important results and
add return them in an immutable structure at the end.

#+name: get-results
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)

(defrecord LpSolution
  [objective-value
   optimal-values
   variable-names
   solution
   status
   model])

(defn model
  "Returns a textual representation of the problem suitable for
  direct input to the =lp_solve= program (lps format)"
  [#^LpSolve lps]
  (let [target (java.io.File/createTempFile "lps" ".lp")]
    (.writeLp lps (.getPath target))
    (slurp target)))

(defn results
  "Given an LpSolve object, solves the object and returns a map of the
  essential values which compose the solution."
  [#^LpSolve lps]
  (locking lps
    (let [status (solve lps)
	  number-of-variables (.getNcolumns lps)
	  optimal-values (double-array number-of-variables)
	  optimal-values (do (.getVariables lps optimal-values)
			     (seq optimal-values))
	  variable-names
	  (doall 
           ;; The doall is necessary since the lps object might
	   ;; soon be deleted.
	   (map
	    #(.getColName lps (inc %))
	    (range number-of-variables)))
	  model (model lps)]
      (LpSolution.
       (.getObjective lps)
       optimal-values
       variable-names
       (zipmap variable-names optimal-values)
       status
       model))))

#+end_src

Here I've created an object called =LpSolution= which stores the
important results from a session with =lp_solve=.  Of note is the
=model= function which returns the problem in a form that can be
solved by other linear programming packages.

*** Solution Status of an LpSolve Object

#+name: solve
#+begin_src clojure :results silent       
(in-ns 'pokemon.lpsolve)

(defn static-integer?
  "does the field represent a static integer constant?"
  [#^java.lang.reflect.Field field]
  (and (java.lang.reflect.Modifier/isStatic (.getModifiers field))
       (integer? (.get field nil))))

(defn integer-constants [class]
  (filter static-integer? (.getFields class)))

(defn constant-map
  "Takes a class and creates a map of the static constant integer
  fields with their names.  This helps with C wrappers where they have
  just defined a bunch of integer constants instead of enums."
  [class]
     (let [integer-fields (integer-constants class)]
       (into (sorted-map)
	     (zipmap (map #(.get % nil) integer-fields)
		     (map #(.getName %) integer-fields)))))

(alter-var-root #'constant-map memoize)
	     
(defn solve
  "Solve an instance of LpSolve and return a string representing the
  status of the computation. Will only solve a particular LpSolve
  instance once."
  [#^LpSolve lps]
  ((constant-map LpSolve)
   (.solve lps)))

#+end_src

The =.solve= method of an =LpSolve= object only returns an integer code
to specify the status of the computation.  The =solve= method here
uses reflection to look up the actual name of the status code and
returns a more helpful status message that is also resistant to
changes in the meanings of the code numbers.

*** The Farmer Example in Clojure, Pass 1

Now we can implement a nicer version of the examples from the
[[http://lpsolve.sourceforge.net/][=lp\_solve= website]]. The following is a more or less
line-by-line translation of the Java code from that example.

#+name: farmer-example
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)
(defn farmer-example []
  (linear-program
   (results
    (doto lps
      ;; name the columns
      (.setColName 1 "wheat")
      (.setColName 2 "barley")
      (.setAddRowmode true)
      ;; row 1 : 120x + 210y <= 15000
      (.addConstraintex 2
			(double-array [120 210])
			(int-array [1 2])
			LpSolve/LE
			15e3)
      ;; row 2 : 110x + 30y <= 4000
      (.addConstraintex 2
			(double-array [110 30])
			(int-array [1 2])
			LpSolve/LE
			4e3)
      ;; ;; row 3 : x + y <= 75
      (.addConstraintex 2
			(double-array [1 1])
			(int-array [1 2])
			LpSolve/LE
			75)
      (.setAddRowmode false)

      ;; add constraints
      (.setObjFnex 2
		   (double-array [143 60])
		   (int-array [1 2]))
            
      ;; set this as a maximization problem
      (.setMaxim)))))

#+end_src

#+begin_src clojure :results output :exports both 
(clojure.pprint/pprint
 (:solution (pokemon.lpsolve/farmer-example)))
#+end_src

#+results:
: {"barley" 53.12499999999999, "wheat" 21.875}


And it works as expected!

*** The Farmer Example in Clojure, Pass 2
We don't have to worry about memory management anymore, and the farmer
example is about half as long as the example from the =LpSolve=
website, but we can still do better.  Solving linear problems is all
about the constraint matrix $A$ , the objective function $c$, and the
right-hand-side $b$, plus whatever other options one cares to set for
the particular instance of =lp_solve=.  Why not make a version of
=linear-program= that takes care of initialization?



