\section{Algorithmic complexity for short strings}
-In their paper, they introduced a method to compute the complexity of D(n).
+In their paper, they introduced a method to compute the complexity of $D(n)$.
With D(4), 5 970 768 960 machines halt. There are 1832 different strings.
-\section{PRNG preserve distribution of short strings}
+\section{Property: a PRNG must preserve the distribution of short strings}
In this section, we propose to measure the quality of a PRNG by considering that
a good PRNG must preserve the distribution of short strings. More precisely, the
original Busy Beaver. Then, machines obtained by such a construction can be
executed and if the PRNG is good, machines should generate the same strings with
the same distribution than the theoretical Busy Beaver (with all the enumeration
-of states).
+of states). In practice, PRNGs will not generate all the strings with exactly
+the same probabilities. Nevertheless, the distance between the theoretical
+probabibilities and the obtained probabilities with a PRNG is a good indication
+of the quality of the PRNG.
+
+In the following we report our experiments. Let $S(n)_i$ be the string number
+$i$ obtained by the busy beaver $D(n)$, $S(n)$ is the set of strings of
+$D(n)$. With $D(4)$ there are 1832 strings, so $|S(4)|=1832$. $N_{bb}(S(n)_i)$ is
+the number of occurrences of the string $S(n)_i$ obtained by the Busy Beaver and $N_{PRNG}(S(n)_i)$ is the
+number of occurrences of the string $S(n)_i$ obtained with the execution of a given
+PRNG. In order to measure the distance between the theoretical result and the
+pratical one, we use this measure:
+\begin{equation}
+Dist(PRNG,BB)=\sum_{i=1}^{|S(n)|} \frac{|N_{bb}S(n)_i-N_{PRNG}S(n)_i|}{N_{bb}S(n)_i}
+\end{equation}
+
+Here are the result for some PRNGS: bbs, lcg, xor128, rand, xorshift, devrandom,
+newprng, isaac. These results are reported for experiments with different
+numbers of random number generation.
+
+
+\begin{table}
+\begin{tabular}{|l|l|}
+\hline
+ & Number of experiments\\
+\hline
+Name of PRNG & 500000000\\
+\hline
+BBS & 358.38\\
+\hline
+lcg & 491.78\\
+\hline
+xor128 & 380.69\\
+\hline
+rand & 382.04\\
+\hline
+xorshift & 372.68\\
+\hline
+dev random & 376.78\\
+\hline
+new prng & 367.42\\
+\hline
+isaac & 366.91\\
+\hline
+\end{tabular}
+\caption{Distance for some PRNG with different number generations}
+\end{table}
+
+
+In the following table, we report the same experiment with small occurence strings (i.e. less than 50).
+\begin{table}
+\begin{tabular}{|l|l|}
+\hline
+ & Number of experiments\\
+\hline
+Name of PRNG & 500000000\\
+\hline
+BBS & 322.88\\
+\hline
+lcg & 435.89\\
+\hline
+xor128 & 347.61\\
+\hline
+rand & 348.02\\
+\hline
+xorshift & 338.33\\
+\hline
+dev random & 341.16\\
+\hline
+new prng & 335.00\\
+\hline
+isaac & 333.39\\
+\hline
+\end{tabular}
+\caption{Distance for some PRNG with different number generations for small occurrence strings}
+\end{table}
+
\end{document}
