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Wednesday, May 19, 2010

Nim Chimpsky Plays A Game: A Simple Game Played With Simple Recursion

Nim Chimpsky, named as a pun of Noam Chomsky the brilliant linguist, cognitive scientist and philosopher, was a chimpanzee that researcher Herbert S. Terrace of Columbia University claimed had learned a form of sign language and was able to communicate intelligently. Whether or not this was in fact true is still hotly debated in the research community.

If one believes that Nim had the intelligence to learn and use language, one might think the chimp could play one of the simplest of children's games, also called Nim. The game of Nim is a game of extensive form and perfect information that comes in many flavors. We will be using perhaps the simplest form, often called the subtraction game. This version consists of  two players starting with a number (usually represented as a pile of objects like coins or pebbles), where each player in turn removes one, two, or three objects. The player to take the last object is the loser.

It can be trivially shown that such a game reduces to a player avoiding being left with n 1 (mod 4) objects, in other words, a player should try to remove enough objects to leave the opponent with 1 (mod 4) objects. In the usual game, where the moves are limited to {1,2,3...m), this becomes 1 (mod m+1), i.e., if a player can take one to six objects in a turn, they want to leave the opponent with 1 (mod 7) objects. If a player finds themselves in such a position already, they cannot move to a winning position, and the usual strategy is to remove only one object in the hope that the more objects that remain, the more likely the opponent will make a mistake.

When she was a toddler, the game amused my daughter. It didn't take long for her to figure out the strategy however, and then it became boring. I came up with a variation that I called "Nim with holes". In this variation, instead of the moves being the positive integers with some upper limit, the moves can be any member of some set X of positive integers agreed upon by the players. So a game might start off with a pile of objects of size fifty, with "legal" moves being the removal of say n{2,4,7,9} objects. Or perhaps "Prime Nim" where the moves are limited to n∈{2,3,5,7,11}. The "holes" version of the game can be analyzed in a similar fashion to the original, the closed form strategy is left as an exercise for the reader.

Instead, let's look at solving the problem using a simple recursive program. Common in the functional programming style (and usually preferred over imperative/iterative methods for this style of programming), a recursive program (or routine or function) is one that references itself in its body.

For example, a simple function to calculate the factorial of a number n, usually denoted as n! and equal to the product of all positive integers less than or equal to n could be written in the iterative and functional styles (using c for this example) as:

Factorial function in C language, written in iterative style (left) and functional style (right).

As can be seen, the iterative version of the function "iterates", or steps incrementally, toward the goal, starting with a value of one for the variable fact and stepping through each integer and multiplying the value of fact by that integer, until the integer of n, the input value has been reached. The functional style, on the other hand, does away with the need for any intermediate variable. Instead, it "winds up" to the return value by taking the input value n and multiplying it by the value of the factorial function of n-1. In a recursive style of programing, the "stop" or "base case" must be defined to allow the "unwinding" of the recursion. In our example, we see that when the factorial function is passed a value less than or equal to one, it simply returns 1. This returned value is used in the multiplication by the caller, and so on back up the recursion chain until the final, initial caller returns the ultimate result.
The same factorial function, written using Scheme, would have the following form:
(define factorial (lambda (n) (if (= n 1) 1 (* n (factorial (- n 1))))))
Recursion can be a difficult concept to get one's head around initially, especially for non-programmers (and even some programmers!) An excellent introduction to the thought process can be found in Thinking Recursively by Eric Roberts. A deeper examination can be had in the landmark book by Douglas Hofstadter titled Godel, Escher, Bach: An Eternal Golden Braid. Anyone interested in recursion should have these books on their shelves. They are certainly must-haves for any self-respecting programmer.
We can see how solving our game of "Nim with holes" lends itself to a recursive solution by thinking about the possible plays going backwards, that is, starting with a losing position. Let's consider a game of "Nim with holes" where the players have agreed the legal moves must belong to the set {2,3,5}. We immediately see that a player finding themselves with one or two objects is in a losing position, and a player finding themselves with 3,4,5,6, or 7 objects can leave their opponent in that losing position. As we move away from the goal, reversing the possible game play steps, we would find that having eight or nine objects, or fifteen or sixteen are "losing positions", that is, the player faced with these cannot remove a number of objects that allows them to leave their opponent with a losing position, nor leave a position where the opponent cannot leave them the next losing position closer to the end game.
With a simple set of legal moves, this is easy to do in one's head. As the set of legal moves becomes more complex, the difficulty in determining winning and losing positions in the game becomes more difficult. Algorithmically it is trivial, and using a recursive and functional style of programming, elegant.
We want to build a simple function that can take the game state (the number of objects) and the set of legal moves as its arguments, and return "True" if we've won already, "False" if we're in a losing position (meaning we'll just move the minimum allowed), and the position we need to move to (if we can) that will put our opponent in a losing position (that we can then maintain for the rest of the game: once there, perfect play means the opponent is ultimately doomed.)
A trivial realization of this using Scheme is shown below.
"Nim with holes" move analysis function written in Scheme with example results (click to enlarge).
The winning position (winpos) function takes the game's current state (number of objects left) and a list of the legal moves. If we've already won (no objects left) #t (true) is returned. Otherwise, for each move in the legal move list (using the map function in Scheme), the function tests if any of the reachable positions are a losing move by testing each of them for the ability to reach a losing position, by testing each of them...until it winds recursively down to the base case: we have won. If the recursion is able to reach this state, when it is "unwound", the final result is the first position presently reachable that can lead us to the ultimate winning position. If the recursion results in not being able to reach the ultimate winning position, it returns #f (false) indicating we a presently in a losing position, and should take the minimum allowed move.

The operation of this recursion can perhaps be most simply imagined by first thinking about the game in what surely must be the simplest case: the only legal move is to take one object. Clearly, the winning positions alternate, with all odd numbers of objects being a losing position, and all even numbers being the winning positions. Pick some position, say five objects. The function tests for zero, which fails, resulting in the function calling itself for a value of four objects, which goes through the same sequence to call itself for three objects...until we arrive at the call to itself for zero objects.

This is our "base case", where the function returns #t (true). As this return is "unwound" or "percolated" back to the original first caller, note that it will have the logical not applied for each return for each nested call. This is where in our function the move positions are alternated between player and opponent: what is a winning position for a player is a losing position for their opponent and vice versa. In our example case, we  are in effect seeing the result of not(not(not(not(not(#t))))), resulting in #f (false), meaning our position of 5 objects is unfortunately a losing position in the case of the only legal move being removal of one object.
We can see in the lower part of the IDE the results from asking the function to evaluate a game state of 17 objects with legal moves of 2,3 or 5. The result indicates taking 2 objects and leaving 15 will put us in the winning position. The second example uses a simple map to map the winpos function onto a list of possible game states from 0 to 19 using 2,3 and 5 as the legal moves. The result is the list of results from the winpos function.
Readers can download the excellent (and free) Dr. Scheme environment from PLT Scheme should they want to experiment with Scheme, functional programming and recursion. The site has excellent tutorial materials and references for those interested in learning the details of this superb Lisp dialect.The trivial game solving function example (which can of course be extended to any similarly solvable game) can be found at NimWithHoles.scm.

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