#+name: lp-solve
#+begin_src clojure :results silent       
(in-ns 'pokemon.lpsolve)
(defn initialize-lpsolve-row-oriented
  "fill in an lpsolve instance using a constraint matrix =A=, the
  objective function =c=, and the right-hand-side =b="
  [#^lpsolve.LpSolve lps A b c]
  ;; set the name of the last column to _something_
  ;; this appears to be necessary to ensure proper initialization.
  (.setColName lps (count c) (str "C" (count c)))

  ;; This is the recommended way to "fill-in" an lps instance from the 
  ;; documentation. First, set row mode, then set the objective
  ;; function, then set each row of the problem, and then turn off row
  ;; mode. 
  (.setAddRowmode lps true)
  (.setObjFnex lps (count c)
	       (double-array c)
	       (int-array (range 1 (inc (count c)))))
  (dorun
   (for [n (range (count A))]
     (let [row (nth A n)
	   row-length (int (count row))]
       (.addConstraintex lps
			 row-length
			 (double-array row)
			 (int-array (range 1 (inc row-length)))
			 LpSolve/LE
			 (double (nth b n))))))
  (.setAddRowmode lps false)
  lps)


(defmacro lp-solve
  "by default:,
   minimize (* c x), subject to (<= (* A x) b),
   using continuous variables.  You may set any number of
   other options as in the LpSolve API." 
  [A b c & lp-solve-forms]
  ;; assume that A is a vector of vectors
  (concat
   (list 'linear-program
	 (list 'initialize-lpsolve-row-oriented 'lps A b c))
   `~lp-solve-forms))
 #+end_src

Now, we can use a much more functional approach to solving the
farmer's problem:

#+name: better-farmer
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)
(defn better-farmer-example []
  (lp-solve  [[120 210]
	      [110 30]
	      [1 1]]
	     [15000
	      4000
	      75]
	     [143 60]
	     (.setColName lps 1 "wheat")
	     (.setColName lps 2 "barley")
	     (.setMaxim lps)
	     (results lps)))
#+end_src

#+begin_src clojure  :exports both :results verbatim
(vec (:solution (pokemon.lpsolve/better-farmer-example)))
#+end_src

#+results:
: [["barley" 53.12499999999999] ["wheat" 21.875]]

Notice that both the inputs to =better-farmer-example= and the results
are immutable.

* Using LpSolve to find Immortal Types
** Converting the Pok\eacute{}mon problem into a linear form
How can the original question about pok\eacute{}mon types be converted
into a linear problem?

Pokemon types can be considered to be vectors of numbers representing
their susceptances to various attacking types, so Water might look
something like this.

#+begin_src clojure :results scalar :exports both
(:water (pokemon.types/defense-strengths))
#+end_src 

#+results:
: [1 0.5 0.5 2 2 0.5 1 1 1 1 1 1 1 1 1 1 0.5]

Where the numbers represent the susceptibility of Water to the
attacking types in the following order:

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.types/type-names))
#+end_src 

#+results:
#+begin_example
[:normal
 :fire
 :water
 :electric
 :grass
 :ice
 :fighting
 :poison
 :ground
 :flying
 :psychic
 :bug
 :rock
 :ghost
 :dragon
 :dark
 :steel]
#+end_example


So, for example, Water is resistant (x0.5) against Fire, which is
the second element in the list.

To combine types, these sorts of vectors are multiplied together
pair-wise to yield the resulting combination.  

Unfortunately, we need some way to add two type vectors together
instead of multiplying them if we want to solve the problem with
=lp_solve=.  Taking the log of the vector does just the trick.

If we make a matrix with each column being the log (base 2) of the
susceptance of each type, then finding an immortal type corresponds to
setting each constraint (the $b$ vector) to -1 (since log_2(1/2) = -1)
and setting the constraint vector $c$ to all ones, which means that we
want to find the immortal type which uses the least amount of types.

