X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/turing4complexity.git/blobdiff_plain/ca6d9b4c0a40ec6c5ca72ce6879d28229f351579..04294db1691e857013d56ff5de268ee122ea7cc6:/turing4complexity.tex?ds=inline

diff --git a/turing4complexity.tex b/turing4complexity.tex
index bbc2e69..c3898bb 100644
--- a/turing4complexity.tex
+++ b/turing4complexity.tex
@@ -16,10 +16,10 @@ blabalbabla
 
 \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
@@ -27,6 +27,82 @@ pseudo-random  numbers obtained  by a  PRNG can  form the  states of  the Rado's
 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}