Exercise 7

Added Exercise 7 to Repository

Informatik_I/Exercise_7/Task_1/README.org

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+#+title: Task 1: Recursive function analysis
+
+#+author: JirR02
+/This task is a text based task. You do not need to write any program or C++ file: the answer should be written in main.md (and might include code fragments if questions ask for them)./
+
+* Task
+:PROPERTIES:
+:CUSTOM_ID: task
+:END:
+For each of the following recursive functions:
+
+1. 
+
+#+begin_src cpp
+bool f(const int n) {
+if (n == 0) return false;
+return !f(n - 1);
+}
+#+end_src
+
+2. [@2] 
+
+#+begin_src cpp
+void g(const int n) {
+if (n == 0) {
+std::cout << "*";
+return;
+}
+g(n - 1);
+g(n - 1);
+}
+#+end_src
+
+1) Formulate pre- and post conditions.
+
+2) Show that the function terminates. *Hint*: No proof expected, proceed
+   similar as with the lecture example of the factorial function.
+
+3) Determine the number of functions calls as mathematical function of
+   parameter n. *Note*: include the first non-recursive function call.
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src markdown
+1. i) Pre- and post conditions
+
+```c++
+// PRE: n is a positive integer
+// POST: returns true if n is even and false if n is odd.
+```
+
+ii) The function =plain|f= terminates because it has a termination
+   condition. It will terminate once n reaches 0. If the number is
+   negative, it will be in an infinite loop.
+
+iii) $\text{Calls}_{f}(n) = n$
+
+2. i) Pre- and post conditions
+
+```cpp
+// PRE: n is a positive integer
+// POST: print 2^n *
+```
+
+ii) The function =plain|g= terminates because it has a termination condition. It will terminate once n reaches 0. If the number is negative, it will be in an infinite loop.
+
+iii) $\text{Calls}_{g}(n) = 2^n$
+
+#+end_src