#+name: pokemon-lp
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)

(defn log-clamp-matrix [matrix]
  ;; we have to clamp the Infinities to a more reasonable negative
  ;; value because lp_solve does not play well with infinities in its
  ;; constraint matrix.
  (map (fn [row] (map #(if (= Double/NEGATIVE_INFINITY %) -1e3 %) 
                      (map #(/ (Math/log %) (Math/log 2)) row)))
	(apply mapv vector ;; transpose
               matrix))) 

;; constraint matrices
(defn log-defense-matrix []
  (log-clamp-matrix
   (doall (map (pokemon.types/defense-strengths)
	       (pokemon.types/type-names)))))

(defn log-attack-matrix []
  (apply mapv vector (log-defense-matrix)))

;; target vectors
(defn all-resistant []
  (doall (map (constantly -1) (pokemon.types/type-names))))

(defn all-weak []
  (doall (map (constantly 1) (pokemon.types/type-names))))

(defn all-neutral []
  (doall (map (constantly 0) (pokemon.types/type-names))))

;; objective functions
(defn number-of-types []
  (doall (map (constantly 1) (pokemon.types/type-names))))

(defn set-constraints
  "sets all the constraints for an lpsolve instance to the given
  constraint. =constraint= here is one of the LpSolve constants such
  as LpSolve/EQ."
  [#^LpSolve lps constraint]
  (dorun (map (fn [index] (.setConstrType lps index constraint))
	      ;; ONE based indexing!!!
	      (range 1 (inc (.getNrows lps))))))

(defn set-discrete
  "sets every variable in an lps problem to be a discrete rather than
  continuous variable"
  [#^LpSolve lps]
  (dorun (map (fn [index] (.setInt lps index true))
	      ;; ONE based indexing!!!
	      (range 1 (inc (.getNcolumns lps))))))

(defn set-variable-names
  "sets the variable names of the problem given a vector of names"
  [#^LpSolve lps names]
  (dorun
   (keep-indexed
    (fn [index name]
      (.setColName lps  (inc index) (str name)))
    ;; ONE based indexing!!!
    names)))

(defn poke-solve 
  ([poke-matrix target objective-function constraint min-num-types]
     ;; must have at least one type
     (let [poke-matrix
	   (concat poke-matrix
		   [(map (constantly 1)
			 (range (count (first poke-matrix))))])
	   target (concat target [min-num-types])]
       (lp-solve poke-matrix target objective-function
		 (set-constraints lps constraint)
		 ;; must have more than min-num-types
		 (.setConstrType lps (count target) LpSolve/GE)
		 (set-discrete lps)
		 (set-variable-names lps (pokemon.types/type-names))
		 (results lps))))
  ([poke-matrix target objective-function constraint]
     ;; at least one type
     (poke-solve poke-matrix target objective-function constraint 1)))

(defn solution 
  "If the results of an lpsolve operation are feasible, returns the
  results. Otherwise, returns the error."
  [results]
  (if (not (= (:status results) "OPTIMAL"))
    (:status results)
    (:solution results)))
#+end_src

With this, we are finally able to get some results.

** Results
#+name: results
#+begin_src clojure :results silent  
(in-ns 'pokemon.lpsolve)

(defn best-defense-type
  "finds a type combination which is resistant to all attacks."
  []
  (poke-solve 
   (log-defense-matrix) (all-resistant) (number-of-types) LpSolve/LE))

(defn worst-attack-type
  "finds the attack type which is not-very-effective against all pure
  defending types (each single defending type is resistant to this
  attack combination"
  []
  (poke-solve
   (log-attack-matrix)  (all-resistant) (number-of-types) LpSolve/LE))

(defn worst-defense-type
  "finds a defending type that is weak to all single attacking types."
  []
  (poke-solve
   (log-defense-matrix) (all-weak) (number-of-types) LpSolve/GE))

(defn best-attack-type
  "finds an attack type which is super effective against all single
defending types"
  []
  (poke-solve
   (log-attack-matrix) (all-weak) (number-of-types) LpSolve/GE))

(defn solid-defense-type
  "finds a defense type which is either neutral or resistant to all
  single attacking types"
  []
  (poke-solve
   (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/LE))

(defn solid-attack-type
  "finds an attack type which is either neutral or super-effective to
  all single attacking types."
  []
  (poke-solve
   (log-attack-matrix) (all-neutral) (number-of-types) LpSolve/GE))

(defn weak-defense-type
  "finds a defense type which is either neutral or weak to all single
  attacking types"
  []
  (poke-solve
   (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/GE))

(defn neutral-defense-type
  "finds a defense type which is perfectly neutral to all attacking
  types."
  []
  (poke-solve
   (log-defense-matrix) (all-neutral) (number-of-types) LpSolve/EQ))

