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zQj=%%f0UX%TYs;_wGH|kG&Iy1{d@Fx(f@-Ug?dDLf?mdmXP?$~zWm!y`wuf{Vpx}v S@B1$, then + $a,b < n'$ and + $$ + a | n' \wedge n' | n \Rightarrow a | n + $$ + contradicting the minimal property. +\end{proof} + + The first algorithm we will be concerned with the (more or less) opposite problem. Given a natural number $n$, we are interested in answering the question: @@ -204,10 +223,13 @@ number $n$, we are interested in answering the question: \text{Describe the set }\, P(n) := \{p\in\mathbb{N}: 2 \leq p\leq n \wedge p \text{ is prime}\}.\tag{{\rm Q1}} \end{equation} The cardinality of the set $P$ is in classic textbooks referred by $\pi(n)$. -The famous prime number theorem gives information about the asymptotic growth +The famous prime number theorem gives information about the asymptotic growth of $\pi(n)$ as $n$ tends to +infinity. +\begin{thm}[Prime Number Theorem] $$ \lim\limits_{n \rightarrow \infty}\pi(n) \sim \frac{n}{\log(n)} $$ +\end{thm} Given an natural number $n\geq 2$, the answer to question \eqref{eq:q1} is given by the \emph{Sieve of Erathostenes}. The classic algorithm proceeds by starting with the set $N := \{2,\ldots,n\}$ successively erasing non-prime numbers. Although this algorithm is not particular performant (compared to tuned versions, such as \emph{Aitken's Sieve}), @@ -266,6 +288,10 @@ would proceed as follows. \caption{Integer-Square-Root} \label{alg:1} \end{algorithm} +\begin{thm} + Algorithm \ref{alg:1} is correct, and terminates in a finite number of steps. +\end{thm} +\begin{proof} The first step of Algorithm \ref{alg:1} is the initialization of the result $x$ and two temporary variables $y$ and $z$. As for the While-Loop, we observe that initially \begin{align} @@ -294,6 +320,7 @@ $$ y > n \Leftrightarrow (x^2+z) > n \Leftrightarrow x^2+2x+1 > n \Leftrightarrow (x+1)^2 > n, $$ hence we have shown $x^2\leq n < (x+1)^2$, and taking square roots shows the claim. +\end{proof} \subsection{The Sieve of Erathostenes I} We have described the algorithm for computing the set of prime numbers less equal $n$ in section \ref{sec:math}. @@ -310,7 +337,7 @@ While this seems obvious when written, we must not forget, that the computer can mathematically inclined people would already provide \emph{per se}, as a formal proof system could not use intuition.\\ -However, we now present one possible answer to question \eqref{eq:q1}, by considering the following algorithm. +Based on that, we now present one possible answer to question \eqref{eq:q1}, by considering the following algorithm. \begin{algorithm}[H] \begin{algorithmic}[1] \Require{$n\in\mathbb{N}$ such that $n>2$} @@ -318,17 +345,47 @@ However, we now present one possible answer to question \eqref{eq:q1}, by consid \State $P \leftarrow \emptyset$; \State $Q \leftarrow \{2,\ldots,n\}$; \While{$Q \neq \emptyset$} -\State $P\leftarrow P \cup \{\min(Q)\}$; -\State $Q\leftarrow Q - \{m\in Q: \min(Q)|m\}$; +\State $r \leftarrow \min(Q)$; +\State $P\leftarrow P \cup \{r\}$; +\State $Q\leftarrow Q - \{s\in Q: r|s\}$; \EndWhile \State\Return $P$. \end{algorithmic} \caption{Sieve of Erathostenes I} \label{alg:2} \end{algorithm} -Algorithm \ref{alg:2} provides several points for discussion. First of all, both sets, $P$ and $Q$ are -at all steps obviously subsets of $\{2,\ldots,n\}$. As this condition holds at each execution of the while -loop, we identify this as a \emph{loop invariant}. +Let us now discuss the steps of Algorithm \ref{alg:2}. +We want to identify steps that hold at each execution step of the while loop, +and are therefore \emph{loop invariants}. + +\subsubsection*{Identifying Loop Invariants} +First of all, both sets, $P$ and $Q$ are subsets of $\{2,\ldots,n\}$. Therefore, +at all stages of the algorithm, we have +$$ +P, Q\subseteq \{2,\ldots,n\}. +$$ +The set $P$ holds prime numbers only, as this is true initially (i.e. $P$ is the empty set and every +element of the empty set is a prime number). For the inductive step, if $P$ holds prime numbers only and +is extended by the least element of the set $Q$, it retains this property (in clear correspondence +with Lemma \ref{lem:leastelemisprime}). Hence, a second invariant describing $P$ is, that every element of $P$ +is prime (and hence, the same applies for the intermediate sets $P$ obtained within this loop).\\ + +Let us now quantify the numbers $2,\ldots,n$ in more detail. Each number $k$ is either prime or composite. +If $k\in P$, then by the preceeding discussion $k$ is prime. Conversely, if $k\in \{2,\ldots,n\}$ is prime, +it is never removed from the set $Q$, and therefore after finitely many steps the minimal element of $Q$ and +therefore transferred to the set $P$. Hence, the algorithm ensures the equivalence $x\in P \Leftrightarrow x \text{ is prime}$.\\ + +As a final observation, we remark, that every element of $P$ is smaller than any element of the set $Q$, +as follows immediately from $a | n \Rightarrow a\leq b$ and the fact that $P$ contains only prime numbers +and therefore proper divisors. + +\subsubsection*{Termination of the Algorithm} +The termination of the algorithm is ensured by the so called \emph{termination term}. +In the beginning, the set $Q$ is not empty, but at each execution step, elements of $Q$ are removed +(as the set $\{s\in Q: \min(Q)|s\}$ is not empty), and therefore, the cardinality of $Q$ decreases. +Therefore, the termination of the algorithm is a consequence of the fact that $|Q|$ decreases, +and this descent can only be performed finitely often. + \subsection{The Sieve of Erathostenes II} @@ -338,16 +395,24 @@ loop, we identify this as a \emph{loop invariant}. \begin{thebibliography}{9} +\bibitem{FuriaMeyer} +C. A. Furia and B. Meyer. \textit{Inferring Loop Invariants Using Postconditions} +A. Blass, N. Dershowitz, and W. Reisig (Eds.): Gurevich Festschrift, LNCS 6300, pp. 277–300, 2010. + +\bibitem{FuriaMeyer2} +C. A. Furia, B. Meyer and S. Velder. \textit{Loop Invariants: Analysis, Classification, and Examples} +ACM Computing Surveys, Vol. 46, No. 3, Article 34, January 2014. + \bibitem{hardylittlewood} G.H. Hardy and E.M. Wright. \textit{An Introduction to the Theory of Numbers}. Oxford University Press, 6. edition. 2008. \bibitem{JML} -\text{Java Modelling Language JML} -\url{http://www.eecs.ucf.edu/~leavens/JML//index.shtml} +\text{The Java Modelling Language (JML)} Home page, February +2013 \url{http://www.jmlspecs.org} \bibitem{Larch} -\text{The Larch System}. +\text{The Larch System} Home page, May 2017 \url{https://www.cs.cmu.edu/afs/cs/project/larch/www/home.html} \bibitem{LidlNiederreiter} -- 2.1.4