\chapter{Conclusions}

\label{chap:conclusions}

This thesis saw developments in fields both applied and abstract.
We found that we could use a very simple procedure to convert strategies
for games into Boolean formulas.
We developed a method to convert disjunctive-normal form 
Boolean formulas into equivalent formulas with predefined disjunct shapes.
We developed a counting method for strategies for the game of tic-tac-toe.
We constructed feasible, favorable strategies for the game of tic-tac-toe.

\section{Strategy Conversions}

%We now know a lot more than we did in the beginning about how to
%convert a strategy into Boolean formulae.
This research has demonstrated a method to convert strategies for a certain
type of game into Boolean formulas.
When this research began, we were in posession of a simple example of a
strategy implemented in MAYA.
At the conclusion of this research, we can now describe the mechanisms
behind the correct functioning of MAYA, and (more importantly) 
a general method for generating Boolean formulas from a game strategy.

By generating and converting strategies for the game of tic-tac-toe into 
Boolean formulas, we have demonstrated that our simple method of strategy
conversion is robust enough to handle somewhat complex strategies.

\section{Boolean Manipulation}

We have developed tools that are useful in determining whether the
CARE-set of a Boolean function admits a form of a given shape.
For instances of games in which the number of potential moves by the opponent
is rather small, we can enumerate all the possible disjuncts with those inputs.
With this list of all disjuncts, we can try to fit them to the function.

\section{Tic-Tac-Toe Results}

%We have developed tools to generate and evaluate tic-tac-toe
%strategies.
We have developed C programs that can generate strategies 
for the game of tic-tac-toe that are 
feasible, favorable, both feasible and favorable, 
and neither feasible nor favorable.
At the outset of this research we were not sure whether the set of all
feasible, favorable strategies for the game of tic-tac-toe was empty or not.
We now know that there exist strategies that are both feasible and favorable.
In performing this research we developed a method to count the number of
strategies for the game of tic-tac-toe.
This counting method was applied, almost trivially, to counting the number of
favorable strategies.

At the outset of this research, the goal was to produce a feasible, favorable
strategy for the game of tic-tac-toe and convert this strategy into
Boolean formulas that can then be simulated in chemistry.
To this end, we used a technology loosely designed around the original
MAYA chemistry.
With MAYA, conjunctions could be simulated with up to two positive and one
negative input.
We tested a technology that could be described as ``3-anded MAYA''.
We tried to map our formulas to conjunctions with at most six positive
inputs and three negative inputs.
We tested this technology against over fifty million strategies to no avail.
None of the generated formulas could be manipulated into a form that would fit
with our technological limitation.

We also tested against a technology with no real basis in the demonstrated
chemistry.
After working with the formulas for a while, we hypothesized that it might be
possible to convert formulas to formulas with conjunctions of (at most) five
literals.
Our hypothesis was confirmed almost immediately as one of the first one
hundred thousand strategies we tested proved to be mappable to a set of
formulas with conjunctions of five or fewer literals.\footnote{An example
of such a strategy may be found in Appendix \ref{chap:atree}.}
Similar tests with a technology including only four literals were 
not successful.
We conjecture that to represent the full game of tic-tac-toe, it is necessary
to have conjunctions of five inputs of arbitrary polarity.