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 ====== Laborator AA ======
+===== Part 1 - Decidability - 4 labs =====
+==== 1. Turing Machine ====
+Key concepts:
+  - computation (the output of the tape) and acceptance (acceptare);
+  - mechanical description of an algorithm
+  - (Answers online - discussion) A **Turing Machine** consists of:
+    * an **alphabet** $math[\Sigma]
+    * a set of **states** $math[K]
+    * an **initial** state $math[q_0]
+    * a **transition function** $math[\delta : K \times \Sigma \rightarrow K \times \Sigma \times \{L,H,R\}]
+    * a set of **final** states $math[F \subseteq K]
+Which of the following components of an **assembly language** would best correspond to the above? $math[K,\Sigma, \delta, q_0, F]
+  * the processor
+  * the memory
+  * registers
+  * assembly instructions
+  - (Answers online) What does the following TM do? (**bitwise complement**)
+  - (Answers online) Write a TM which **accepts** only if the **input** is a binary encoding of a **even** natural number.
+  - (Answers online) Write a TM which adds **5** to a number encoded in binary on the tape. The machine will always accept.
+  - (Answers online) Check if a symbol is present on the tape.
+  - (Discussion) How would the following algorithm be represented as a Turing Machine:
+<code>
+Algorithm(vector V, integer M) {
+   integer s = 0
+   for-each x in V
+      s += x
+   if (x > 1000)
+   then  return 1
+   else  return 0
+}
+</code>
+Helpful questions:
+  * how should the tape be organised?
+  * when should the machine accept?
+  * how would ''foreach x in V'' be implemented?
+  * how would ''s += x'' be implemented?
+  * how would ''if (x > 1000) then ... else ...'' be implemented ?
+Homework:
+  * Write a TM which verifies if a string has the **same number** of ones and zeroes. Give hints - live (what should the machine do?)
+  * write a TM which **accepts** a given regular expression
+  * write a TM which **reverses** a given binary string (always accepts)
+==== 2. Turing Machines and Solvability ====
+Key concepts:
+  * **acceptance** vs **decision**
+  * complement of a problem.
+  * Can the following problem be **accepted** by a TM? (f(x) = 0)
+    * What is the complement of this problem?
+  * Can a problem be accepted by two different TMs? Can a TM accept two different problems?
+  * If a problem is accepted by some TM, can its complement also be accepted?
+  * If a problem is **decided** by some TM, can its complement also be decided?
+  * Write a TM which accepts //is-odd// problem but which does not decide it.
+  * Which of the following problems you **think** can be **accepted** and which can be **decided**? Use pseudocode instead of writing a TM.
+      * a) V [[https://arxiv.org/pdf/1902.10188.pdf | Undecidable example 1]]
+      * b) V [[https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem | Hilbert undecidable ]]
+      * c) V [[https://en.wikipedia.org/wiki/Wang_tile | Wang Tile]]
+      * e) k-color
+      * f) Linear Integer Programming
+==== 3. The Universal Turing Machine ====
+Key concepts:
+  * simulation
+Exercises:
+  * The Von Newmann architecture - explained.
+  * Which of the components of Von Newmann arch. corresponds best to the TM?
+  * Write a TM pseudocode which verifies if a word is the proper encoding of a TM.
+    Discussion on the pseudocode.
+  * Write a TM pseudocode which accepts if **there exists** a word which is accepted by a given TM in **k steps**.
+    Discussion on the pseudocode
+  * Which of the following is a suitable pseudocode for a TM:
+<code>
+Algoritm(M,w){
+   if size(w) > 10
+     then if M halts for w in k steps
+          accept.
+}
+</code>
+<code>
+Algoritm(M1,M2,w){
+   k = 0
+   while true
+       if M1(w) has the same behaviour as M2(w) after k steps
+          then accept
+       else k = k + 1
+}
+</code>
+<code>
+Algorithm(M,A) {
+   // A is a finite set of words
+   for each w in A
+       if M(w) halts  //undecidable! Pseudocode is ok, but this machine may not terminate
+          then accept
+}
+</code>
+<code>
+Algorithm(M,w) {
+   build the machine M' such that M(x) accepts iff M'(x) does not accept, for all words x
+   if M'(w) in 1000 steps
+      accept
+}
+</code>
+<code>
+Algorithm(M1,M2) {
+    if M1 always halts then     //we know of no procedure, terminating or not, which can achieve this. This is not a proper TM/algorithm.
+       if M2 always halts then
+          accept
+}
+</code>
+  * Write the problem which is accepted by each of the above machines.
+  * Write a TM pseudocode which accepts if a **given** word is accepted by two given TMs. Explain the dovetailing technique.
+Homework:
+  * Write a TM pseudocode which accepts if **there exists** a word which is accepted by two given TMs.
+  * Write a TM pseudocode which accepts if **there exists** a TM which accepts a given word.
+  * Write a TM pseudocode which accepts if a given TM accepts **some** word of a given finite set A.
+  * Write a TM pseudocode which accepts if a given TM accepts **all** words of a given finite set A.
+[[https://www.bbc.co.uk/bitesize/guides/zhppfcw/revision/3#:~:text=Von%20Neumann%20architecture%20is%20the,both%20stored%20in%20primary%20storage | Von Newmann Model]]
+==== 4. Undecidable problems ====
+Key concept:
+  * reduction
+  * proving a problem is not in R
+  * proving a problem is not in RE
+===== Part 2 - Measuring algorithm performance (3 labs) =====
+==== Notatii asimptotice ====
+  * (Homework) Implement mergesort and insertionsort in python. Use a large dataset (provided by us) to test your implementation. Plot the execution times together with the functions $math[n^2] and $math[n\cdot \log{n}] using ''gnuplot''. What do you observe? Adjust the constants for the previous functions so that the rate of growth can be better observed.
+  * Exercitii clasice
+==== Recurente ====
+  * Cativa algoritmi si recurentele lor
+    * Merge-sort,
+    * Quick-sort (curs)
+    * Exemplul cu sqrt(n) al lui Sebi.
+  * Exercitii clasice
+==== Ammortised Analysis ====
+  * Classical exercises
+===== Part 3 - Algorithm complexity (4 labs) =====
+==== NP completitudine ====
+  * Implement a SAT solver which encodes formulae as matrices and iterates over interpretations treating them as binary counters. Plot execution times.
+  * Implement a better SAT solver which uses BDDs to encode a formula. The variable ordering is known in advance. Plot execution times.
+  * Implement a k-Vertex-Cover solver using a reduction from SAT, and any of the above solvers.
+  * Exercitii clasice cu choice si reduceri
+===== Part 4 - Abstract Datatypes (2 labs) =====
+==== TDA-uri ====
+  * Conceptul de operator vs cel de functie (exercitiu in C, exercitiu in Haskell, pe Liste)
+  * (Homework) Implementare LinkedList si ArrayList in Python, impreuna cu operatii. Implementare Haskell a operatiilor, dupa o discutie la curs despre acestea.
+  * Exercitii clasice