#+end_src

*** Strongest Attack/Defense Combinations
    
#+begin_src clojure :results output :exports both 
(clojure.pprint/pprint
 (pokemon.lpsolve/solution (pokemon.lpsolve/best-defense-type)))
#+end_src

#+results:
#+begin_example
{":normal" 0.0,
 ":ground" 1.0,
 ":poison" 2.0,
 ":flying" 1.0,
 ":fighting" 0.0,
 ":dragon" 0.0,
 ":fire" 0.0,
 ":dark" 1.0,
 ":ice" 0.0,
 ":steel" 1.0,
 ":ghost" 0.0,
 ":electric" 0.0,
 ":bug" 0.0,
 ":psychic" 0.0,
 ":grass" 0.0,
 ":water" 2.0,
 ":rock" 0.0}
#+end_example

# #+results-old:
# : [[":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]]


This is the immortal type combination we've been looking for.  By
combining Steel, Water, Poison, and three types which each have complete
immunities to various other types, we've created a type that is resistant to
all attacking types.

#+begin_src clojure :results output :exports both 
(clojure.pprint/pprint
 (pokemon.types/susceptibility 
  [:poison :poison :water :water :steel :ground :flying :dark]))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 0,
 :dragon 1/2,
 :fire 1/2,
 :ice 1/2,
 :grass 1/2,
 :ghost 1/4,
 :poison 0,
 :flying 1/2,
 :normal 1/2,
 :rock 1/2,
 :electric 0,
 :ground 0,
 :fighting 1/2,
 :dark 1/4,
 :steel 1/8,
 :bug 1/8}
#+end_example

# #+results-old:
# : {: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}


Cool!

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/solution (pokemon.lpsolve/solid-defense-type)))
#+end_src

#+results:
#+begin_example
{":normal" 0.0,
 ":ground" 0.0,
 ":poison" 0.0,
 ":flying" 0.0,
 ":fighting" 0.0,
 ":dragon" 0.0,
 ":fire" 0.0,
 ":dark" 1.0,
 ":ice" 0.0,
 ":steel" 0.0,
 ":ghost" 1.0,
 ":electric" 0.0,
 ":bug" 0.0,
 ":psychic" 0.0,
 ":grass" 0.0,
 ":water" 0.0,
 ":rock" 0.0}
#+end_example

Dark and Ghost are the best dual-type combo, and are resistant or
neutral to all types.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.types/old-school
  (pokemon.lpsolve/solution (pokemon.lpsolve/solid-defense-type))))
#+end_src

#+results:
#+begin_example
{":normal" 0.0,
 ":ground" 0.0,
 ":poison" 0.0,
 ":flying" 0.0,
 ":fighting" 0.0,
 ":dragon" 0.0,
 ":fire" 0.0,
 ":ice" 0.0,
 ":ghost" 1.0,
 ":electric" 0.0,
 ":bug" 0.0,
 ":psychic" 1.0,
 ":grass" 0.0,
 ":water" 0.0,
 ":rock" 0.0}
#+end_example

Ghost and Psychic are a powerful dual type combo in the original games,
due to a glitch which made Psychic immune to Ghost type attacks, even
though the game claims that Ghost is strong against Psychic.

#+begin_src clojure :results verbatim :exports both
(pokemon.lpsolve/solution (pokemon.lpsolve/best-attack-type))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :results verbatim :exports both
(pokemon.lpsolve/solution (pokemon.lpsolve/solid-attack-type))
#+end_src

#+results:
: INFEASIBLE


#+begin_src clojure :results verbatim :exports both
(pokemon.types/old-school
 (pokemon.lpsolve/solution (pokemon.lpsolve/best-attack-type)))
#+end_src

#+results:
: INFEASIBLE


#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.types/old-school
  (pokemon.lpsolve/solution (pokemon.lpsolve/solid-attack-type))))
#+end_src

#+results:
#+begin_example
{":normal" 0.0,
 ":ground" 0.0,
 ":poison" 0.0,
 ":flying" 0.0,
 ":fighting" 0.0,
 ":dragon" 1.0,
 ":fire" 0.0,
 ":ice" 0.0,
 ":ghost" 0.0,
 ":electric" 0.0,
 ":bug" 0.0,
 ":psychic" 0.0,
 ":grass" 0.0,
 ":water" 0.0,
 ":rock" 0.0}
#+end_example

The best attacking type combination is Dragon from the original games.
It is neutral against all the original types except for Dragon, which
it is strong against.  There is no way to make an attacking type that
is strong against every type, or even one that is strong or neutral
against every type, in the new games.


