✎ laborator [books]

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====== Laborator AA ======

==== Masina Turing ====

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)

 
==== Turing Machines and Solvability ====







  * Exercitii simple cu MT
  * Exercitii cu MTU si conceptul de simulare
[[https://www.bbc.co.uk/bitesize/guides/zhppfcw/revision/3#:~:text=Von%20Neumann%20architecture%20is%20the,both%20stored%20in%20primary%20storage | Von Newmann Model]]

  * Algoritmi

==== Decidabilitate ====

  * Ce este o problema de decizie?
  * Problema si rezolvare;
  * Lista de probleme care (pot/nu pot) fi rezolvate;
  * Reduceri
  * Further work: coRE, complement, dovetailing

==== Notatii asimptotice ====
  * Implementari care sa ilustreze faptul ca constanta conteaza/nuconteaza
  * Implementari care sa ilustreze faptul ca log n >> n >> n2 >>> n3
  * Grafice (in ?!)
  * Exercitii clasice

==== Recurente ====
  * Cativa algoritmi si recurentele lor
  * Exercitii clasice

==== NP completitudine ====
  * Implementare care sa ilustreze exponentiala (backtracking); legatura cu MTN.
  * Reduceri pentru SAT solvere
  * Exercitii clasice cu reduceri

==== TDA-uri ====
  * Conceptul de operator vs cel de functie (exercitiu in C, exercitiu in Haskell, pe Liste)
  * Exercitii clasice