From: Raphael Couturier <raphael.couturier@univ-fcomte.fr>
Date: Thu, 19 Apr 2012 16:14:27 +0000 (+0200)
Subject: suite
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/turing4complexity.git/commitdiff_plain/04294db1691e857013d56ff5de268ee122ea7cc6?ds=sidebyside

suite
---

diff --git a/turing4complexity.tex b/turing4complexity.tex
index 2146039..c3898bb 100644
--- a/turing4complexity.tex
+++ b/turing4complexity.tex
@@ -16,7 +16,7 @@ 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{Property: a PRNG must preserve the distribution of short strings}
@@ -32,11 +32,77 @@ 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 $S_i$ be  the string number $i$
-obtained by the busy beaver, $S(n)$ is the set of strings of $D(n)$. With $D(4)$
-there  are  1832  strings,  so  $|S(4)|=1832$.  $P_t(S_i)$  is  the  theoretical
-probability of  the string $S_i$ and  $P_{PRNG}(S_i)$ is the  probability of the
-string $S_i$  obtained with the execution of  a given PRNG. Let  For $D(n)$, the
-number of strings is
+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}