*** Weakest Attack/Defense Combinations

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.types/old-school
  (pokemon.lpsolve/solution (pokemon.lpsolve/worst-attack-type))))
#+end_src

#+results:
#+begin_example
{":normal" 5.0,
 ":ground" 0.0,
 ":poison" 0.0,
 ":flying" 0.0,
 ":fighting" 0.0,
 ":dragon" 0.0,
 ":fire" 1.0,
 ":ice" 2.0,
 ":ghost" 1.0,
 ":electric" 1.0,
 ":bug" 1.0,
 ":psychic" 0.0,
 ":grass" 3.0,
 ":water" 2.0,
 ":rock" 0.0}
#+end_example

# #+results-old:
# : [[":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]]

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-attack-type)))
#+end_src

#+results:
#+begin_example
{":normal" 4.0,
 ":ground" 1.0,
 ":poison" 1.0,
 ":flying" 0.0,
 ":fighting" 1.0,
 ":dragon" 0.0,
 ":fire" 0.0,
 ":dark" 0.0,
 ":ice" 4.0,
 ":steel" 0.0,
 ":ghost" 1.0,
 ":electric" 3.0,
 ":bug" 0.0,
 ":psychic" 1.0,
 ":grass" 1.0,
 ":water" 1.0,
 ":rock" 2.0}
#+end_example

# #+results-old:
# : [[":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]]


This is an extremely interesting type combination, in that it uses
quite a few types.

#+begin_src clojure :results verbatim :exports both
(reduce + (vals (:solution (pokemon.lpsolve/worst-attack-type))))
#+end_src

#+results:
: 20.0

20 types is the /minimum/ number of types before the attacking
combination is not-very-effective or worse against all defending
types. This would probably have been impossible to discover using
best-first search, since it involves such an intricate type
combination.

It's so interesting that it takes 20 types to make an attack type that
is weak to all types that the combination merits further
investigation.

Unfortunately, all of the tools that we've written so far are focused
on defense type combinations.  However, it is possible to make every
tool attack-oriented via a simple macro.

#+name: attack-oriented
#+begin_src clojure :results silent
(in-ns 'pokemon.lpsolve)

(defmacro attack-mode [& forms]
  `(let [attack-strengths# pokemon.types/attack-strengths
	 defense-strengths# pokemon.types/defense-strengths]
     (binding [pokemon.types/attack-strengths
	       defense-strengths#
	       pokemon.types/defense-strengths
	       attack-strengths#]
       ~@forms)))
#+end_src

Now all the tools from =pokemon.types= will work for attack
combinations.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.types/susceptibility [:water]))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 1,
 :dragon 1,
 :fire 1/2,
 :ice 1/2,
 :grass 2,
 :ghost 1,
 :poison 1,
 :flying 1,
 :normal 1,
 :rock 1,
 :electric 2,
 :ground 1,
 :fighting 1,
 :dark 1,
 :steel 1/2,
 :bug 1}
#+end_example


#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/attack-mode 
  (pokemon.types/susceptibility [:water])))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 1,
 :dragon 1/2,
 :fire 2,
 :ice 1,
 :grass 1/2,
 :ghost 1,
 :poison 1,
 :flying 1,
 :normal 1,
 :rock 2,
 :electric 1,
 :ground 2,
 :fighting 1,
 :dark 1,
 :steel 1,
 :bug 1}
#+end_example

Now =pokemon.types/susceptibility= reports the /attack-type/
combination's effectiveness against other types.

The 20 type combo achieves its goal in a very clever way.

First, it weakens its effectiveness to other types at the expense of
making it very strong against flying.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/attack-mode
  (pokemon.types/susceptibility
   [:normal :normal :normal :normal
    :ice :ice :ice :ice
    :electric :electric :electric
    :rock :rock])))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 1,
 :dragon 2,
 :fire 1/4,
 :ice 1/4,
 :grass 2,
 :ghost 0,
 :poison 1,
 :flying 512,
 :normal 1,
 :rock 1/16,
 :electric 1/8,
 :ground 0,
 :fighting 1/4,
 :dark 1,
 :steel 1/1024,
 :bug 4}
#+end_example

Then, it removes it's strengths against Flying, Normal, and Fighting
by adding Ghost and Ground.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/attack-mode
  (pokemon.types/susceptibility
   [:normal :normal :normal :normal
    :ice :ice :ice :ice
    :electric :electric :electric
    :rock :rock
    ;; Spot resistances
    :ghost :ground])))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 2,
 :dragon 2,
 :fire 1/2,
 :ice 1/4,
 :grass 1,
 :ghost 0,
 :poison 2,
 :flying 0,
 :normal 0,
 :rock 1/8,
 :electric 1/4,
 :ground 0,
 :fighting 1/4,
 :dark 1/2,
 :steel 1/1024,
 :bug 2}
#+end_example

Adding the pair Psychic and Fighting takes care of its strength
against Psychic and makes it ineffective against Dark, which is immune
to Psychic.

Adding the pair Grass and Poison makes takes care of its strength
against poison and makes it ineffective against Steel, which is immune
to poison.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/attack-mode
  (pokemon.types/susceptibility
   [;; setup
    :normal :normal :normal :normal
    :ice :ice :ice :ice
    :electric :electric :electric
    :rock :rock
    ;; Spot resistances
    :ghost :ground
    ;; Pair resistances
    :psychic :fighting
    :grass :poison])))
#+end_src

#+results:
#+begin_example
{: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}
#+end_example

Can you see the final step?

It's adding the Water type, which is weak against Water, Dragon, and
Grass and strong against Rock and Fire.

#+begin_src clojure :results output :exports both
(clojure.pprint/pprint
 (pokemon.lpsolve/attack-mode
  (pokemon.types/susceptibility
   [;; setup
    :normal :normal :normal :normal
    :ice :ice :ice :ice
    :electric :electric :electric
    :rock :rock
    ;; Spot resistances
    :ghost :ground
    ;; Pair resistances
    :psychic :fighting
    :grass :poison
    ;; completion
    :water])))
#+end_src

#+results:
#+begin_example
{:water 1/2,
 :psychic 1/2,
 :dragon 1/2,
 :fire 1/2,
 :ice 1/2,
 :grass 1/2,
 :ghost 0,
 :poison 1/2,
 :flying 0,
 :normal 0,
 :rock 1/2,
 :electric 1/4,
 :ground 0,
 :fighting 1/2,
 :dark 0,
 :steel 0,
 :bug 1/2}
#+end_example

Which makes a particularly beautiful combination which is ineffective
against all defending types.


# #+begin_src clojure :results scalar :exports both
# (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])))))
# #+end_src

# #+results:
# | [: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] |


Is there anything else that's interesting?

#+begin_src clojure :exports both
(pokemon.lpsolve/solution (pokemon.lpsolve/worst-defense-type))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :exports both
(pokemon.types/old-school
 (pokemon.lpsolve/solution (pokemon.lpsolve/worst-defense-type)))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :exports both
(pokemon.lpsolve/solution (pokemon.lpsolve/weak-defense-type))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :exports both
(pokemon.types/old-school
 (pokemon.lpsolve/solution (pokemon.lpsolve/weak-defense-type)))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :exports both
(pokemon.lpsolve/solution (pokemon.lpsolve/neutral-defense-type))
#+end_src

#+results:
: INFEASIBLE

#+begin_src clojure :exports both
(pokemon.types/old-school
 (pokemon.lpsolve/solution (pokemon.lpsolve/neutral-defense-type)))
#+end_src

#+results:
: INFEASIBLE

There is no way to produce a defense-type that is weak to all types.
This is probably because there are many types that are completely
immune to some types, such as Flying, which is immune to Ground. A
perfectly weak type could not use any of these types.

* Summary

Overall, the pok\eacute{}mon type system is slanted towards defense
rather than offense.  While it is possible to create superior
defensive types and exceptionally weak attack types, it is not
possible to create exceptionally weak defensive types or very powerful
attack types.

Using the =lp_solve= library was more complicated than the best-first
search, but yielded results quickly and efficiently. Expressing the
problem in a linear form does have its drawbacks, however --- it's
hard to ask questions such as "what is the best 3-type defensive combo
in terms of susceptibility?", since susceptibility is not a linear
function of a combo's types. It is also hard to get all the solutions
to a particular problem, such as all the pokemon type combinations of
length 8 which are immortal defense types.

* COMMENT main-program
#+begin_src clojure :tangle ../src/pokemon/lpsolve.clj :noweb yes :exports none
<<intro>>
<<body>>
<<declares>>
<<memory-management>>
<<get-results>>
<<solve>>
<<farmer-example>>
<<lp-solve>>
<<better-farmer>>
<<pokemon-lp>>
<<results>>
<<attack-oriented>>
#+end_src